## PHYS 550 Class Info

TEST 1 will be Tues Oct 2 and will cover Chap 1 through end of Chap 4. Bring your book and a "dumb" calculator (one that can't connect to internet).

TEST 2 will be Thur Nov 15 and will cover Chap 5 through end of Chap 8. Bring your book and a "dumb" calculator (one that can't connect to internet).

FINAL will be Mon Dec 10 from 8:00-10:00 am in the usual class room. The final will be comprehensive. Bring your book and a "dumb" calculator (one that can't connect to internet).

The text book for the class is Quantum Physics 3

^{rd}Ed. by Stephen Gasiorowicz. The publisher's link has some nice supplementary material as well as errata which I've converted to pdf here. 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); printout of relevant pages are OK as long as they only have info from the book.

We will cover most of Chaps 1-10 but I hope to cover some of hte practical aspects

Syllabus

Office hours: Monday 2:30-3:30, Tuesday 10-11 (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.

Class
Notes

Chap
1; Chap
2;
Chap
3;
Chap
4;
Chap
5;
Chap
6;
2D
Notes;
Chap
7; Legendre Polynomials; Associated Legendre Polynomials; Spherical Harmonics;

Chap 8; Spherical Bessel Functions; Radial energies/wave functions; Chap 10; Chap 20;

Homework

HWK 1 (Due Tues Aug 28, 7 problems total): Problems 1-5 are Gasiorowicz Chap 1 probs 4, 6 (do for 100 keV photon AND for 1 eV photon), and Chap 2 probs 1, 5 (assume the lengths given [1 micron, 1 angstrom, and 1 cm] are the standard deviation), 11

Problem 6: An atom of

^{23}Na is at rest in free space and in its first excited state which is 2.102 eV above the ground state. 100 ns later a photon is emitted in the +y direction. (a) Determine the velocity of the

^{23}Na after the photon is emitted. (b) Compare the de Broglie wavelength of the Na to that of the photon. (c) Comment on any relationship there is between the two wavelengths and why that relationship holds.

Problem 7 is numerical: Define k = 15 pi and dk = pi/3. For the range x = -3 to x = 3, PLOT the SQUARE of each of the four functions:

f

_{1}(x) = cos(k x)

f

_{2}(x) = [cos(k x) + cos((k+dk) x) + cos((k-dk) x)]/3

f

_{3}(x) = [cos(k x) + cos((k+dk) x) + cos((k-dk) x) + cos((k+2dk) x) + cos((k-2dk) x)]/5

f

_{4}(x) = [cos(k x) + cos((k+dk) x) + cos((k-dk) x) + cos((k+2dk) x) + cos((k-2dk) x) + cos((k+3dk) x) + cos((k-3dk) x)]/7

(These look complicated but you should see the pattern.) These functions oscillate very fast so you might want to have a small spacing of points in x. What do these functions inform you about an "uncertainty relation" between the spread in k and the spread in x? (This last question is squishy so a qualitative answer will suffice.)

You can check by comparing to the plot below which is the f

_{4}(x) but has k = 11 pi and dk = pi /5.0

HWK 2 (Due Tues Sep 4, 7 problems total):

Problem 1: Gasiorowicz prob 2-16

Problem 2: Gas. probs 2-17,18 (see the integral on pg 3 of Chap2 notes or here and here; in 18 should the > be =?)

Problem 3: Gas. probs 2-19, 20

Problem 4: Gas. prob 2-22

Problem 5: Gas. probs 3-5 and 6. See Eq. (3-20)

Problem 6: Gas. prob 3-9. See Eq. (3-31).

Problem 7 is numerical: See the discussion around Eq. (3-35). An electron is in an infinite square well with walls at x = 0 and x = L = 100 a (where a is the Bohr radius). The wave function (Eq. 3-32 Gasiorowicz) at t=0 has

A

_{n}= C exp(-{[n-30]/4}

^{2}) exp(i (n pi/L) x0) with x0 = L/2 = 50 a.

(a) Give the numerical value of C to 6 significant digits. (b) Plot the probability, P

_{n}, that a measurement of the energy will give the value E

_{n}vs. n.

For the case, A

_{n}= C exp(-{[n-35]/5}

^{2}) exp(i (n pi/L) x0), I found C = 0.39947079 and the plot looks like

HWK 3 (Due Tues Sep 11, 7 problems total):

Problem 1: Gasiorowicz prob 3-10 (there is a typo in this problem; in part (b) the sum given in the book is only over odd integers)

Problem 2: Use the results of the previous problem to compute <H>. You should get an "interesting" result. Can you give a physical reason for why you get this result???

Problem 3: For infinite square well between 0 and a, the wave function at t=0 is equal to psi(x,0) = A x (a - x). (a) Compute A. (b) Compute <H>. (c) Compute the probability to be in the ground state and in the 1st excited state.

Problems 4-6: Gas 3-14 (see Example 3-6), 15 (use Eq. 3-52), 16

Problem 7 is a continuation of the numerical problem from HWK 2. Use the coefficients from HWK2 prob 7 to compute and plot |Psi (x,t)|

^{2}and J(x,t) at times t = 0, T/4, T/2, (3/4)T, and T where T = L/v with v = 30 pi hbar/(L M). M is the mass. Interpret your results.

For the case, c

_{n}= A exp(-{[n-35]/5}

^{2}) exp(i (n pi/L) x0), the plots at t = 0.4T where T = L/v with v = 35 pi hbar/(L M) are:

HWK 4 (Due Tues Sep 18, 7 problems total):

Problem 1: Gas. prob 4-1 (Hint: set the flux in the left region equal to that in the right region and use the fact that the flux must be the same in the two regions for any combination of A and D).

Problem 2: If the potential in the 1st problem is shifted in space (change V(x) to V(x + a)), what are the new S_ij in terms of the original S_ij?

Problem 3: A step potential (Fig 4-1) for an electron (m = 9.11E-31 kg) has a

height of 1.6E-19 J. Use hbar = 1.0545E-34. Plot the energy eigenstate from -10 nm to 10 nm for E = 0.5E-19, 1.0E-19, and 1.5E-19 J. So we all have the same normalization, normalize u(x) so that u(0) = 1 (Hint: divide by T).

Problem 4: For the parameters of Problem 3, plot the reflection probability vs. E from 0 to 5.0E-19 J.

Problem 5: For the potential well (Sec. 4-2), determine the A and B for the cases where R=0. Write the u(x) in the region |x|<a in terms of cos[q (x+a)] and sin[q (x+a)].

Problem 6: Gas. prob 4-4(a) and write down the equations that result from the boundary conditions at x=-a and x=-b (you don't need to write them down for x=c and x=d).

Problem 7: For the wave function defined in Eqs. 4-6 and 4-8, what does the integral of energy eigenstates at energies E_1 and E_2 give? That is solve

integral u_1^*(x) u_2(x) dx = ???

where u_1 is at energy E_1 = (hbar k_1)^2/2M and u_2 is at energy E_2 = (hbar k_2)^2/2M. (I think this is a very tricky problem with a surprising answer. Note that if V_0 = 0, then the answer is

integral u_1

^{*}(x) u_2(x) dx = 2 pi delta(k_1 - k_2)

)

HWK 5 (Due Tues Sep 25, 7 problems total):

Problem 1: For the potential Eq. (4-65), derive the normalized bound state wave function.

Problem 2: For the potential Eq. (4-65), compute the reflection and transmission probability vs. E.

Problem 3: Gas. prob 4.6.

Problem 4: Use Eq. (4-111) to derive Eq. (4.109) (Hint: Plug 4-111 into dH/dy) and Eq. (4.107) (Hint: first show d

^{n}[yF(y)]/dy

^{n}= y d

^{n}F(y)/dy

^{n}+ nd

^{n-1}F(y)/dy

^{n-1}and use d

^{n+1}F(y)/dy

^{n+1}=d

^{n}(dF(y)/dy)/dy

^{n}).

Problem 5: (For some reason Gas. does not write down the eigenstates for the harmonic oscillator. Near the bottom of pg 9 of Chap. 4 notes gives the expression [but mislabeled in the notes as psi

_{n}(x) instead of u

_{n}(x)].) A particle is in the potential of Sec 4-7. At t=0 psi(x) = A x

^{2}u

_{0}(x). (a) Determine A. (b) Write psi(x) = C

_{0}u

_{0}(x) + C

_{1}u

_{1}(x) + ... and determine all the C

_{n}. (Hint: use a recursion relation for Hermite polynomials to figure out what x u

_{0}(x) equals; then multiply that result by x and use the recursion relation again.) (c) Write psi(x,t) for all other t.

Problem 6: A constant force F is added to the harmonic oscillator Hamiltonian so V(x) goes to (1/2) m w

^{2}x

^{2}- F x. (a) Use a change of variables to show that this gives a harmonic oscillator centered at a different position. (b) Find all of the eigenenergies for this new potential. (c) Give the expression for all of the eigenstates.

Problem 7 is numerical: Plot the n=14 harmonic oscillator wave function for the case where the scale factor sqrt(m omega/hbar) = 1. On the same graph plot the two curves F(x) = +/- psi(0) * sqrt(p(0)/p(x)) where p(x) is the classical momentum at position x for the energy of the n=14 state. (Note p(x) is only defined in the classically allowed region.) Why do you expect the wave function to oscillate with an amplitude proportional to 1/sqrt(p(x))? Once you have your code working for n=14 you might try calculations for other n. Unless you're really careful in how you program, your code will give you garbage starting at some large n.

Make sure your graph doesn't have ranges so big the grader can't actually see the plot. Below is my calculation for n=12 (not 14 like I'm asking you). Languages like Mathematica have Hermite polynomials as defined functions. If you don't have access, then you can compute it using recursion. In my program, I computed the 14th Hermite polynomial using Eq. 4-108. My code look liked this for each value of y

herm[0] = 1 ;

herm[1] = 2*y ;

for(j = 1 ; j < nfin ; j++) herm[j+1] = 2*y*herm[j] - 2*j*herm[j-1] ;

psi = coef*herm[nfin]*exp(-y*y/2) ;

HWK 6 (Due Thur Oct 11, 7 problems total):

Problems 1-6 are Gasiorowicz Chap 5: 1, (2 and 3), (4 and 5), 6, (9 and 10), 11 (to 11 add a part (d) where you give an expression for [A,BC] in the style of part (b))

Problem 7 is numerical and shows how

F(x,y) = u

_{1}(x) u

_{1}(y) + u

_{2}(x) u

_{2}(y) + u

_{3}(x) u

_{3}(y) + ...

converges to the Dirac delta function: delta(x - y).

For the infinite square well with a = 1 nm and y/a = 0.4, plot F(x,y) vs. x/a for a sums that stop at n = 10, 20, 40, and 80. (Make sure you have enough x-points to trace out the curve.) There are other ways of doing the sum that converge more smoothly and faster to a delta function; ask me if you're interested. My answer for sums that stop at 10 (red) and 20 (green) are:

The oscillations are an example of Gibbs' phenomenon.

HWK 7 (Due Tues Oct 16, 7 problems total):

Problems 1-4 are Gasiorowicz Chap 5: 12 (show the first 3 terms), (7 and 13), (15 and 16 but set w

_{2}and C to 0), (17 and 18 [for 18 don't solve for <H>])

Problems 5,6 are Gasiorowicz Chap 6: 1, 2

Problem 7 is numerical and is to find the ground state energy to 4 significant digits for a particle in a finite well. Use Eq. 4-54 in the form alpha a = q a tan(q a). Use a = 1 nm, M = 1E-30 kg, hbar = 1.0545E-34 J s, and V

_{0}= b (hbar hbar/[2 M (2 a) (2 a)]) where b = 1/2. (Use exactly these values to the digits shown otherwise you will get a different answer.)

(My algorithm used the fact that alpha a - q a tan(q a) > 0 if E is less than the correct energy and is <0 if E is greater than the correct energy but less than the value that gives q a = pi/2 (see Fig 4-7). So I start with one energy, E1, lower and one energy, E2, higher than the correct energy. You know E > -V

_{0}and you know E < the case q a = pi/2 so choose these for E1 and E2. I then compute the average. I then compute alpha a - q a tan(q a) at this value. If it is > 0 then that E must be less than the correct energy so swap it for E1, else swap it for E2. Every iteration decreases the difference, E2-E1, by a factor of 2.)

Once you've solved this, you can think about how to get the odd eigenstates and the even higher energy states.

You can ask me about other algorithms that converge faster...

e1 = -v0 ;

e2 = (hbar*hbar*0.5/M) *(pi*0.5/a)*(pi*0.5/a) - v0 ;

if(e2 > 0.0) e2 = 0.0 ;

while((e2-e1) > (-e2-e1)*1.e-8)

{

etrial = 0.5*(e1 + e2) ;

alpha = sqrt(-2.0*M*etrial)/hbar ;

q = sqrt(2.0*M*(etrial+v0))/hbar ;

ftrial = alpha*a - q*a*tan(q*a) ;

print out the etrial and ftrial to make sure ftrial is going to 0

if(ftrial > 0.0) e1 = etrial ;

else e2 = etrial ;

}

To test your code, I give E for other values of b:

b = 0.1 gives E = -3.363340E-24 J

b = 0.2 gives E = -1.303859E-23 J

b = 0.3 gives E = -2.846671E-23 J

HWK 8 (Due Tues Oct 23, 7 problems total):

Problems 1-2 are Gasiorowicz Chap 6: 4, (5 and 6)

Problem 3: Use the result of Gas. (5 and 6) to write the matrix for the p and x operators. Solve for the matrix product p x and the matrix product x p. Solve for the matrix that results from the difference of the matrix products p x - x p.

Problems 4-7: Gas. (10 and 11), 12, 13, 16

HWK 9 (Due Tues Oct 30, 7 problems total):

Problems 1-3 are Gasiorowicz Chap 7: (8 and 9), 3, 5

Problem 4: For the 2D Harmonic oscillator with potential

V(rho) = (1/2) M omega

^{2}rho

^{2}

(a) Show that separation of variables in cartesian coordinates turns this into two one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. (b) Determine the degeneracy d(n) of E

_{n}.

Problem 5: Use the results from Prob 4. (a) Is the ground state also an eigenstate of L

_{z}? (b) Are the two eigenstates with an energy increased by hbar omega from the ground state, eigenstates of L

_{z}? (c) Are the three eigenstates with an energy increased by 2 hbar omega from the ground states, eigenstates of L

_{z}? Show work. (Hint: Just do it: using the states in terms of x,y and the differential form for L

_{z}OR do the commutator of L

_{z}with the raising and lowering operators.)

Problem 6: (a) Define A

_{x}, A

_{y}to be the lowering operators in the x,y coordinates and A

_{+}= [A

_{x}- i A

_{y}]/sqrt(2) and A

_{ -}= [A

_{x}+ i A

_{y}]/sqrt(2). (a) Determine the hermitian conjugate of these two operators. (b) Determine the Hamiltonian in terms of these two operators and their Hermitian conjugates. (c) Determine all of the commutators of these operators with the Hermitian conjugates (plus with plus, minus with minus, and plus with minus). (d) Starting with the ground state use these raising operators to find all eigenstates. Be clear with your logic. (e) Are these eigenstates also eigenstates of L

_{z}? (f) If yes, determine m in terms of n

_{+}and n

_{ -}.

Problem 7 is numerical: The 2D Schrodinger equation for V=0 can be written as

-d

^{2}F(s)/ds

^{2}- (1/s)dF(s)/ds + (m

^{2}/s

^{2}) F(s) = F(s)

where s = k rho. Determine what k is in terms of E, hbar, M, etc. For m=0, write F(s) as a power series in s. Give the recursion relation that lets you get c

_{n}from c

_{n-1}and c

_{n-2}. Give the coefficient of each power as a column of numbers starting with c

_{0}= 1 up to n=20 using 6 significant digits. A plot of F(s) is below which might help you check your coefficients.

HWK 10 (Due Tues Nov 6, 7 problems total):

Problems 1-3 are Gasiorowicz Chap 7: 2 (hint: use x/r, y/r, z/r), 11 (hint: use results from previous problem), 12 (hint: what is the minimum angle you need to rotate before you get to the same place [it isn't 2 pi])

Problem 4: (a) For the potential V(r) = 0 for r<a and infinity otherwise, find the eigen-energies and states in terms of spherical Bessel functions and their zeros. (b) For this problem find the ordering of the first 10 energy levels in terms of n

_{r}and L. (Hint: this needs to be done numerically [follow the link to a Bessel function zero calculator]; the first three are (n

_{r},L) = (0,0), (0,1), (0,2))

Problem 5: Due to their spin, you can put 2 neutrons in each spatial eigenstate. At temperature about equal to 0, the neutrons must fill the energy levels from lowest energy to highest. How many neutrons are needed to fill the 10 levels in the preceeding problem? (Hint: how many spatial eigenstates make up the level (n

_{r},L) = (0,2))

Problem 6: For the isotropic harmonic oscillator, use the results on pgs 2 and 3 of the Chap 8 notes to get the radius where the probability density |psi(r,theta,phi)|

^{2}is maximum for the case where there is no radial nodes as a function of L. How does this compare to the minimum of V

_{eff}(r)?

Problem 7: This is a numerical problem. Use the results from pg 3 of the Chap 8 notes to plot u(y) = h(y) exp(-y

^{2}/2) for 0<y<4 for L= 0 and the eigenstate epsilon/2 = 3/2. Take a

_{1}= 1. Repeat for epsilon/2 = 3/2 + 0.005 and for epsilon/2 = 3/2 - 0.005 (a common mistake is not to include enough terms in the series for the a

_{n}, depending on the difference from epsilon/2 = 3/2, you might need n up to 60 or more!). Repeat for epsilon/2 = 3/2 + 0.0025 and epsilon/2 = 3/2 - 0.0025. You can put all plots on one graph. Comment on the trends and why they occur. (Below is epsilon/2 = 3/2 and epsilon/2 = 3/2 + 0.01)

HWK 11 (Due Tues Nov 13, 7 problems total):

Problems 1-4: Gasiorowicz Chap 8 probs 9, 12, (6 and 10), 7 (and calculate the probability it ends in the 2p state)

Problem 5-6: Gas. Chap 10 probs (6 and 9), 10

Problem 7: For Gas. Chap 10 prob 10 take vec(a) = (0,a,0) with a > 0 (that is in the y-direction). Taking the operator Q(a) = exp(i a sigma_y). (i) Is Q hermitian? Is it unitary? (ii) Compute Q(a) sigma_j Q

^{dagger}(a) for j = x, y, z. Interpret your results. (iii) If I have a state |psi> that is an eigenstate of sigma_z, what operator is the state Q(a)|psi> an eigenstate of?

HWK 12 (Due Tues Nov 27, 7 problems total):

Problems 1: Gas 10-1 and find the eigenvectors of S

_{x}

Problem 2: Find the 2X2 matrix that is the projection of the eigenstates of the S

_{z}operator (bra) on the eigenstates of the S

_{y}operator (kets). The ordering of states is +hbar/2 then -hbar/2 for both. What kind of matrix is it (hermitian, unitary, or neither)? Show which one. (Hint: it is not the unit matrix.)

Problem 3: Find the 3X3 spin matrices (S

_{x}, S

_{y}, and S

_{z}) for a spin 1 particle in terms of the eigenstates of the S

_{z}operator. Order the states -hbar, 0 hbar, hbar. (Hint: find the matrix for the raising or lowering operator and get the other from hermitian conjugate. The S

_{x}(S

_{y}) is the sum (difference).)

Problem 4: In class, I said that the quadrupole operator 2S

_{z}S

_{z}- S

_{x}S

_{x}- S

_{y}S

_{y}is 0 for spin 1/2 but not for spin 1. Thus, spin 1/2 particles can't have a quadrupole momentum but a spin 1 can. Give the 2X2 matrix representation of this operator for spin 1/2 and the 3X3 matrix representation for spin 1. Was I right?

Problem 5: The H atom in a weak electric field is hardly changed in its ground state but the n=2 states are strongly changed. Show that the 4 n=2 states can be written as two separate 1X1 (m=1 and m=-1) and one 2X2 matrix (m=0). The potential energy for the external E-field part of the Hamiltonian is e E z. Give the Hamiltonian of the n=2, m=0 states using the Pauli spinors and the unit matrix (don't forget the Hamiltonian when the E-field is 0). Find the eigenenergies and eigenvalues. (Eqs 11-42 through 47 will let you short cut some of the integrations.)

Problem 6: See online notes for Chap 10 pgs 4 and 5. If the system starts in the down state at t=0, plot the probability to be in the up state as a function of time for a final time 6 pi/Omega. Do the plot for delta omega = 0, 0.25, 0.5, 1, 2 times Omega.

Problem 7: On pg 8 of the Chap 10 notes, I give C

_{n}(infinity). Find the expression for C

_{n}(t). Take omega_0 to be 0 and plot |C

_{n}(t)|

^{2}/(delta omega) vs omega_n for Gamma = 10

^{7}s

^{-1}and delta omega = Gamma/20. Do plots at the times t =1/Gamma, 2/Gamma, 4/Gamma, and infinity. Have the x-range of your plot go from -10 Gamma to 10 Gamma. Below is the plot for t = 0.75/Gamma from -20 Gamma to 20 Gamma to show some of the interesting features. Give a physical explanation for how the plots are evolving with time.

HWK 13 (Due Tues Dec 4, 7 problems total):

Problem 1: At the top of pg 3 of the Chap 10 notes, I give a spin up wave function in terms of the angles alpha, beta and gamma. Determine the sum <S

_{x}>

^{2}+ <S

_{y}>

^{2}+ <S

_{z}>

^{2}divided by [hbar

^{2}/4]. (This is the sum of the squares of the expectation values of the Pauli spinors.)

Problem 2: For the entangled wave function C_1|1>|k>+C_2|2>|k'> (middle of pg 6 Chap 20 notes), repeat Prob 1 assuming <k|k>=<k'|k'>=1 and <k|k'> = 0 using C_1 = cos(alpha/2) and C_2 = sin(alpha/2) e

^{i beta}.

Problem 3: Repeat Prob 2 assuming <k|k>=<k'|k'>=1 and <k|k'> = sin(theta). There isn't a good definition of entanglement in general. Propose a measure of the entanglement as a function of alpha and/or beta and/or theta for this case.

Problem 4: A wavefunction psi(x,y) is not entangled in x,y. For two operators, Q

_{x}(only acts on the x coordinate but can be any combination of x and p

_{x}) and Q

_{y}(only acts on the y coordinate but can be any combination of y and p

_{y}) compare <Q

_{x }Q

_{y}> to <Q

_{x}><Q

_{y}>. Comment on whether the system is correlated in x,y.

Problem 5: A 1D harmonic oscillator at t=0 is in the state (u

_{0}(x)+u

_{1}(x))/sqrt(2). (a) Write the wave function for times t>0. (b) Compute <x>(t). (Hint: using raising and lowering operators makes the matrix elements simple.)

Problem 6: An isotropic 2D harmonic oscillator at t=0 is in the state (u

_{0}(x)u

_{0}(y)+u

_{1}(x)u

_{1}(y))/sqrt(2). (a) Write the wave function for times t>0. (b) Compute <x>(t). Comment on why the answer to this problem is qualitatively different from the previous problem.

Problem 7: See pg 1 of Chap 20 notes. A

^{85}Rb atom has a kinetic energy equivalent to 100 microKelvin (this is easily achieved by laser cooling). (a) What is the classical speed of this atom? (b) What is the de Broglie wavelength? (c) Suppose it goes through two holes which have a tiny radius and a separation of 100 nm in the y-direction. A screen is 10 micrometers from the holes. Taking z = 0, plot the interference pattern as a function of y (take y=0,z=0 to be between the two holes). Normalize the plot so that the value at the center is 2. Below is for the case with the screen 20 micrometers from the holes.

Tests/Solutions