## PHYS 461 Class Info

TEST 1 will be Tues. Feb 20 and will cover Chap 6 through Chap 7. Bring your book and a calculator.

TEST 2 will be Thurs. Apr 12 and will cover Chap 8 through Chap 9. Bring your book and a calculator.

FINAL will be Tues May 1 from 8-10 am. The final will be comprehensive.

The text book for the class is Introduction to Quantum Mechanics 2nd Edition by David J. Griffiths. The tests are open textbook so you must have access to a physical copy (not electronic because you can't have open laptop during test).

Errata from Griffiths web page: Errata 1 ; Errata 2; Errata 3

A good, free online quantum book by Paul Berman is here.

We will cover most of Part 2 of Griffiths (Chapters 6-12). Course description.

Syllabus

Office hours: Monday 4-5, Tuesday 10:30-11:30 (Contact me by email if you want to arrange a special time.)

Homework will be due by midnight on Tuesdays. You must show enough work on the homework so the grader can understand what you're doing.

Notes, old tests, homework, etc for PHYS 460 is here.

Class
Notes

Chap 6, H-energies(c++), H-energies(values), Clebsch-Gordon notes; Chap 7, Linear Potential, Chap 8, Chap 9, Chap 11

Homework

HWK 1 (Due Tues Jan 16, [Orals this week] 7 problems total): Problems 1-4 are Griffiths Chap 6 probs 1(a only), 3, 4(a only), 32(a only)

Problem 5: A charged particle, q, of mass M is confined to a ring of radius R in the xy plane. A uniform electric field of strength E is in the x-direction. Obtain the ground state energy including the corrections of order E

^{1}and of order E

^{2}.

Problem 6: Using the potential in Fig. 6.3 of Griffiths, obtain the 2nd order correction to the ground state energy. I don't know how to do the resulting sum analytically. Do the sum numerically to better than 0.1%; it converges quickly.

Problem 7: A particle is in the ground state of the potential V(x) = -alpha delta(x) (with delta(x) the Dirac delta function). Define L = hbar

^{2}/(M alpha). The particle is perturbed by an extra potential U(x) = U

_{0}exp(-x

^{2}/w

^{2}) where you can think of U

_{0}as the strength of the potential and w as the spatial width. (a 2 pts) Obtain the expression for the first order change in terms of an integral times U

_{0}(Caution: don't forget that the wave function has |x| NOT x.). You don't need to get the analytic form of the integral, but, for the curious, the integral gives a complementary error function times other factors. (b 6 pts) Give numerical values for the integral for the cases L = 2 nm and w = 1/4, 1/2, 1, 2, 4, 8, 16 nm to 3 significant digits. (To test your code L = 3 nm, w = 4 nm gives 8.298267E-01). (c 2 pts) Figure out the limits L << w and L >> w.

HWK 2 (Due Tues Jan 23, 7 problems total): Problems 1-4 are Griffiths Chap 6 probs 6.5(a only), 6.6, 6.7, 6.8

Problem 5: A 1-D harmonic oscillator is perturbed by the potential C x

^{4}. Find the first order shift in the n=0 energy and the n=1 energy by doing the integral with the wave function, Eq. [2.85].

Problem 6: Find this shift for any n using the expression for x

^{2}in terms of the raising and lowering operators just below Eq. [2.69]. (Hint: first find x

^{2}psi

_{n}in terms of a sum of psi

_{n-2}, psi

_{n}, psi

_{n+2}and then use <psi

_{n}|x

^{4}|psi

_{n}> = <x

^{2}psi

_{n}|x

^{2}psi

_{n}>.) You might want to check that your answers for this and the previous problem match.

Problem 7: The isotropic 2D harmonic oscillator (the x and y directions have the same frequency) is perturbed by the potential V(x,y) = C x y. Find the first order shift for the ground state at energy hbar omega and for the two states at energy 2 hbar omega.

HWK 3 (Due Tues Jan 30, [Orals this week] 7 problems total): Problems 1-4 are Griffiths Chap 6 probs 6.9, 6.16, 6.19, 6.20

Problem 5: Treating the proton as a uniform sphere of charge, find the shift of the Hydrogen 1s, 2s, and 2p energies. Comment on why the shifts have their relative sizes. (Hint: A uniform sphere of charge Q has an electric potential V(r) = kQ/r for r>R and V(r) = kQ(3 - r

^{2}/R

^{2})/(2R) for r<R.)

Problem 6: Numerically compute the 1S through 5S energies using the relativistic correction (Eq. 6.67, but you need very accurate values for the fundamental constants: 13.6 eV is not accurate enough) and compare to the experimental energies (in cm

^{-1}, the experimental energies are 0.0000000000, 82258.9543992821, 97492.221701, 102823.8530211, 105291.63094 with a threshold 109678.77174307). You might look at the code above for how to convert these values and the errors for 1s, 2s, 3s to make sure your program is correct. Plot the errors vs. n. See if you can find a formula like C/n

^{p}that predicts the error vs n; if there is, give the constant and the power, p.

Problem 7: An object of mass M and charge q experiences an isotropic 2D potential (1/2) M omega

^{2}(x

^{2}+ y

^{2}) AND a uniform magnetic field B in the z direction. (You might look at the first page in Sec. 6.4 but ignore the spin.) (a) What is the condition that the magnetic field is a perturbation? (b) For magnetic fields in the perturbative limit, obtain the energy shift for the ground state at energy hbar omega and for the two states at energy 2 hbar omega. (Hint: there are several different ways to solve this problem.) (c) Also give the correct wave functions from degenerate perturbation theory.

HWK 4 (Due Tues Feb 6, [Orals this week] 7 problems total): Problems 1-4 are Griffiths: 6.14, 7.1(b only), 7.3, 7.5

Problem 5: An isotropic 2D harmonic oscillator is perturbed by the Hamiltonian

H

^{(1)}= hbar w

_{1}(a

_{+,x}a

_{ -,y}+ a

_{+,y}a

_{ -,x})

Find the energies and eigenstates for the ground state, the two excited states at energy 2 hbar omega, and the three excited states at energy 3 hbar omega.

Problem 6: In the notes, I did an infinite square well with a trial wave function proportional to x (a - x). The main problem with this wave function is that the 2nd derivative does not go to 0 at x=0 and x=a. Use the trial wave function

psi

_{tr}= A (x - (1/3) (2/a)

^{2}x

^{3}) for 0<x<a/2

= A ([a-x] - (1/3) (2/a)

^{2}[a-x]

^{3}) for a/2<x<0

(a 2 pts) Show the trial wave function is continuous and 2nd derivative is continuous at x=a/2. Show the 1st derivative is 0 at x=a/2 and the 3rd derivative is not continuous. (b 6 pts) Find the variational ground state energy (Hint: since the function is symmetric, you only need to do integrals in the region from 0 to a/2.) I found the variational energy was approximately 0.13% too high. (c 2 pts) Plot the error in this trial wave function and the one from class to show you expect this one to be more accurate.

Problem 7: Use the trial wave function

psi

_{tr}= A (L

^{2}- x

^{2}) for |x| < L

= 0 otherwise

to get a variational estimate of the harmonic oscillator ground state energy as a function of L. Find the best L and evaluate the ground state energy for that value.

HWK 5 (Due Tues Feb 13, 7 problems total): Problems 1-3 are Griffiths: 7.7, 7.13, 7.14 (Hint: Use the trial function sqrt(Z

^{3}/[pi a

^{3}]) exp(-Z r/a). If you're careful, you can use a lot of the integrals in Eqs 7.27-7.30.)

Problem 4: Repeat 7.14 but using 1st order perturbation theory. Compare the answer using the two methods. Which one is more accurate? Explain why.

Problem 5: For the potential from pg 4 of the notes (V = F x for x>0 and infinity otherwise), get the best ground state energy with a trial wave function of the form A x exp(-b x

^{2}). Make sure to find the best value for b. Is your answer more accurate than the one from the notes? Explain how you know.

Problem 6: A molecule is composed of 6 identical units in a ring. The spacing between the units is the same. If you add one electron to make an ion, the electron can be treated as being on each of the units giving 6 basis functions. Find the form for the 6 energies and wavefunctions using the ideas from pages 8-10 of the notes.

Problem 7: Repeat Griffiths 7.1(b) but use the trial wave function

psi

_{tr}= A (L

^{2}- x

^{2}) for |x| < L

= 0 otherwise

to get a variational estimate of the ground state energy as a function of L. Find the best L and evaluate the ground state energy for that value. Compare to the answer you got for 7.1(b) (3/4)

^{4/3}(hbar

^{4/3}alpha

^{1/3}/m

^{2/3}) and explain this trial function is better (or worse) than the Gaussian.

HWK 6 (Due Tues Feb 27, [Orals this week] 7 problems total): Problems 1-5 are Griffiths: 8.1, 8.2, 8.3, 8.16

Problem 5: For a potential V(x) = F x for x > 0 and infinity for x<0, get the exact wave function in terms of Airy functions. What is the boundary condition that gives the eigenenergies? Write the energies in terms of the zeros of the appropriate Airy function.

Problem 6: Repeat problem 5 but using the WKB approximation for one wall, Eq. [8.47]. Compute the fractional error in the first 5 energies.

Problem 7: In the classically allowed range, plot the n=20 harmonic oscillator wave function using units where hbar, m and omega are all 1. On the same graph, plot the WKB approximation in the classical region (Use Eq [8.13] with C

_{1}= 0 and C

_{2}= psi(0)*sqrt(p(0)). (Hint: See HWK 4 of the PHYS460 page for calculating the n=20 state also for roughly what the plot of the wave function should look like.) The plot below is for n=10.

HWK 7 (Due Tues Mar 6, 7 problems total): Problems 1-4 are Griffiths 8.5, 8.6, 8.7, 8.14

Problem 5: A particle of mass M has a potential energy V(x) = (1/2) M w

^{2}x

^{2}for x<0 and 0 for x> 0. In the region x>0, the wave function has the form

psi(x) = A sin(k x + delta)

where delta is the phase shift. Get the WKB approximation of the phase shift as a function of energy.

Problem 6: A particle of mass M starts at negative x with positive velocity. It interacts with a potential which is V(x) = V

_{0}(1 - x

^{2}/L

^{2}) for |x| < L and 0 otherwise. For 0<E<V

_{0}obtain the transmission probability vs E.

Problem 7: A potential is positive infinite for x<0, is equal to V

_{0}from x = a to x = b, and is 0 everywhere else. In the region, 0<x<a the wave function is written as

psi(x) = A sin(k x)

In the region a<x<b, the wave function is written as

psi(x) = B exp(K x) + C exp(-K x)

and in the region b<x, the wave function is written as

psi(x) = D sin(k x) + F cos(k x)

(a) Give the formula for k and K. (b) Get the formula for B and C vs A, k, K, a. (c) Get the formula for D and F vs B, C, k, K, b. DO NOT BOTHER GETTING D AND F vs A, k, K, a, b, etc (it's super complicated and I'm not interested.) (d) Plot A/sqrt(D

^{2}+ F

^{2}) vs. E for E < V

_{0}. You should write E = E

_{sc}hbar

^{2}pi

^{2}/(2 M a

^{2}) and plot vs E

_{sc}. between 0 and 1. Choose the following parameters: V

_{0}= hbar

^{2}pi

^{2}9/(2 M a

^{2}) and b = 1.5 a. There should be a narrow feature between 0.80 and 0.82. You will need a lot of points to resolve the feature. (Hint: You can check your results on the following values: when V

_{0}= hbar

^{2}pi

^{2}16/(2 M a

^{2}) and b = 2.0 a, the feature is at E

_{sc}= 0.85685.)

HWK 8 (Due Tues Mar 20, [Orals this week] 7 problems total):

Prob 1: Griffiths 8.14 but use Langer correction replace L(L+1)

Prob 2: Find the WKB eigenvalues for the isotropic, 3D harmonic oscillator PE = (1/2) M w

^{2}r

^{2}using the Langer correction. (Hint: in the integral you might try a change of variables r

^{2}= (hbar/[M w]) s and look at Eq. [8.56].)

Prob 3: In the notes, I used r = exp(s) as the transformation of variables. Derive the "Robicheaux correction" using r = s

^{2}as the transformation of variables. Does it give as simple a correction as Langer? Does it give the correct correction?

Prob 4: Griffiths Problem 1.16 but for a time dependent Hamiltonian.

Prob 5: For an infinite square well, you have a wave function which starts as a superposition of two states Psi(t=0) = [psi

_{n}+ psi

_{n+1}]/sqrt(2). What is the smallest nonzero time T>0 that ALL observables are the same as at T=0? Relate this time to a physical process or parameter.

Prob 6: For the Rabi model (starting bottom of pg 2 of the notes), the system starts 100% in state 1. What is the time average of probability to be in state 2 as a function of time? Plot the probability to be in state 2 for a reasonable amount of time when Omega = 2 pi 1 GHz and delta = 2 pi F GHz with F = 0, 1/2, 1, 2, 3.

Prob 7: For the Rabi model, the system starts 100% in state 1. Compute the expectation value of sigma_x, sigma_y, and sigma_z (the Pauli spinors). Do the case for 0 detuning.

HWK 9 (Due Tues Mar 27, 7 problems total): Probs 1-4: Griffiths 9.1, 9.3, 9.7, 9.12(Hint: I found it easier to use [L

^{2},z] = i hbar (L

_{y}x + x L

_{y}- L

_{x}y - y L

_{x})

Prob 5: Solve 9.3 using first order time dependent perturbation theory. Give the range of interaction strength where you would expect perturbation theory to be accurate and compare to the exact answer.

Prob 6: A particle of mass M is in an infinite square well where V=0 for 0<x<L. The particle starts in the state n at t = -infinity. There is a time dependent perturbation which is V(x) = alpha delta(x - L/2) for the times -T<t<T. What are the units of alpha? Compute the probabilities to be in states n' not equal to n at t = infinity. Are there initial states, n, where the probability to be in a different state is exactly 0 for all n'? Explain why/why not.

Prob 7: The same as Prob 6 except the perturbation is

V(x) = alpha delta(x - L/2) exp(-t

^{2}/T

^{2}) for -infinity < t < infinity. Why does the probability for making a transition go to 0 as T gets larger for Prob 7, but not for Prob 6? (Hint: you might read the first paragraph of Chap 10.)

HWK 10 (Due Tues Apr 3, [Orals this week] 7 problems total): Probs 1-2: Griffiths 9.10, 9.13

Prob 3: A laser is linearly polarized in the z-direction. The frequency is chosen to be nearly resonant with the n=1 to n=2 transition in hydrogen. Which of the four n=2 states can be excited? What is the wavelength of the light accurate to 4 digits? The electric field has the form E(t) = E

_{0}cos(w t) exp(-t

^{2}/T

^{2}) where T = 10 ns. What value of E

_{0}gives 1% transition probability? Plot the transition probability vs. laser detuning in frequency units (for example MHz).

Prob 4: A particle of mass M is in a harmonic oscillator potential with PE = (1/2) M w

^{2}x

^{2}. At t<0, the particle is in the ground state. A perturbation of the form V(x) = C x is on from 0 to T. Obtain the wavefunction at time T to second order in C.

Prob 5: See pg 11 of the notes. Plot the P(E) for finite times (t = 0.1/Gamma, 0.2/Gamma, 0.5/Gamma, 1/Gamma, 2/Gamma, 5/Gamma) and for the final time equal infinity. Take hbar Gamma = 1 and E

_{r}= 0. You should find the integral of the probability to be smaller at earlier time. Explain. You should find that the energy width is larger for earlier time. Explain. See below for the plot at t = 0.05/Gamma.

Prob 6: The system consists of a large Harmonic oscillator (coordinate x) with PE = (1/2) M w

^{2}x

^{2}and an infinite number of small oscillators with mass mu and the j-th oscillator (j goes from 1 to infinity) has angular frequency w

_{j}= j Delta w where the Delta w is much, much, much smaller than w. There is an additional PE that couples the large oscillator with each of the small oscillators and is PE = C sqrt(M mu) w

^{3/2}sqrt(Delta w) x x

_{j}where C is a dimensionless constant. What is the decay rate when the large Harmonic oscillator starts in the n=1 state and all of the small oscillators start in their ground states?

Prob 7: Repeat Prob 6 but start the large Harmonic oscillator in the state n. Is it possible to write an equation for the average energy in the large Harmonic oscillator as d<E>/dt = -Gamma <E - (hbar w/2)>? How is this related to friction? (This is basically the Caldeira-Leggett model.)

Plot for Prob 5

HWK 11 (Due Tues Apr 17, [Orals this week] 7 problems total): Probs 1-4: Griffiths 11.2, 11.3 (Hint: is Eq. 11.33 correct?), 11.4, 11.6

Prob 5: Classically, cos(theta) = -1 for b=0 and monotonically increases to 1 at b = b

_{0}and stays 1 for b > b

_{0}. Compute the total cross section using Eqs. [11.4] and [11.7] for this generic situation. A specific example might be cos(theta) = -1 + 2 b/b

_{0}for b < b

_{0}and 1 otherwise. Plot the d sigma/d Omega vs cos(theta) for this example.

Prob 6: Classically, cos(theta) = 1 for b=0, decreases to the value cos(theta)

_{m}at b=b

_{m}, and then increases back to 1 at b

_{0}and stays 1 for b>b

_{0}. Compute the total cross section using Eqs. [11.4] and [11.7] for this generic situation. A specific example might be cos(theta) = 4 (b - b

_{0}/2)

^{2}/b

_{0}

^{2}. Plot the d sigma/d Omega vs cos(theta) for this example. (HINT!!!: you need to consider how to treat the fact that two different impact parameters lead to the same final cos(theta).)

Prob 7: The phase shift for a potential as a function of L is delta_L = 5 exp(-L). Give the value for the total cross section in units of 1/k

^{2}and plot the differential cross section vs. cos(theta) in units of 1/k

^{2}. (Hint: If you don't have access to a Legendre polynomial function use P

_{0}(x) = 1 and P

_{1}(x) = x and P

_{n+1}(x) = [(2 n +1) x P

_{n}(x) - n P

_{n-1}(x)]/(n+1).) The plot below is for delta_L = 7 exp(-L) which has a total cross section of 70.17/k

^{2}.

HWK 12 (Due Tues Apr 24, 7 problems total): Probs 1-4: Griffiths 11.8, 11.10, 11.17, 11.19

Prob 5: Show that the second derivative of |x| is proportional to the delta function and get the proportionality factor. Defining G(x) = exp( i k |x|), use this result to show that [d

^{2}G(x)/dx

^{2}+ k

^{2}G] is proportional to the delta function and get the proportionality factor. Use this last result to show Eq. 11.102.

Prob 6: In 1D, a potential is given by V(x) = V

_{0}cos(pi x/a) exp(-x

^{2}/L

^{2}) where V

_{0}, a, and L are constants with a << L. Find the reflection probability as a function of k when V

_{0}is small. Explain why the reflection is peaked. (Hint: you should do the book problems first.)

Prob 7: Suppose the potential is the sum of two potentials: V(vec{r}) + V(vec{r}-vec{a}). At the 1st Born level, show that the resulting differential cross section is the product of the differential cross section for one potential and an interference term. Generalize to the case where there are N identical potentials at N different places.

Tests/Solutions

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