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The symbol ζ (lowercase zeta) denotes courses currently being taken.
Single Variable Calculus
A swift introduction to calculus - cursory glances so you understand a lot of the things in WIL.
There are so many great videos on calculus online that this seems a little useless, but here
I'll keep things as cut and dry as possible.
The derivative (of a function): Another function that tells you the slope of the original function
at every point. You can take derivatives of derivatives. Notable simple rules: derivative of x^n
is n*x^(n-1). Derivative of nx is n. Derivative of n is 0. (n for all of these is just a number).
More complicated rules: product rules (a function multiplied by another function), chain rules
(a function of a function, such as sin(x^2)), and quotient rules (a function divided by another
function).
Product rule: derivative of f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x). You can see this holds for even
the simple expressions. x^2, which can be expressed as x*x, becomes 2x. This is consistent with
n*x^(n-1) from earlier. Something like 5x, expressed as 5*x, becomes 5 for the same reason. This
works with more complicated expressions too, as long as the variable they're dealing with is the
same.
Chain rule: derivative of f(g(x)) is g'(x)*f'(g(x)). The internal function's derivative is taken
and is then multiplied by the derivative of the external function. So something like sin(x^2) has
a derivative of 2x*cos(x^2).
Quotient rule: derivative of f(x)/g(x) is a bit more complex, but I remember it with a mnemonic.
"Low d high - high d low over low squared." This is expressing (g(x)*f'(x) - f(x)*g'(x))/(g(x)^2).
It's a useful mnemonic and is worth memorizing.
More basic derivative things: the trigonometric functions. They go (in order of derivatives) sin(x),
cos(x), -sin(x), -cos(x), sin(x). There are more complicated ones but sin(x) and cos(x) come up a lot.
The integral (of a function): This is another function that is basically the opposite of derivative,
but with it comes some set of challenges. You can't integrate everything - there are some
expressions that are just unintegrable. Basic rule is integral of x^n becomes (x^n+1)/(n+1), nx
becomes nx^2/2, n becomes nx. There's also two types of integrals: definite (has defined bounds) and
indefinite (the entire function is integrated). So what is the integral visually, with its definite
bounds, and considering that the derivative is the slope of a function. Well, the integral is the
area under the curve of a function, where above the axis is positive and below is negative.
There are a lot of integration techniques, but the most basic one is "u-substitution," where you
can simplify an integral by setting some part of it equal to u, making it easier to integrate.
That sounded confusing so let me clarify. Let's say you have an integral of xsin(x^2) dx. You can plug
in x^2 for u, so du/dx = 2x, and dx = 1/(2x). This lets you transform the expression to the intermediary
integral of (1/2x)(xsin(u))du. The multiple variables would be a problem, but they cancel out and you
end up with integral of (1/2)sin(u)du, which becomes (1/2)(-cos(u)). Plug the x equivalent of u back
in to end up with the answer: (1/2)(-cos(x^2)) + C. C is any numerical constant, because if you think
of the integral as the opposite of a derivative, you could take the derivative of any function + C and
C would disappear. You may be wondering why cos is negative - this is because the trigonometric
derivative path is reversed for integrals, and goes sin to -cos to -sin to cos to sin.
Definite integrals are now convenient to explain - once you evaluate an integral, you then just plug in
the definite integral bounds, and subtract lower from upper. There's no +C in the final expression since
you have a numerical value.
A Dive into Discrete Math [More TBA 2023 Spring]
A quick preface before I start - I am unofficially sitting in on this lecture and only taking the most
rudimentary notes. There will be no heavy problem solving of this topic on WIL (for now) because of my
lack of textbook access as well as course prioritization (since this is just a class I sit in on, the
notes won't be as good as those for Physical Chemistry or Linear Algebra + Differential Equations).
Truth Tables, Negations, Disjunctions, Conjungtions, Conditionals, and Biconditionals:
Starting with the very basics, a truth table is the simplest Interpretaion of a true/false statement.
Put succinctly, the statement 3 > 2 is true while 2 > 3 is false. Trutb Tables minimize this further by
replacing the statement with a variable "p," which can either be true or false. More complex statements
can have more variables, but for now we'll deal with two: "p" and "q," where the latter has the same true
- false properties as the former. However, p and q ARE distinct statements. They can be put together into
a truth table to show what all the other terms mentioned are.
Negations are basically an inversion of the variable's values. If p is true, false, then ~p is false, true.
Negations are typically the first operation applied in discrete math statements.
Disjunctions, represented by v, as in p v q, are a way of mathematically making an "OR" statement. Consider
this truth table: p is true, true, false, false; q is true, false, true, false. Ergo p v q would be true,
true, true, false, where the only false disjunction result is when both p and q are false.
Conjunctions, represented by ^, as in p ^ q, are a way of mathematically making an "AND" statement. Taking
the same table as with the disjunction but with p ^ q, the results would be true, false, false, false, where
the only true statement is when both p and q are true.
Conditionals are pretty abstract (more in depth explanation later on). They are represented by -->, as in
p --> q. Using our truth table from above, the results of the conditional would be true, false, true, true.
The reason for the "false" result is because a true statement cannot "cause" a false statement. It's less
muddled if you look at p as a hypothesis and q as a conclusion. If your hypothesis is right and your conclusion
is too, that makes sense. If both are wrong, it also makes sense. If your hypothesis is false but your
conclusion is true, you're lucky and as long as the conclusion is right, it's ok. But if your hypothesis is
right but your conclusion isn't, that conditional makes no sense - there must've been some sort of error.
One way of expressing the conditional is by ~p v q, this will yield the same truth table results.
Biconditionals are conditionals where p implies q and q implies p. They are represented by <-->, as in p <-
-> q. Their truth table results are true, false, false, true, meaning that it is true when the values of p
and q are the same. [END OF LEC1 NOTES]
Linear Algebra and Differential Equations ζ
Lecture 1/2: Basics of Differential Equations
Differential equations should be a somewhat familiar topic from Calculus 2, as they frequently come up as
separable variable problems. Separable variables are fairly straightforward - they simply require algebraic
manipulation and some integration skills. Have some f(y)dy = g(x)dx, integrate both sides, and finish.
These differential equations have something called 'orders.' They are pretty straightforward: the highest
derivative defines the order of the differential equation. For example, y''' (or the 3rd derivative of y)
- 7y' + 22 = x^2 is a 3rd order linear differential equation. The distinction of linear differential equation
is important, because nonlinear DEs are completely different in terms of solutions (and our class hasn't
covered them yet). You can identify nonlinear differential equations pretty simply - if the derivative is
multiplied, divided, exponentiated (basically everything except added to or subtracted from), it's not linear.
One of the most fundamental differential equations is the first order linear differential equation (FOLDE).
It's very commonly encountered and you usually see it in computation problems like initial value, which allow
you to solve for the constant left after integration. One of the most useful tools for the evaluation of
differential equations is the slope field - at every point in a cartesian plane, values are plugged in and
result in a detailed field that lets you see how a particle placed in the system would move about, more or
less. One subfield of differential equastions are autonomous DEs, which only have one variable within them.
Something like y' = y+5 is an example, and they're incredibly easy to make a slope field for.
Some important side notes: some initial value DE problems are unsolvable, and others don't necessarily have
unique solutions. An example of the former would be y' = -y/x, y(0)=0. After separation of variables, you end
up with integral of 1/y dy = -integral 1/x dx. So, ln|y|=-ln|x|+ln|c| (this ln|c| is still a constant and is
useful in simplifying the problem). So, ln|y|=ln|c/x|, y = c/x, and y(0) = 0. But this is impossible, because
y(0) is c/0, which cannot ever be 0 - the DE is unsolvable. For a nonunique solution example y' = 3y^(2/3), y(0)
= 0 is a good example. This will be left as an exercise for the reader, but there are two solutions for this
initial value problem, not one.
Some differential equations aren't seperable. The techniques for solving them will be discussed in a later
lesson posting. However, one notable exception to inseperable DEs in the form y' + p(x)y = q(x) is q(x) = 0.
This results in something called homogeneity, and the DE can now be solved by separation of variables.
Finally, going back to slope fields, differential equations have stable, semistable, and unstable solutions.
This is easily visualized in a slope field. Stable solutions are those where a particle placed above or below
the solution (as long as it is within the domain - that is, not high/low enough to be affected by another
solution) will go towards the solution. An unstable solution is the opposite - any deviation from the solution
will cause the particle to never return to that solution. A semistable solution is one that is stable from one
side but not the other (so the slope field points into it at one side and out at the other).
Lecture 25. x"-4x'+3x = 0. x = u_1, x' = u_2, x' = u_1' = u_2, x'' = u_1'' = u_2'. x" = 4x'-3x --> x" = 4u_2 - 3u_1.
u_1' = u_2. u_2' = 4u_2 - 3u_1. This is a system of two eqns and 2 unknowns. [u_1'; u_2'] = [0 1; -3 4][u_1; u_2].
x' = Ax + F. Transforming 2nd order LDE into linear algebra. 2y""-3y"+4y = sin(t). 2y""=0y"'+3y"+0y'-4y+sin(t). 4th
order differential equation. y=u1, y'=u2, y"=u3, y"'=u4, y""=u4'. u4' = 0u4 + (3/2)u3 + 0u2 - 2u1 + (sin(t))/4.
[u1'; u2'; u3';, u4'] = (0 1 0 0; 0 0 1 0; 0 0 0 1; -2 0 3/2 0)(u1; u2; u3; u4)+ (0; 0; 0; sin(t)/2). u' = Au + F.
Homogeneous system X' = AX. y" + ay' + by = f(x) -> yH = c1y1+c2y2, yH + yP.S = yG.S. X = Xc + Xp. Xp = particular
solution. x1 = e^(at)(pcos(bt)-qsin(bt)); x2 = e^(at)(qcos(bt)+psin(bt)). X' = AX, A = (-1 2; -2 -1), lambda =
-1 +- 2i. a = -1, b = 2 --> x1 = e^(-t)(pcos(2t)-qsin(2t)); x2 = e^(-t)(qcos(2t)-psin(2t)).(i, -1; 1 i) -->
(1 i; 1 i) --> (1 i; 0 0). a + ib = 0, b is free. a = -ib. So eigenvector is (a;b) = (-ib;b) = (-i;1)b so v =
(0;1) + i(-1;0). General solution c1e^(-t)((0;1)cos(2t)-(-1;0)sin(2t)) + c2e^(-t)((-1;0)sin(2t)+(0;1)cos(2t)).
So c1e^(-t)(sin2t;cos2t)+c2e^(-t)(-sin2t;cos2t) = x(t). Defective matrix generalized eigenvector. If lambda is an
eigenvalue for A with am = 2 and gm = 1, then the two linearly independent solution for x' = AX are x1(t) =
e^(lambda*t)v, x2(t) = e^(lambda*t)(tv1 + v2). For 3x3, x3(t) = e^(labmda*t)(t^2*v3/2 + tv2 + v1). Defective example:
(1 -4, 4 -7). x1' = x1-4x2, x2' = 4x1-7x2. x1(t) = e^(-3t)(1;1); b(1;1) + (1/4;0) x2(t) = e^(-3t)(t(1;1)+(1;3/4)) =
e^(-3t)(t+1;t+3/4). W(x1, x2) /=/ 0. |x1 x2| /=/ 0. |e^(-3t) e^(-3t)(t+1); e^(-3t) e^(-3t)(t+3/4)| = e^-6t(-1/4) =
-e^(-6t)/4 /=/ 0. Therefore {x1(t);x2(t)} are linearly independent. x(t) = c1x1 + c2x2. x' = Ax where A = (1 -2;2 5)
lambda solving |1-lam -2; 2 5-lam| --> 5 + lam^2 + 4 - 6lam = lam^2 - 6lam + 9 = (lam-3)^2 --> lambda = 3, 3.
(2 0 0; -7 9 7; 0 0 2). Find lambda (l). (2-l)((9-l)(2-l) --> l = 2, 2, 9. For l = 2, (0 0 0; -7 7 7; 0 0 0) -->
(-1 1 1; 0 0 0; 0 0 0) so -a+b+c = 0, b and c are free so v1 = (1;1;0), v2 = (1;0;1). l = 2 has 2 free vars. l = 9
has 1. v3 = (0;1;0) so x(t) = c1x1+c2x2+c3x3 = c1e^(2t)(1;1;0)+c2e^(2t)(1;0;1)+c3e^(9t)(0;1;0). IVP x(0) = (0;1;2)
so (0;1;2)=c1e^0(1;1;0)+c2e^0(1;0;1)+c3e^0(0;1;0), solve for c values. Consider x' = Ax = (1 1 1; 2 1 -1; 0 -1 1)x,
if eigenvalues of A are 2 and -1, find fundamental solutions to system. If x(0)=(-1;2;6), find solution to IVP.
l = 2, am = 2, gm = 1 --> defective. dy/dx = f(y) = y(y-2), equilibrium solutions dy/dx = 0.
Lecture 3: Separation of Variables: More Techniques
So far, the linear differential equations we've been able to solve have all been easily separable by means of
algebraic manipulation. However, not all DEs are like this, and today we're going to be covering some novel
methods of solving problems of the form (dy/dx) + P(x)y = Q(x). When Q is 0, we call the resulting equation a
homogeneous one. Now, we will cover the Method of Integrating Factor for ODE.
Knowing the form y' + py = q, we can generate something called an integrating factor. Lets call it "m" for now.
m = e^(integral of (p)dx). We do this for a special reason - it turns the left side of our equation into an
easy-to-solve product rule. m(y') + pmy = qm. And since m is e^(integral of (p)dx), the left side is equivalent
to d/dx (m * y). Through further integration and algebraic manipulation, the result is that y = (1/m) * integral
of Qmdx. So for a problem such as y' + 2y/x = 9, p = 2/x so pdx = 2ln|x|, m = e^(2ln|x|) = x^2, so y = (1/x^2)
* integral 9x^2dx, ergo the result is y = 3x + c/x^2.
Another way to find soltuions to homogeneous DEs is through the substitution method. This is done by having some
function f(x,y), then creating a function F(y/x) and substituting u for y/x. For example, f(x,y) = (2x-y)/(x+4y).
Both sides are divided by 1/x to get (2-(y/x))/(1+(4y/x)). This is form F(y/x). u = y/x and y = ux, so y' = ux'
+ xu', and y' = u + x(du/dx). Therefore, y' = F(y/x), u + x(du/dx) = F(u), x(du/dx) = F(u) - u, and xdu =
(F(u)-u)dx. Then, du/(F(u)-u) = dx/x.
An example of this new technique is dy/dx = (x-y)/(x+y) = (1-(y/x))/(1+(y/x)) = F(y/x). u + x(du/dx) = (1-u)/(1+u)
so x(du/dx) = (1-u)/(1+u) - (u(1+u))/(u+1). x(du/dx) = (1-2u-u^2)/(1+u), dx/x = (1+u)du/(1-2u-u^2). Integrate
both sides, after some computation you will get an implicit solution of 1-2(y/x)-(y/x)^2 = (xc)^-2.
Biology: The Basics ζ
The essence of biology consists of four key concepts: Principles, Levels of Organization, Unity and
Diversity of Life, and Biology as a Scientific Discipline. A cursory overview of the importance of biology
is best demonstrated through two examples: penecillin and ziconotide. Penecillin was discovered by accident
by Alexander Fleming, who noticed that bacteria was either not growing or growing very poorly around a
yeast culture that he later determined had penecillin. This was inhibiting the growth of other cultures,
and hence the first antibiotic was born. Nowadays, there are thousands, and new challenges to overcome, as
bacteria are capable of rapidly adapting to antibiotics and developing a resistance to them. Ziconotide is
a chemical isolated from sea snail venom and is a very potent pain killer, where only a tiny amount is needed
to induce an effect similar to 100x that amount in morphine. Supposedly, it is also less likely that people
will form a tolerance to it (at least to the same extent as they do with morphine) making it a safer and more
effectuve painkiller.
The core concepts of biology are evolution, structure and function, information flow (and exchange + storage),
Pathways, Production, and Transformation of Matter and Energy, and Systems.
03.07.2022
Biology lecture. Approx. 20-30% of all genes encode transmembrane proteins. Membrane proteins are important
medically and biologically. 70% of medications act on membrane proteins. Efforts focused on identifying genes
that encode membrane proteins and how to predict if a gene will encode a transmembrane alpha-helix. 6000-7000
transmembrane encoding genes in human genome. --- Osmium tetroxide to bond to polar outside of cell. Used for
pulse-chase experiments which used radioactive amino acids. --- Freeze Fraction Electron Microscopy (FFEM) is
sample frozen in liquid nitrogen, then fractured with a knife. Then inside of sample can be analyzed. Leaflets
separate into P and E faces, then spray with heavy metal like platinum to see the membrane protein (which appear
as bumps). --- Membranes are semifluid, since lipids can rotate freely around their long axes and move laterally
within the membrane leaflet. At body temp neighboring lipid molecules switch places 10^7 times per second. Flip-
flop of lipids - lipid from 1 leaflet will move into the other leaflet. This is energetically unfavorable and
uses a transport protein (flippase), process requires ATP. --- Lipid rafts - group of lipids that float together
as a unit within the larger sea of lipids in the membrane. Has high concentration of cholesterol, and can aid
with information transmission. Lipid rafts are dynamic and thus difficult to study. --- Shorter fatty acid
tails make the membrane more fluid. More double bonds and unsaturated acid tails will also be more fluid.
Cholesterol makes membranes less fluid at higher temps and more fluid at lower temps. Transmembrane proteins
can also rotate and move laterally but they are much slower than lipids. They never flip-flop. Frye and Edidin
in 1970 fused mouse and human cell and demonstrated lateral movement of membrane proteins. Not all membrane
proteins can move - depends how tightly it is anchored to other structures in the cell. --- Plasma membrane is
selectively permeable. Passive transport - simple/facilitated diffusion via concentration gradient (for example
how water follows salt). Active transport requires ATP, transports molecules against concentration gradient.
3.21.2022
Biology lecture. Muscle contractions, cellular organization, protein sorting, chemical reactions all require
energy to be performed. No system is fully efficient so some energy will be lost as heat. So enthalpy in cell
is G (usable energy) + entropy. If change in G is negative, free energy is released and the process is
exergonic. If change in G is positive, free energy is consumed (energy is required for the reaction to occur)
and thus the reaction does not occur spontaneously. Change in G for ATP hydrolysis is -7.3kcal/mole (so it is
spontaneous). Enzymes and coupling regulate ATP usage so excess ATP is not used up spontaneously.
Glucose + Phosphate --> Glucose-6-Phosphate + H2O, change in G is 3.3kcal/mole so not spontaneous. Humans
hydrolize 100 pounds of ATP per day. ATP --> ADP + energy, ADP gets recycled back into ATP. 20% of proteins
bind ATP. Allows us to predict if a new protein uses ATP or not. Most chemical reactions in cells require
a catalyst to proceed faster. Catalyst does not get consumed during the reaction and can be reused. Example:
catalase in peroxisomes causes peroxide decomposition to occur 10^15 times faster. Catalysts lower the
activation energy of a reaction. Emil Fischer came up with the lock and key metaphor for substrate-enzyme
bonding, where the substrate snugly fits into the the enzyme in such a way that other molecules wouls not be
able to. More and more substrate causes more engagement with enzyme molecules, plateau occurs because the
quantity of enzyme is not enough to handle all the substrate (this state is called saturation). Michaelis
constant Km is substrate concentration (molar) where velocity is half of the maximum value. Inhibition of
enzymatic reactions: two types. Reversible or irreversible. Reversible has two sub-parts: competitive and
noncompetitive inhibition, where binding between enzyme and inhibitor is noncovalent. Irreversible has an
enzyme-inhibitor bond that is covalent. Thus, the enzyme is permanently deactivated. Competitive inhibition:
inhibitor competes for the active site of the enzyme with the substrate. Noncompetitive inhibition: inhibitor
changes the 3D structure of the protein by binding to the allosteric site of an enzyme. Irreversible
inhibition: DIPF (nerve gas) deactivates acetylocholinesterase, which is in charge of breaking down ACh and
limits muscle contraction.
3.23.2022
Biology lecture. Ribozymes - until 1980s scientists thought all enzymes were proteins. Discovery of
ribonuclease P (RNapse P) changed this view. This enzyme is found in every live organism. Has two subunits, RNA
and protein. RNA alone can catalyze the chemical reaction, so RNA can also function as a catalyst. Catabolic
pathways are the breakdown of cellular components, is exergonic. Anabolic pathways build cellular components,
are endergonic. Glucose oxydation requires NAD+, NAD+ becomes NADH. Hydrolysis of NADH is exergonic, change in
G is -158.2kJ. NADH used to make ATP. Metabolic pathways are regulated so the cell doesn't overproduce or over-
consume. Three methods of regulation: gene regulation, cellular regulation, biochemical regulation. Gene
regulation example: E. coli introduced to lactose, gene activates to make special proteins for bringing in and
breaking down lactose. When no more lactose left, gene turns off. Ubiquitin is used to tag proteins for them to
be broken apart by proteasomes. Cellular respiration: aerobic, anaerobic, fermentation. Aerobic: glucose gets
oxidized to form CO2 and H2O. Glucose has a lot of potential energy so a lot of free energy is released on break-
down. Glycolysis, breakdown of pyruvate, citric acid cycle, oxidative phosphorylation convers the glucose energy
into ATP. Glucose -> 2 pyruvate -> into mitochondria -> breaks down into 2 CO2 and 2 acetyl -> citric acid cycle
-> 2 acetyl into 4 CO2. During the citric acid cycle, NADH and FADH formed, chemiosmosis done to form a lot of ATP.
4.4.2022
Biology lecture. Cell cycle. Eukaryotic cell cycke: human body contains 10-50 trillion cells, all derived from
fertilized egg. Forms from fusion between sperm and egg cell after many cell divisions. Mitosis: one mother cell
divides into two daughter cells, daughter cells have equal amount of genetic material as mother chromosomes. Meiosis
is mother cell dividing into two daughter cells, daughter cells have half the genetic material of the mother cell.
Bacteria, mitochondria, chloroplasts divide through binary fission. Cytogenetics: the microscopic examination of
chromosomes and cell division. Karyotype: reveals number, size, and for of the chromosomes in an actively dividing
cell. 23 pairs of chromosomes --> 46 chromosomes per human cell. But sperm and egg cell have only 23 chromosomes.
This is why sperm and egg cells are referred to as gametes, or haploids. Most cells are diploids (2n) with 46
chromosomes. MTFHR gene: sequence snippet: ACAGC -- Single Nucleotide Polymorphism --> ATAGC, C[change location]T.
Folic acid absorption varies - MTFHR gene improves, polymorphism means that it slows down. Folic acid prevents neural
problems with babies. Homologues: OCA2 Chromosome 15 determines eye color, one copy may carry the form that confers
blue eyes, the other copy may carry the form that confers brown eyes. X and Y chromosomes are not homologous. Eye
color sometimes changes from blue to some other color in early life due to pigment production. Cell cycle. Interphase:
cell grows and prepares for cell division. In the interphase: G1 (gap), synthesis of DNA, G2 (gap). Then, mitosis.
G1 phase: cell becomes committed to divide, cell growth occurs. Cell checks if it has any mutations. Synthesis phase:
chromosomes replicate, after replication two copies stay joined together and are called chromatids. G2 phase: cell
synthesizes proteins needed during mitosis and cytokinesis (spindle proteins). Cyclins: amount varies through the
cell cycle. Cdks are kinases controlling the cell cycle, must bind to cyclin to be active.
Quiz Review Brief: 10 questions. Glucose in cellular respiration in aerobic respiration forms CO2 and H2O. Exergonic
rxn. Energy intermediates ATP, NADH, FADH2. Glycolysis breaks down glucose into two pyruvates in cytosol. Pyruvate
broken down into two acetyl in mitochondrial matrix. Citric acid cycle in mitochondrial matrix turns the two acetyl
into 4 CO2. Oxidative phosphorylation occurs in inner mitochondrial membrane. Proton gradient is created, then flow
of protons across gradient through ATP synthase creates ATP.
4.6.2022
Biology Lecture. Control of the cell cycle. Interphase includes G1, S, G2. Interphase comprises most of the time the
cell spends in the cell cycle. M (Mitosis) is the shortest part of the cell cycle. G1 checkpoint (restriction pt):
determines if conditions are favorable for cell division and if the DNA is damaged. G1 cyclin is made in response to
sufficient nutrients and growth factors. G1 cyclin bonds to cdk to become activated as a complex. If DNA is damaged,
cyclin/cdk complex is not formed, progression of cell cycle stops. Once S phase is entered, G1 cyclin is degraded,
cdk remains. Mitotic cyclin accumulates in G2, also forms complex with cdk. G2 checkpoint checks for DNA damage,
determines if all of the DNA is replicated, monitors protein levels needed for M phase. Then G2 cyclin degrades and
cdk remains. Metaphase checkpoint is next - determines if all chromosomes are attached to the spindle apparatus.
What are the molecules that drive the cell from one phase cycle to the next? Masui and Markert's study of frog
oocyte. Frog oocytes are dormant in G2 up to 8 months, then when progesterone enters oocyte, G2 progresses to M.
This process is called maturation. Hypothesized maturation promoting factor (MPF) - long term exposure to
progesterone molecule caused MPF to be made (2 hours was not enough). Mitotic cell division is important due to it
allowing asexual reproduction, and the development and growth of multicellular organisms (recovering from injury).
When DNA is replicated, two identical copies of the double helix are produces. These copies are side by side and are
called sister chromatids. Easy to see under microscope. Mitotic spindle ensures that each daughter cell will have
the correct number and types of chromosomes. Astral microtubules position spindle in the cell, polar microtubules
separate the two poles, kinetochore microtubules attach to the kinetochore. Mitosis phases: prophase, prometaphase,
metaphase, anaphase, telophase. Then cytokinesis, which is not part of mitosis. Prometaphase: mitotic spindle fully
formed, centrosomes move apart and demarcate the two poles, once nuclear envelope has dissociated, spindles can
allow chromatids to interact. Microtubules grow out from the two poles. Metaphase: pairs of sister chromatids have
aligned halfway across the cell, attached to kinetochore microtubules. Anaphase: connection between sister
chromatids are broken, each individual chromatid is linked to only one pole by kinetochore tubules, kinetochore
microtubules shorten to pull chromosomes towards pole, polar microtubules lengthen so the two poles move further
from each other. Telophase: chromosomes have reached respective poles, nuclear membranes reform to produce two
separate nuclei, cell membrane folds inwards for cytokinesis to occur. Cytokinesis in animals: cleavage furrow
constricts to separate the cells. Cytokinesis in plants: cell plate forms a cell wall between the two daugther cells.
Bacteria divide via binary fission - after chromosome replication, each copy of the chromosome becomes anchored
to the plasma membrane. FtsZ protein forms ring, constricts bacterial cell, creates two daughter cells. FtsZ is
evolutionarily linked to tubulin. Tubulin in eukaryotes, forms linear microtubules, and acquired additional roles
in cell division: division of the nucleus, sorting of chromosomes. Meiosis: homologous pairs form a bivalent,
crossing over. Homologous pairs of sister chromatids positioned side by side in a bivalent way (4 chromatids).
Process is called synapsis. Synaptonemal complex connects sister chromatid pairs. Physical exchange between
chromosome segments of the bivalent. This may increase the genetic variation of a species. Chiasma shape.
Humans have an average of 2 crossovers per chromosome.
4.11.2022
Biology Lecture. Meiosis division. Prometaphase 1: spindle apparatus is complete, chromatids attach to kinetochore
microtubules. Each pair of sister chromatids attaches to a single pole. (In mitosis, each pair of sister
chromatids attaches to both poles). Metaphase in meiosis: bivalents organized along metaphase plate as double row.
Arrangement of sister chromatids is random with respect to the homologs - this promotes genetic diversity. 23
chromosomes per set means 2^23 options just from chromosomes (not including crossing over). 1 diploid cell
produces 4 haploid cells. Short arm of the chromosome is p (petit), long arm is q. Metacentric, submetacentric,
acrocentric, telocentric. Chromosome structural changes: deletions, inversions, duplications, translocations.
Deletions happen when a region of the chromosome is deleted, likewise for duplication. Translocation is exchange
of one chromosome fragment with another chromosome. Euploid vs polyploid: polyploid is when cell has three or more
sets of chromosomes. If 3 copies, triploidy, if 4, tetraploidy, etc. Aneuploidy refers to when the number of
chromosomes of one pair of chromosomes is changed instead of all of the chromosomes. Trisomy 2 (2n+1), monosomy 3
(2n-1). Nondisjunction: chromosomes do not sort propely during cell division. Can happen in meiosis 1 or 2. If in
meiosis 1, all 4 products are chromosomally abnormal. If in meiosis 2, only 2 products are chromosomally abnormal.
Alleles are different variants on homologous chromosomes. Spermatogenic cells, sustentacular cells. Interstitial
cells are in between seminiferous tubules, create and release testosterone. Myoid cells surround seminiferous
tubules. Spermatogenisis starts in spermatogenic cells, spermatogonia (divide mitotically before puberty). At
puberty, spermatogonia become fully functional stem cells and divide by mitosis. One daughter cell remains a stem
cell, while the other daughter cell becomes primary spermatocyte. Spermatocyte undergoes first meiotic division,
forms two secondary spermatocytes (haploid). Secondary spermatocytes undergo meiosis 2, form 4 haploid spermatids.
Sustentacular cells surround spermatogenic cells and provide support, extend from basement membrane to seminiferous
tubule lumen. This anatomical arrangement forms the blood-testes barrier.
4.18.2022
Biology Lecture. Oogenesis begins before female infant is born. Primary oocyte makes two cells: first polar body and
secondary oocyte. 1st polar body and secondary oocyte are each haploids. 1st polar body degenerates because of lack
of cytoplasm. Secondary oocyte is arrested in metaphase 2, if fertilized will complete the second meiotic division.
Both spermatogenesis and oogenesis produce gametes (23 chromosomes). Different cell sizes: ovum receives most of
nutrient rich cytoplasm, sperm cell does not recieve much cytoplasm. Chromosomes contain DNA and proteins. In late
1920s Fredrick Griffith was working with pneumococcus, concluded that "S" type bacteria is virulent, "R" type is
benign. Heat-killed "S" type bacteria were also benign. Mix of live "R" type and heat-killed "S" type is virulent.
Therefore there is some material that the dead "S" bacteria transferred to the "R" bacteria, making them virulent.
1940s: Avery, MacLeod, McCarty used purification methods to reveal that DNA is the genetic material. Incubate
isolated DNA with "R" type bacteria. Various trials. Result is that DNA is found to be responsible as the genetic
material.
4.25.2022
Biology Lecture. Telomerase functions - cells without telomerase result in senescence, cells with telomerase result
in no senescence. Cancer cells found a way to overcome senescence, and 99% of all types of human cancers have high
telomerase levels. Stem cells also have high telomerase. How does DNA get packaged into the cell? Two conditions:
it has to fit, it has to be easy to find specific genes. 3.2 billion base pairs. Chromosomes are the morphological
meaning, chromatin is biochemical meaning (DNA + Proteins that have organized DNA). DNA wrapping around histone
proteins forms nucleosomes (beads). Shortens DNA 7-fold. Histone octamer (4 types, 2 copies of each) H2A, H2B, H3,
H4. 146-147 base pairs of DNA wrapped around octamer. If DNA is tightly wrapped around the histone it is more
difficult to read. Histone proteins have tails (N-terminus) that protrude from the octamer. Lysine and arginine
residues on N-terminus attract negative charges of phosphates. Lysine acetylation forms acetyl-lysine, neutralizes
positive charge, histone tails no longer attract DNA. Acetylation is reversible. Chromatin compaction - studies in
animals and humans show that Alzheimer's disease has lots of histone tail deacetylation years before disease
becomes apparent in specific brain regions. H1 histone linker - 20-100 base pair DNA connects nucleosomes, depends
on species and cell type. 30 nm fiber (zig zag structure) additional 7-fold compaction. Radial loop domains
created by the interaction between 30 nm fibers and proteins in the nuclear matrix (nuclear lamina). Radial loop
domains organize chromosome territories. On chromosomal level compaction is not identical - euchromatin is less
compacted, heterochromatin is highly compacted. Inborn errors of metabolism: in 1908 proposed relationship btwn
genes and enzyme production. Phenylalanine - amino acid found in human diet, phenylalanine hydroxylase breaks it
down into tyrosine. Mutations cause phenylketonuria. Alkaptonuria - homogentisic acid builds up in the body, gets
deposited into various tissues. Leads to arthritis, ears, connective tissues, etc. Disease caused by missing
enzyme - inborn error of metabolism. Beadle and Tatum experiment - neurospora crassa. If we mutate a gene that
makes an amino acid, we must give it that amino acid to grow. One-gene one-enzyme hypothesis. DNA -transcription->
RNA -translation-> Protein. Structural genes make an mRNA which gets translated. Gene is an organized unit of DNA
sequences that enables a segment of DNA to be transcribed into mRNA and ultimately results in the formation of a
functional product.
4.27.2022
Biology Lecture. Genes - promoter signals beginning of transcription, transcribed region contains info on amino
acid sequence, terminator signals the end of transcription. Regulatory sequences bind regulatory proteins to change
rate of transcription, inhibit/enhance transcription of a gene. 3 stages of transcription - initiation, elongation,
termination. Initiation is a recognition step, in bacteria, sigma factor causes RNA polymerase to recognize the
promoter region. Completed when the 2 DNA strands separate near the promote to form an open complex. Elongation -
polymerase synthesizes RNA, template 3' to 5' used for RNA synthesis. Coding strand 5' to 3' is not used. mRNA
synthesized to 5' to 3'. mRNA has the same nucleotide sequence as the coding strand except U substitutes T residue.
Termination - RNA polymerase reaches terminator sequence, causes both polymerase and newly made mRNA to fall off.
Synthesis of RNA transcript is always 5' to 3', DNA template strand reads 3' to 5'. Direction of transcription and
which DNA strand is used varies among genes. RNA polymerase II make mRNA, polymerase I and III make structural
RNA. RNA polymerase II in eukaryotes has 5 . Bacterial mRNA can be translated immediately, eukaryotic mRNA made as
longer pre-mRNA, requires processing into mature mRNA. Splicing - removal of introns. Addition of cap, addition of
a tail. Recognition sequences on boundary between introns and exons which allows for splicing out of introns.
Exons are transcribed and translated. 5' cap and 3' poly A tail. Splicing happens in RNA not DNA. Splicing first
described in 1977 - loop like structure for introns before they get cut out. Capping - modified guanosine attached
to 5' end. 7-methylguanosine, recognized by cap binding proteins. Prevents mRNA degradation in the cytosol, helps
mRNA bind to the ribosomes. Poly A tail - 100 to 200 adenine nucleotides added to 3'-end. Helps export mRNA into
the cytosol, increases stability in the cytosol, not encoded in the gene sequence. Average human gene has 9
introns, intron size 10-100k nucleotides. Spliceosome removes introns precisely. Composed of snRNPs (small
nuclear RNAs + proteins). Alpha-tropomyosin - protein found in thin filaments of muscle. Alternative splicing -
do not have to have same resulting protein. Exon 2 used in smooth muscle, Exon 3 used in striated muscle. One
set of genome can create diverse proteome due to alternative splicing. 4 bases in DNA, 2 bases combo makes 16
codons, but 20 amino acids. Combination of 3 bases specify 64 codons. The genetic code is degenerate so more
than one codon can specify the same amino acid. This provides some protection against mutation. ATG/AUG = start
codon. 3 stop codons: TAA/UAA, UAG/TAG, UGA/TGA. tRNAs in bacteria, archaea, and eukaryotes share common features.
Different tRNA molecules present in a cell, encoded by different genes. Amino acid attachment site at the 3'
single stranded region. Anticodon, Stem-loop, hydrogen bonds present. Cell has several tRNA(Ser) molecules.
5.4.2022
Biology Lecture. 98.5% of human genome is non coding, "junk DNA" that doesn't get translated. Several classes:
rRNA, tRNA, lncRNA, etc. In charge of encoding RNA molecules that get formed because of transcription and don't
get translated. MicroRNAs (miRNAs) one class of non-coding RNA molecules, around 22 nucleotides long. Silence the
mRNA (messenger RNA) molecules. Are complementary to mRNA thus capable of binding to it in order to silence the
molecule. mRNA does not get translated as a result. Called RNAi (RNA interference). MicroRNAs provide a path to
inhibit the creation of a protein by silencing the problematic portion of the mRNA. pre-miRNA cut by Dicer to
release the miRNA, associates with cellular proteins to become RNA-induced silencing complex (RISC). Upon binding
mRNA is degraded. 1 miRNA binding to mRNA is sufficient to block translation. Mutation, DNA repair, and cancer.
Mutation is a heritable change in the sequence of the genetic material. Source of variation for natural selection,
to the individual, mutations more likely to be harmful than beneficial. All species have developed ways to repair
mutations. Base substitution (T to G), base insertion/deletion. Silent mutations do not alter the amino acid
sequence, possible due to the degeneracy of the genetic code. Occurs in the 3rd base of codons. Missense mutations
change a single amino acid in a polypeptide. Less likely to alter function if the substituted amino acid is similar
in chemistry to the original one. Sickle cell disease (mutation in the beta-globin gene). Nonsense mutations -
change from a codon encoding an amino acid to a stop codon, produces a truncated polypeptide. Intergenic region is
least likely to cause problems. Germ-line or somatic cell mutation - germ-line give rise to gametes, somatic cells
(rest of the body) gives a genetic mosaic with patches of mutant tissue. Disruption of base pairing due to a
mutagen. Nitrous oxide exposure with cytosine forms uracil, pairs with adenine (C to U, U pairs with A). Mustard
gas alkylates bases, adding methyl or ethyl groups. Guanine modified to ethylguanine, which pairs with thymine.
Base analogues - 5 bromodeoxyuridine (5-BU) can exist as keto or enol form, binds to adenine or guanine. Physical
mutagens via ionizing and nonionizing radiation. X-rays, gamma rays can break one or both strands of DNA. UV has
less energy and can crosslink two thymines. Ames test to determine if a chemical is a mutagen. Uses Salmonella
typhimurium that cannot synthesize histidine due to a point mutation. Bacteria cannot grow unless histidine is
added to the medium or a second mutation occurs that fixes the original, allowing for synthesis to occur. Test
monitors the rate at which mutation occurs. Types of DNA repair - direct repair, MGMT removes methyl group from
methylguanine, "suicide mission." Base exclusion repair (BER) corrects small base lesions that do not significantly
distort the DNA helix. Damage typically results from deamination, oxidation, or methylation. Nucleotide excision
repair best for bulky regions. Uncontrolled cell division --> cancer and other disease, 25% of deaths in the US are
from cancer. Many cancers occur due to exposure to physical or chemical agents that cause mutations (exposure to
mutagen). Carcinogen - can cause cancer but doesn't involve mutation. Cancer happens when something goes wrong with
one cell and that cell loses cell-cycle control, starts replicating into a tumor. Oncogenes normally need to be
silenced, lead to cancer when overactive. Tumor suppressor genes normally need to be active, lead to cancer when
inactivated. Oncogene may promote cancer by keeping the cell division signaling pathway in permanent "on" state.
RAS GTP to GDP hydrolyzation. Some mutatiosn convert Ras to oncogenic Ras, decreased GTP hydrolysis activity,
higher levels of active Ras, faster cell divisions.
Chemistry: The Basics
Info to be added later. Covers stoichiometry, acids & bases, buffers & titrations, basic thermodynamics.
Physics: The Basics
Info to be added later. Covers kinematics, rotational motion, fluid mechanics (lightly), goes up to optics.
Physics 3: Waves, Optics, and Thermodynamics ζ
Decided to take this instead of discrete math as it's a course I will have to take in the future. I think
learning the content sooner rather than later is a fine idea. Expect infrequent but lengthy posts on this
class (and possibly some revision posts next fall, when I actually take it for credit).
Physical Chemistry - Quantum Chemistry and Mechanics ζ
Lecture 1: A Quick Introduction to Waves
Let's begin with some pretty tough physics. Hey, at least it's not quantum...yet. So we're going to begin with
classical waves. Classical waves are common in everyday life. They include sound waves, water waves, and many
more, but today we're going to talk about a specific example of a vibrating string.
A vibrating string is effectively just a deformed string that has a restoring force towards its original
position. However, this restoring force overshoots the original position, and then continues overshooting back
and forth until it finally settles into an equilibrium where the string is no longer vibrating. This decaying
overshoot is referred to as damping, and it adds a lot of complexity, but luckily we can ignore it here (so the
string vibrates infinitely.)
So let's have a string along which there is a length x, and allow the deformation amount to be the variable 'psi.'
We can express psi as a function of x and t, where t is time. We must also remember that for the string to be
vibrating, there must be a restoring force. Use F=ma and convert it into derivative form (so F=m*(d^2(psi)/
dt^2)) [or put simply, the second derivative of psi over the second derivative of time]. Why is this important?
Well, the restoring force is proportional to something called the curvature of the string, and the curvature of
the string can be expressed as the second derivative of psi over the second derivative of x. This allows us to
end up with the expression [(d^2(psi)/dt^2) = k*(d^2(psi)/dx^2)] where k is some constant. However, we can actually
determine what k is. On the left, time is in the denominator while on the right "x" is, we need the expressions to
be equal so we must multiply (d^2(psi)/dx^2) by a term that is (length)^2/(time)^2. So it follows that this is just
velocity^2, or v^2.
So, [d^2(psi)/dt^2 = v^2*d^2(psi)/dx^2]. But not really. I haven't covered calculus 3 here yet but since psi is
a function of x and t, you would use partial derivatives (this means you treat the non-derived variables as some
constants.) I will be using PD for partial derivative denoting but commonly it is a lowercase delta from the Greek
alphabet. A quick example is PD(x) of x^2+5y^3+6yx^5. Treat y as a constant, so you have 2x+0+30yx^4.
Now back to waves and vibrating strings. A common way to express a travelling wave (which is what we're dealing
with) is psi(x,t) = Asin(wt-kx), where w and k are constants. If we take 2 PDs of psi with respect to x, we end up
with -k^2(Asin(wt-kx)). Doing the same with respect to t yields -w^2(Asin(wt-kx)). You may notice that the original
expression of psi is still in both formulas, so we can rewrite them as -k^2(psi) and -w^2(psi.)This is useful
because we've now rewritten the equation from the first sentence, previous chapter. So -k^2*v^2*psi = -w^2*psi.
We can now solve for v^2 - some algebraic manipulation will show that it is w/k.
We have this psi sine wave formula, and we can manipulate either the x or t term to see two different things. If
we go through all the values of time, we will have all of the temporal (time-based) information at some fixed x
value. If we go through all the values of x, we will have all the spatial info - we freeze time and see a snapshot
of the wave equation. We can also use the properties of sines to get more information about the wave equation. A
(regular) sine wave is always uniform - the value at sin(0) is the same at sin(2pi), et cetera. We can use this to
always know when the peak of the sine wave is: the angle is (pi/2)+2n(pi), where n is an integer. Sine of pi/2 is
1, the maximum value, and the period of the sine wave is 2pi, hence the peaks repeat. If we solve for the peak
(pi/2)+2n(pi) such that it is equal to wt-kx, the internal component of the wave equation, we can get more
information on the nature of the wave function.
It is stipulated (pi/2)+2n(pi) = wt-kx. Manipulating this algebraically, x = ((-pi/2)-2n(pi)+wt)/k. And x here
changes values based on n, so x_0 is ((-pi/2)+wt)/k, and x_1 is ((-pi/2)-2(pi)+wt)/k. The difference between these
two values is important: it's 2pi/k, and this value is equal to the wavelength of the wave function. The symbol is
denoted as lambda. 2pi/k = lambda and 2pi/lambda = k, so the units of k are 1/m (since lambda is a length). From
the formula x = ((-pi/2)-2n(pi)+wt)/k we can take a derivative with respect to time to get velocity. This resolves
to v = w/k, which should seem familiar from earlier. This establishes that its the velocity is in the wave equation,
not velocity*(some constant).
A period, represented by tau, is the time between 2 consecutive maxima in the sine wave. Since v = distance/time,
time is distance/v. Therefore tau = lambda/v, since the units work in the same way. But continuing with the algebra,
tau = lambda/v = (k*lambda)/w = 2pi/w. Another term you need to know here is frequency, which is 1/tau and denoted
by the greek letter nu. So, nu = w/2pi, so w = 2(pi)nu which is the angular frequency. We can also express v as
nu*lambda.
We are familiar with Asin(wt-kx) now, and it is necessary to ask what happens if k is negative. Well, the function
travel direction reverses (leftwards instead of rightwards). So this is another solution to the wave function.
Critically, linear algebra dictates that the sum of two solutions to a function will yield another solution: this
is called superposition. So a possible wave function is Asin(wt-kx) + Bsin(wt+kx), where A, B, w, and k can be any
number (as long as w/k = v).
On its own, though, this isn't very useful. We need to set boundary conditions on the vibrating string by fixing
its position at two points (akin to a string in real life). Assuming fixed points are x = 0, L, we can start
plugging numbers into our new wave function. psi(0,t) = Asin(wt-k0) + Bsin(wt+k0) = 0. So, A = -B. psi(L,t) =
Asin(wt-kL) - Asin(wt+kL) = 0. Using a trigonometric identity*, this results in A[sin(wt)cos(kx) - cos(wt)sin(kx)]
- A[sin(wt)cos(kx)+cos(wt)sin(kx)]. *The identity is sin(A+B)=sinAcosB+sinBcosA. Then by cancelling things out,
the result is -2Acos(wt)sin(kx) = psi(x,t). This is now a standing wave - the minima and maxima locations are
fixed and are determined by kx = (pi/2) + n(pi). At the fixed points, kL = n(pi), so k_n = (n(pi)/L). From this,
w_n = (n(pi)v/L). [END OF LECTURE 1]
Lecture 2: Review and Intro to the Schrodinger Equation
As was discussed in the last lecture, a vibrating string obeys the classical wave equation, which is expressed by
(PD^2(psi)/PDx^2) = (1/v^2)(PD^2(psi)/PDt^2), This is a linear partial differential equation, or LPDE for short.
We also went over superposition, which is a property of linear differential equations that states that two LDE
solutions can be summed to create a 3rd solution. With some algebraic manipulation, we were able to get the formulae
for traveling waves and standing waves.
The Schrodinger equation is not very far from the classical wave equation, but one of its most distinct features
is the inclusion of the imaginary number "i," which we will briefly go over. "i" is an imaginary number that is the
equivalent of the square root of -1. It follows that i^2 is -1, i^3 is -i, and i^4 is 1. Complex numbers are a
combination of a real number and an imaginary number. The general form of a complex number is x + iy. The inclusion
of x and y is important, as it is useful for visualization. Imagine a cartesian plane, but instead of the y-axis
being real numbers, they're imaginary (so, real number*i). With this cartesian plane, we can now plot these complex
numbers. We can use the Pythagorean theorem to determine their distance from the origin, and name that term "r,"
because it's essentially a radius. So, r = sqrt(x^2+y^2). We've basically created a triangle, so let's call the
angle it makes with the +x axis "theta." All the trigonometric rules apply: tan(theta)=y/x, sin(theta)=y/r,
cos(theta)=x/r. In other words, x=rcos(theta) and y=rsin(theta). So the complex number can also be expressed as
r(cos(theta)+isin(theta)), and we can use Euler's identity to express that as re^(i*theta). Euler's identity is a
result of the power series expansion of e^x but with (i*theta) instead of x, and is a fun proof if you're studying
Calculus 2. The last bit of knowledge needed on complex numbers is their conjugate: x+iy has a conjugate of x-iy.
This is useful because a complex number * its conjugate is x^2+y^2, which is r^2.
It's time for the Time-dependent Schrodinger Equation. (-h_bar/i)(PD(psi)/PDt) = (-(h_bar^2)/2m)(PD^2(psi)/PDx^2)
+ V(x,t)psi(x,t). Let's definie some terms. V(x,t) is the potential energy function, h_bar is Planck's constant/2pi,
psi(x,t) is the state function, which has all the information about the actual system. This formula is pretty much
underivable - much of it was deduced by experiment. It's also very complex, so we're going to simplify it to the
Time-independent Schrodinger Equation. (-h_bar/i)psi(x)(df(t)/dt) = (-(h_bar^2)/2m)f(t)(d^2(psi)/dx^2)
+ V(x)f(t)psi(x). So what changed? Well, we performed a separation of variables, so now we're dealing with normal
instead of partial derivatives. We can simplify this further though, by dividing by f(t)psi(x). This gives us the
expression (-h_bar/i)(1/f(t))(df(t)/dt) = (-(h_bar^2)/2m)(1/psi(x))(d^2(psi)/dx^2) + V(x). Since both sides are
equal but have different variables, this means both sides are constant. We can call this constant "E" for energy.
Solving the left side for E is simple - isolate 1/f(t)*df(t)/dt and then integrate both sides. You should get
f(t) = Ae^(-iEt/h_bar), where A is just a constant. The useful manipulation for the right side isn't a full
solution - the formula you want to end up with is (-(h_bar^2)/2m)(d^2(psi)/dx^2)+V(x)psi(x) = E*psi(x).
[END OF LECTURE 2]
Lecture 3: Probability Density, Particle in a Box
One of the postulates of Schrodinger's equation is that the state function (or the more simple wave function) can
have its absolute value squared in order to determine the probability that a particle is in a given region of 1
dimensional space. A probability of 0 is 0%, while a probability of 1 is 100%. Thus it must follow that an integral
from -(infinity) to +(infinity) of the probability density equation will yield 1 - the particle is guaranteed to be
in some place in space. This is called normalization. For the simpler wave function, we have the necessary tools to
evaulate these probabilities.
While the probability density formulas have a lot of variety, the proper ones are always normalizable and positive.
You can't have a negative probability, it doesn't really make sense. The probability of finding the particle between
points x = a and x = b is the integral from a to b of |(psi)|^2 dx.
A particle in a box is the simplest solvable quantum mechanics problem in 1D (at least, not including probability).
Earlier, we came up with the formula (-(h_bar^2)/2m)(d^2(psi)/dx^2)+V(x)psi(x) = E*psi(x). For the purposes of the
particle in a box, we will assume V is a constant. So, we can do some simplification. (-(h_bar^2)/2m)(d^2(psi)/dx^2)
= (E-V)psi(x), which leads to d^2(psi)/dx^2 = (-2m(E-V)psi(x))/(h_bar^2). Important thing to note here is that while
in Classical Mechanics, E-V is always positive (the potential energy never exceeds the total energy), in Quantum
Mechanics, this is not the case. In fact, we have a name for times when V exceeds E: quantum tunneling. This makes
sense if you draw a potential energy diagram and then set a random total energy bound. A quantum particle "tunnels"
into the potential energy graph when it exceeds the total energy in the system.
With quantum tunneling out of the way, we can establish more rules for this particle in a box formula we came up with.
For E > V, psi follows a function like Asin(kx), where k is the square root of (2m(E-V)psi(x))/(h_bar^2). For V > E,
psi follows an exponential function like Ae^(+/- kx), where k is sqrt of (2m(V-E)psi(x))/(h_bar^2). This lets us look
at the qualitative behavior of the wave function - how it looks like on various potential energy graphs.
Now let's set up the actual box for the particle: we do the walls by setting V to infinity, By solving for psi there
we get 0. Knowing probability density, this means there is 0 chance of the particle being there, since |0|^2 is still 0.
For the "floor" of the box, we set V to 0. From this we get a familiar equation: psi(x) = Acos(kx)+Bsin(kx). This should
remind you of the vibrating string standing wave - it is basically like that. Next, we can start solving problems for
the particle in the box - for example, energy level problems. [END OF LECTURE 3]
Lecture 4,5: Particle in a Box Cont., Particle in a Rectangular Well, Operators
Going back to last lecture, a Particle in a Box (or PIAB) is described by the potential energy equation V(x) = 0 within
the box and V(x) = infinity outside of the box. This keeps the particle from tunneling into the box walls, and allows
for a solution that is akin to a standing wave from classical physics (more or less). We can use boundary conditions at
x = 0 and x = l to find out more about this particle. The solution for V(x) = 0 can be represented as Acos(kx) + Bsin(kx)
= 0. Ergo, for x = 0, A must = 0. For x = l, we're left with just Bsin(kl) = 0, which means the kl term is a multiple of
pi, also expressed as n(pi). Thus l = (n(pi)/k). Since V(x) = 0, we can express k using a formula from last lecture as
sqrt(2mE/(h_bar^2)). So, we can express l as equal to n(pi)sqrt((h_bar^2)/2mE), where n is a positive integer. If n = 0
we just have an empty box, which isn't useful for solving various problems. Same goes for B = 0: empty box.
We can rearrange this formula for l to solve for E, the energy level. This reveals some interesting information. After
algebraic manipulation, E = (n^2*h^2)/(8ml^2). This formula is critical because it means that on the quantum scale, energy
values are discrete, or in other words, quantized. This is due to n being an integer. In addition, it follows that energy
increases quadratically, because n is squared, while all the other values are constants. So, spacing between the levels
gets larger as n increases.
psi_n(x) = Bsin(kx) = Bsin(n*pi*x/l) within the box. This is what we've determined. But to find functions that work, this
one needs to normalize. So, we need to find what value B is. For a PIAB, normalization is the integral from 0 to l of
|psi(x)|^2 dx, so here, integral from 0 to l of |B^2sin^2(n*pi*x/l)|dx. After doing the computation (email me with any
questions about this, by the way) you end up with |B^2|/2 * l. Hence, B = sqrt(2/l). So, the final solution for PIAB is
psi_n(x) = sqrt(2/l)*sin(n*pi*x/l).
An interesting proprety of these wavefunctions is considering multiplying 2 different psi(x)s (so, psi_n(x) and psi_m(x))
and integrating them over all space in the box (from 0 to l). Well, after using a trigonometric identity, we find out that
the integral is 0. This means that psi_n and psi_m are orthagonal. Using the Kroenecker delta (which put simply, compares
i and j, if i = j then Kroenecker(ij) = 1, if they aren't equal, Kroenecker(ij) = 0) we can state that psi_n and psi_m are
also orthonormal.
Graphing the PIAB is fairly straightforward. psi_1(x) = sqrt(2/l)*sin(pi*x/l), so its effectively a large sinusoid stretching
across the whole box. It has 2 nodes, one at x = 0 and the other at x = l. psi_2 is identical but with n=2 instead of 1, and
has a 3rd node at l/2. So on and so forth. The probability density is also just the absolute value of psi(x), squared. On
probability density, its interesting to point out that the density for a classical particle in a box is just a rectangle. The
particle has an equal chance of being anywhere in the space of the box. Recall that raising n is basically the same as raising
the energy level, and raising n by 1 adds one node to the graph. So as n goes to infinity, psi_n(x) becomes a rectangle that
is centered at y = 0, between 0 and l. Then, the absolute value is just another rectangle - and is in fact identical to the
classical PIAB probability density. This an example of the correspondence principle - a quantum particle with a macroscopic
energy starts behaving like a macroscopic particle.
What if we had a free particle in one dimension: no box, and V(x) is 0 everywhere. Using the time independent Schrodinger eqn
psi must then be sin(theta), cos(theta), or e^(i*theta). Using the time independent equation again and solving for psi(x),
we get C_1(e^(i*sqrt(2mEx/h_bar))) + C_2(e^(-i*sqrt(2mEx/h_bar))). E > 0 everywhere, so oscillatory behavior (as per Lec3)
is found everywhere. Also, E cannot be quantized - theres no "n" term. It can only be greater than 0. So, lone electron
energy is not quantized.
Particle in a Rectangular Well: the infinitely high potential energy of the box is not exactly realistic. There's always going
to be a little bit of tunneling. Rectangular wells have complicated equations because the regions of finite and 0 energy have
to have one continuous* function through them. Thus, psi(x) of region 1 = C*exp[(sqrt(2m/h_bar^2)*sqrt(V_0-E))x], similarly
for region 3 G*exp[-(sqrt(2m/h_bar^2)*sqrt(V_0-E))x]. Region 2, within the actual box, is somewhat familiar, as it should be.
Acos[(sqrt(2m/h_bar^2)*sqrt(E))x] + Bsin[(sqrt(2m/h_bar^2)*sqrt(E))x]. If the well is symmetrical (potential energy of region
1 = region 3) then C = G. The computation of actual quantities is complicated by the 4 boundary conditions that must be applied
for A, B, C, and G. *Function must also be differentiable.
Quantum tunneling is an inherent part of the Particle in a Rectangular Well problem, hence why the formulas for psi(x) of the
different regions are more complicated. But what happens when the total energy of the system exceeds the energy 'walls'
that are imposed by the well? Well, this results in something called an unbound energy state, and allows the particle to be
found very far from the box. If we let N be the number of bound states, where a bound state occur when V_0 > E and an unbound
state occurs when E > V_0 (where V_0 is the "height" of the potential energy of the well), then we can state the following
inequality: N >= b/pi > N-1, where b = sqrt(2mV_0)/h_bar, also expressed as b = 2pi*sqrt(2mV_0)/h.
A common question is to present a real world example of quantum tunneling. There are two good cases: alpha decay and ammonia
inversion. Alpha decay occurs via tunneling, because the helium nucleus that breaks off the the core atom can break off without
having enough energy to counteract the nuclear forces holding the atom together. Ammonia inversion is an ammonia molecule
inverting in three dimensions, much like an umbrella in the wind. The core nitrogen stays in place, while the lone pair switches
sides with the hydrogen atoms. This process occurs spontaneously at most temperatures, despite the actual energy required for a
inversion being higher than the energy within the system. [END OF LECTURE 4/PART 1 OF LECTURE 5, END OF CH2]
Lets talk about Operators and the start of chapter 3. When we look at one of the time independent Schrodinger equations, such as
-(h_bar^2/2m)(d^2psi/dx^2)+V(x)psi(x) = E*psi(x), we notice that psi(x) shows up on both sides. This basically means that the
-(h_bar^2/2m)(d^2psi/dx^2)+V(x) part 'operates' on psi(x), and transforms it into the function E*psi(x). But before we get too
deep into quantum operators, we first must cover the basic rules that govern them.
Let D be the differentiation by x operator. Df(x) = f'(x). Let "3" be an operator that multiplies a function by 3. "3"f(x) = 3f(x).
We'll come back to these soon. You can add operators, so (D+3)f(x) = Df(x) + 3f(x). Subtraction works the same, (D-3)f(x) = Df(x)
- 3f(x). For example, (D+3)(x^3+5) = 3x^3+3x^2+15. Products of operators: (D*3)f(x) = D(3(f(x))). CRITICALLY, order matters here.
If we have an operator x that multiplies a function by x, for example, (Dx)f(x) = xf'(x) + f(x). This is determined by the chain
rule. However, (xD)f(x) = xf'(x). This means (xD) and (Dx) do not commute. (3D and D3 commute, though). We also have some special
operators: 1 is the unit operator, so (1)f(x) = f(x), and 0 is the null operator (just turns f(x) into 0). We can perform algebra
with operators too. Knowing Dx of f(x) is xf'(x) + f(x) and xD of f(x) is xf'(x), we can state that Dx - xD = 1, the unit op.
The associative law is also a part of operator algebra, so operators A(BC) = (AB)C. The main difference between operator algebra
and regular algebra is the lack of a commutative law. We have a term called a commutator, though. A commutator [A,B] is defined
as AB - BA, where if the result is 0 the 2 operators commute, if its not 0, they don't. Operator exponents work as expected, so
A^2 = AA, A^n = AAAA.....A n times.
We're going to be dealing with linear operators for the most part. "A" is a linear operator if and only if A[f(x)+g(x)] = Af(x) +
Ag(x), and A[cf(x)] (where c is a constant) = cAf(x). So operator x, x^2, D, D^2 are all linear operators. However, something like
sqrt, ()^2, sin, and cos are not. The linearity of the function is not preserved. Linear operators are related closely to
eigenvalues and eigenfunctions. Eigen is German for characteristic, so an eigenfunction is the core function that results from a
linear operator, while an eigenvalue is the value that affects that function. For example, applying D(e^(2x)) results in 2e^(2x),
so e^(2x) is the eigenfunction and 2 is the eigenvalue. For all eigenfunctions and eigenvalues of D, a separation of variables can
be done. D is just d/dx, so let df/dx (where f is just a function) = cf. Via separation of variables, the result is df/f = kdx.
Integrating both sides and rearranging for f(x), we get f(x) = Ce^(kx). k can be a complex number (this will be useful in later
chapters).
So back to our original Operator, -(h_bar^2/2m)(d^2psi/dx^2)+V(x). This is called the Hamiltonian operator (H). So, H*psi(x) =
E*psi(x). Ergo, E is the eigenvalue. To understand what exactly the Hamiltonian is, we have to go back to classical mechanics. The
Hamiltonian there represents the total energy of a system - kinetic + potential energy. It is described in terms of coordinates q,
commonly x, y, z for our 3-dimensional problems. We take the conjugate momentum P_q, and can then calculate the total kinetic
energy of a system using P^2/2m. For 3 dimensions, its just (P_x^2+P_y^2+P_z^2)/2m. The Hamiltonian for 1 particle in 1D in classical
mechanics is similar to the one in quantum. Classical: H = P_x^2/2m + V(x); Quantum: H = -(h_bar^2/2m)(d^2psi/dx^2)+V(x). It can be
proven that the quantum one can be derived from the classical one.
With what we've learned here, we can generalize. Let B be a quantum mechanical operator associated with a physical property B. Bf_i
= b_i*f_i, where i is a natural number, and f_i is an eigenfunction of B with eigenvalue b_i. Quantum Mechanics postulates that a
measurement of property B must yield one of the eigenvalues b_i of operator B. So if we let the B operator = Hamiltonian operator,
then the Hamiltonian of psi_i = E_i*psi_i.[END OF LECTURE 5]
Lecture 6,7: More Operators; Expanding into the Third Dimension
Quickly touching upon operator B's purpose - it simply interlinks physical properties such as momentum to quantum mechanics. It's
there so that we can derive more properties of quantum particles and systems. Now, another postulate: if the state function of the
system Psi (capital P, this is in terms of x and t) is the eigenfunction of B operator with eigenvalue b_k, then the measurement
property of B must yield b_k. If we suppose Psi(x,t) = e^-(iEt/h_bar)*psi(x) where psi(x) is the solution to the time-independent
Schrodinger equation, then is Psi(x,t) an eigenfunction of the Hamiltonian operator? Well, the Hamiltonian of Psi(x,t) is equal to
the Hamiltonian of e^-(iEt/h_bar)*psi(x). And the Hamiltonian has no derivative with respect to time, so from its "perspective" the
e^-(iEt/h_bar) term "looks" like a constant. Thus it can be extracted out of the operator, so e^-(iEt/h_bar)*H(psi(x)) = e^-(iEt/h_bar)
*E*psi(x). We can factor out E and our result checks out - the state function Psi is an eigenfunction of the Hamiltonian.
We can consider another property - momentum p_x, which can also be expressed as (d/dx)*(h_bar/i). What are its eigenfunctions and
eigenvalues? Well, lets say p_x*g(x) = k*g(x) [k is a constant] so (d/dx)*(h_bar/i)*g(x) = k*g(x). Rearranging, dg/dx = ikg(x)/h_bar
so a general way to express g(x) is as Ae^(ikx/h_bar) [a is also a constant]. Using Euler's law, this can be split into g(x) =
A[cos(kx/h_bar)+i*sin(kx/h_bar)]. We can conclude from this that k must be a real number, or else g(x) can go to infinity, which is
not possible. Now lets consider the momentum of the PIAB. Psi(x,t) = e^-(iEt/h_bar)*sqrt(2/l)*sin(n*pi*x/l). E = n^2h^2/8ml^2. Now we
can use the momentum operator p_x on Psi. This yields (h_bar/i)*e^-(iEt/h_bar)*(n*pi/l)*sqrt(2/l)*cos(n*pi*x/l). Critically, this is
NOT an eigenfunction of p_x operator because the result is different from our expected k*Psi function. Therefore, we cannot predict
the eigenvalues when p_x is measured. However, (p_x)^2 is an eigenfunction of Psi. The end result is (h_bar)^2*e^-(iEt/h_bar)*sqrt(2/l)
*(n*pi/l)^2*sin(n*pi*x/l). The eigenvalue here is (h_bar)^2*(n*pi/l)^2 = (nh/2l)^2. This is consistent with what we already know about
the PIAB: kinetic energy is the momentum squared divided by 2m, and potential energy is 0. So kinetic energy (or total energy E) is just
(n^2*h^2/8ml^2). What's notable here is that in classical mechanics, p_x = +/- sqrt((p_x)^2). This is not true in quantum mechanics.
[END OF LECTURE 6]
Let's examine 1 particle, 3D Schrodinger equation. Once again H*psi=E*psi. But this time, the Hamiltonian is the sum of the energies in
the x, y, and z directions. Expressed as momentum equations, it's ((p_x)^2+(p_y)^2+(p_z)^2)/2m + V(x,y,z), where V is potential energy.
So, after rewriting p_x,y,z, we end up with (-h_bar^2/2m)*(PD^2/PDx^2 + PD^2/PDy^2 + PD^2/PDz^2) + V(x,y,z). This partial derivative is
called a Laplacian Operator, and is expressed by a point-down triangle symbol called del. For the purposes of these lessons, it will be
written out as del or del^2, the latter is what is written as PD^2/PDx^2 +... So, the Hamiltonian is (-h_bar^2/2m)*del^2+V(x,y,z).
For 2 particles in 3D, you have 2 sets of coordinates for each one, so x1,y1,z1 and x2,y2,z2. The particles can also have different
masses. So the formula for them is (-h_bar^2/2m1)*del1^2+V(x1,y1,z1) + (-h_bar^2/2m2)*del2^2+V(x2,y2,z2). Generalizing, for n particles
in 3D, you have -1*[sum from i = 1 to n of (h_bar^2/2mi)*deli^2+V(x1,y1,z1,x2,y2,z2,...xn,yn,zn)] = Hamiltonian.
Lecture 12: Various Hydrogen Atom Tidbits
The 1s orbital of a hydrogen atom is larger than the 1s orbital of a Helium +1 ion atom. This is due to the nuclear attractive force
of the helium being twice as big. Zeeman effect - discovered experimentally in 1896. If applying external magnetic field to hydrogen
gas spectra, some spectral lines split and shifted. Quantum mechanics explains this. A charged particle moving through space at a
constant velocity produces a magnetic field (this is how electromagnets work). Magnetic field vector B at a point P and charge Q moving
at velocity vector v. B = (mu_0)*Q/(4pi) * (v x r)/(r^3). r is vector from charge Q to point P. mu_0 is magnetic permeability with
value 4pi * 10^-7 N*C^-2*s^2. So an electric current travelling through a loop of wire will cause a magnetic moment to be generated
perpendicular to the loop. Magnetic moment vector |M| = current * loop area (I*A). If loop is circular with radius r, circumference is
2pi*r, area is pi*r^2. If charged particles that make up the current travel at velocity v, time to get around the loop will be distance
travelled/velocity, which is just 2pi*r/v. Current I is charge per unit time, which ends up being Q*v/(2pi*r). Velocity of a particle
is related to linear momentum, so p = mv, thus v = p/m. |M| = Qvr/2 = Qrp/2m. Since that is the magnitude of the magnetic moment, the
vector is just Q*(r x p)/2m = (Q/2m)L, where L is the angular momentum vector.
Now lets apply this information to the hydrogen atom. For a hydrogen atom, Q = -e, m = m_e, so magnetic moment M = (e/2m_e)*|L|. This
is intrinsic if the electron in the atom is travelling in a circular orbit. We know L^2 = l(l+1)h_bar^2. So absolute value of L is
sqrt(l(l+1)h_bar^2) = ((e*h_bar)/2m_e)[l(l+1)]^(1/2). (e*h_bar/2m_e) = mu_b, called Bohr magneton. Value of 9.274*10^-24 J/T. In the
1s state, there is no magnetic moment associated with it, as l = 0. Now lets apply an external magnetic field. There will be
interaction energy between the magnetic dipole and external field, which is given by E_B = -M ⋅ B = (e/2m_e)L ⋅ B. Assume B is pointed
in the z-direction, so B vector = B*k vector. E_B = (e/2m_e)(L_x*i+L_y*j+L_z*k) ⋅ B*k vector. This simplifies to (e/2m_e)B*L_z =
(mu_b/h_bar)*B*L_z. Now lets find the quantum mechanical operator for this interaction. Hamiltonian_B = (mu_b/h_bar)*B*(Operator L_z)
so Schrodinger equation becomes (H + H_B)psi = E*psi. Since Hamiltonian_B = (mu_b/h_bar)*B*(Operator L_z) has an operator we already
know works, our old wave functions are still eigenfunctions of this new Schrd equation. Using psi = R_nl(r)y(l,m)(theta, phi),
(H + H_B)R_nl(r)y(l,m)(theta, phi) = H*R_nl*y(l,m) + H_B*R_nl*y(l,m) = E_n*R_nl*y(l,m)+(mu_b/h_bar)B*(Op L_z)(R_nl*y(l,m)(theta, phi)).
This is just E_n*R_nl*y(l,m)+(mu_b/h_bar)B*(R_nl*(Op L_z)y(l,m)(theta, phi)) = E_n*R_nl*y(l,m) + m*h_bar*R_nl*y(l,m), which simplifies
to (E_n + mu_b*B*m)R_nl*y(l,m). This removes the n-degeneracy of the hydrogen atom wavefunctions. For example, for n = 2, we have
2s, 2p0, 2p1, 2p-1. Corresponding m values: 0,0,1,-1. So 2s, 2p stay at E_2. But as B gets larger, 2p1 rises energetically while 2p-1
falls (2p splits into 2p0, 2p1, 2p-1). Starting at the 1s state and moving to the 2s, 2p state is a simple transition. With a magnetic
field, there are 3 transitions: s to 2p-1, to 2p0, and to 2p1.
(-=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=--=|^|^|=-)
Theorems of Quantum Mechanics: Particle in a box, harmonic oscillator, rigid rotor, hydrogen atom. Atoms with 2+ electrons cannot be
solved exactly. Thus we must develop approximate methods for solving other problems: variation method, pertubation theory. Notation
changes. Integrals are tedious to write --> bracket notation. Integral of conj f_m*(Operator A)*f_n d(tau) is defined 〈f_m|Op(A)|f_n〉
or just 〈m|Op(A)|n〉. For integral (conj f_m*f_n)d(tau), corresponding notation is 〈f_m|f_n〉. For m = n, 〈m|m〉* = 〈m|m〉 thus 〈m|m〉 is real.
All Quantum Mechanical operators are Hermitian operators. Recall average value of A = integral of conjugate of state function*Op(A) *
state function d(tau). Since A represents a physical quantity, its average value must be real. Thus, complex conjugate of the average
of A = average of A. Any linear operator that obeys this equation is called a Hermitian operator. In bracket notation, 〈f_i|A|f_j〉 =
〈f_j|A|f_i〉* --> 〈i|A|j〉 = 〈j|A|i〉*. In matrix notation, A_ij = (A_ji)*. Linear momentum operator, potential energy operator, Hamiltonian
operator are all Hermitian. Theorems about Hermitian operators: eigenvalues of Hermitian operators are real numbers (proof - let g_i
and a_i be eigenfunctions and eigenvalues of A respectively. Then Op(A)g_i = a_i*g_i. Integral(g* * Op(A) * g d(tau)) = Int(g(Ag)* d(tau))
then simplifying and substituting g_i for g, and applying a_i --> Int(g_i* * a_i * g_i) = Int(g_i * (a_i*g_i)*). a_i and a_i* are
constants, so we can just pull them from the integral. Thus, a_i = a_i*); eigenfunctions of Hermitian operators are mutually orthagonal,
meaning Op(B)F = sF, Op(B)G = tG therefore 〈F|Op(B)|G〉= 〈G|Op(B)|F〉* thus t〈F|G〉 = s*〈G|F〉*. s is real though, so s* = s, so (t -s)〈F|G〉 = 0
A set of functions can always be found that are orthogonal (eg. Psi(2p1), Psi(2p0), Psi(2p-1) or Psi(2px), Psi(2py), Psi(2pz)).
Normalization: Int((g_i)* * g_i)d(tau) = 1. Orthogonality: Int((g_i)* * g_j)d(tau) = 0. Thus, 〈i|j〉 can be orthonormal.
Lecture 13: Expanding in terms of Eigenfunctions
Fourier, Taylor series expansion. In general, an arbitrary function that satisfies the b.c of a physical problem can be expanded as a
linear combination of the eigenfunctions of that quantum mechanical problem. Example: expansion of function using PIAB eigenfunctions.
psi_n(x) = sqrt(2/l)*sin(n*pi*x/l). n is a natural number, for x between 0 and l. psi = 0 outside region. Any function f(x) that
satisfies b.c can be described as f(x) = sum_n=1toInfty[a_n*psi_n] = sqrt(2/l)sum_n=1toInfty(a_n*sin(n*pi*x/l)). Now to calculate a_n
multiply both sides by psi_m* and integrate. Int(0tol)[psi_m* * f(x)dx] = Int(0tol)[psi*[sum_n=1toInfty(a_n*psi_n)]dx] = sum_n(1toInfty)
[a_n * Kroenecker(mn)] = a_m. Specific example: f(x) = x between 0, l/2; f(x) = l-x between l/2, l. a_n = Int(0tol)[psi_n* * f(x)dx] =
sqrt(2/l)Int(0tol)[sin(n*pi*x/l)f(x)dx] = sqrt(2/l)Int(0tol/2)[sin(n*pi*x/l)xdx] + sqrt(2/l)Int(l/2 to l)[sin(n*pi*x/l)(l-x)dx] = ...
= [(2l)^(3/2)/(n^2*pi^2)]*sum(2n+1 to Infty)sin(n*pi/2). A set of functions is called complete if it can be used to expand an arbitrary
function. The set of eigenfunctions of any Hermitian operator that represents a physical quantity forms a complete set. Eigenfunctions
of the momentum operator p_x. g_k = e^(ikx/h_bar), f(x) = Int(-infty to infty)[a(k)e^(ikx/h_bar)]dk. Theorem: let g1, g2, g3... be a
complete set of eigenfunctions of Op(A). Let F be particular eigenfunction of A with an eigenvalue k. If F is expanded as F =
sum(a_i*g_i) then the only non-zero coefficients a_i are those for which g_i has the same eigenvalue k. F is bound state of H-atom and
its energy eigenvalue is E_2. F = a_1*psi(2s) + a_2*psi(2p0) + a_2*psi(2p1) + a_2*psi(2p-1). Eigenfunctions and commuting operators.
Theorem: if Op(A) and Op(B) share a common complete set of eigenfunctions, g_i then Op(A) and Op(B) commute. Parity operator Op(PI):
Op(PI) f(x,y,z) = f(-x,-y,-z). Op(PI)^2 f(x,y,z) = f(x,y,z). Op(PI)^2 = unit operator Op(1).
Lecture 20: Chapter 10 Electron Spin and Pauli Principle
H' operator is the perturbation. E(1)_n = int(psi_n* * H * psi_n)d(tau), H' = Ae^-(bx^2), psi_0 = Ae^-(ax^2/2). E_n ~=~ E(0)_n + E(1)_n.
Sodium vapor lamp - bulb with sodium, high voltage across sodium to electronically excite sodium, making it emit light. Visible emission
line - one notable one is called sodium D line (yellow), gives lamps yellowish hue. Wavelength mainly at 589 nm. Shine this light thru
a prism to disperse into various wavelengths - separates into two very closely spaced lines with slightly different wavelengths. Fine
structure of atomic spectra - light breaks up into doublets or triplets that are very closely spaced. 1925 - Uhlenbeck and Goudsmit
proposed that electron has an intrinsic angular momentum in addition to orbital angular momentum. Imperfect analogy to Earth orbiting
sun, Earth spinning on its axis. Electron does not spin like Earth does because of vast disparity in size - called a non-classical
effect because no good classical analogy. Analogous to orbital angular momentum of an electron (operators L^2, L_x, L_y, L_z), we can
postulate existance of spin angular momentum operators (operators S^2, S_x, S_y, S_z). S^2 = (S_x)^2+(S_y)^2+(S_z)^2. [S_x, S_y] =
i*h_bar*S_z, etc. S^2 has eigenvalues s(s+1)(h_bar)^2. s = 0, 1/2, 1, 3/2, etc... S_z eigenvalues = m_s(h_bar), m_s = -s, -s+1,...+s.
s for an electron is restricted to 1/2, so m_s can only be -1/2 or 1/2. S^2 = (3/4)(h_bar)^2, S_z = -(h_bar)/2 or (h_bar)/2. |S| =
sqrt(S^2) = sqrt(3)/2 * h_bar. Let a be spin eigenfunction of S^2 and S_z, with eigenvalue m_s = 1/2. Let b be spin eigenfunction of
S^2 and S_z, with eigenvalue m_s = -1/2. S^2(a) = a(3/4)(h_bar)^2, S_z(a) = a(h_bar)/2, S^2(b) = b(3/4)(h_bar)^2, S_z(b) = -a(h_bar)/2.
Wavefcn psi is a function of spatial coordinates x,y,z. psi(x,y,z). a and b are functions, but of what coordinates? Coordinates m_s.
psi(x,y,z) spatial normalization --> 1. a and b are orthogonal, spin normalization done via summation. a(1/2) = 1, a(-1/2) = 0, b(1/2)
= 0, b(-1/2) = 1. a and b can be thought of as 2-component vectors a = (1, 0) and b = (0, 1). Now how is this related to the hydrogen
atom? To a good approximation H for H-atom is independent of spin variables, therefore can use separation of variables. Psi = Psi(x,y,z)
* g(m_s). H[psi(x,y,z)g(m_s)] = g(m_s)Hpsi(x,y,z) = g(m_s)Epsi(x,y,z) = Epsi(x,y,z)g(m_s). Energies are the same as before we included
spin. Without spin, H-atom degeneracy was n^2, now with spin, degeneracy = 2n^2. Psi(1s)a or Psi(1s)b. Pauli principle - cannot put two
electrons with the same spin - one must be spin up, other must be spin down. In classical mechaincs, all particles are distinguishable,
since you can always follow the trajectory of each particle (e.g billiard balls). In QM we cannot follow individual electron trajectory
due to uncertainty principle. Thus they are indistinguishable. If we exchange two electrons in a system. the probability distribution
should remain unchanged. Define exchange or permutation operator P_12: intnerchange all the coordinates of particle 1 and 2. P_12 W+ is
symmetric, P_12 W- is antisymmetric. He atom, electron 1 in 1s orbital with spin up, electron 2 in 3s orbital with spin down. Psi =
1s(1)a(1)3s(2)b(2) = psi_1s(x,y,z)a(1)psi_3s(x2,y2,z2)b(2) --> P_12 psi = 1s(2)a(2)3s(1)b(1). Pauli principle: for a system of electrons
the wavefunction psi must be antisymmetric with respect to any permutation operator. More generally, fermions have a half integer spin,
and any system of fermions must be antisymmetric. Bosons have integer spin and must have symmetric wavefunctions. There is zero
probability of finding 2 electrons at the same point in 3D space (due to Pauli replusion). This is an additional repulsion on top of
Coulomb repulsion. 2 electrons inn 1D PIB of length l, psi(x1, x2). x1 is at x = 2, x2 is at x = 7. Then interchange location. Psi(2,7)
= -Psi(7,2) due to antisymmetry. If the 2 particles are H-atoms, e+p, p has spin 1/2, e has spin 1/2 so H-atom spin is either 1 or 0.
H-atom is therefore a boson. Deuterium - neutron has a spin of 1/2 so possible spins: 3/2 or 1/2 therefore deuterium is fermion, so
1D PIB wavefcn will change sign if interchanged deuterium atoms. He atom spin possibilities: a(1)a(2), a(1)b(2), b(1)a(2), b(1)b(2).
a(1)a(2) annd b(1)b(2) are ok, other 2 violate indistinguishability. So applying P_12 here, 1 and 4 are symmetric, 2 and 3 are neither
symmetric nor antisymmetric. (1/sqrt(2))[a(1)b(2) +- b(1)a(2)], + sign gives symmetric, - gives antisymmetric. So 3 symmetric, 1 anti-
symmetric. Pauli principle states that the total wavefunction must be antisymmetric with respect to exchange. Ground state of helium:
space part is to put each electron in a 1s orbital. 1s(1)1s(2) symmetric, so multiply by (1/sqrt(2))[a(1)b(2) - b(1)a(2)]. Therefore
the spin part must be antisymmetric.
Guest Lecture: CBE1002
Sustainable Electrochemical Manufacturing - Prof. Modestino. Avoidance - Mitigation - Remediation - Adaptation. Electricity,
manufacturing make up around 60% of emissions. Tech must be created to decarbonize industries. Transportation solutions: batteries
(short term energy storage). Chemical fuels like hydrogen can be used to provide seasonal storage of electricity. Hydrogen is energy
dense by weight but not by volume. 5 kg H2O roughly equivalent to 125,000 kg of dammed water. Combine photovoltaics with electrolyzer
to create solar hydrogen device (14% efficiency). How to transform thermochemical processes into electrochemical processes. Goal of
lab is to make electrochemical processes for simple organic compounds (butadiene, ethylene, toluene, benzene, propylene, etc). Focus
on adiponitrile, precursor to nylon. Eletrohydrodimerization of AN to ABN. High selectivity is key to achieve most of useful product.
Mass transport limitation - can be mitigated by pulsed electrolysis. Data-driven optimization: build regressional model and neural
network to further optimize rate of ADN.