PHY 741 Quantum Mechanics

MWF 9-9:50 AM OPL 107 http://www.wfu.edu/~natalie/f10phy741/

Instructor: Natalie Holzwarth Phone:758-5510Office: 300 OPL e-mail:natalie@wfu.edu

Homework Assignments

Problem Set #1 (8/25/2010)
Problem Set #2 (8/30/2010)
Problem Set #3 (9/01/2010)
Problem Set #4 (9/03/2010)
Problem Set #5 (9/06/2010)
Problem Set #6 (9/08/2010)
Problem Set #7 (9/10/2010)
Problem Set #8 (9/13/2010)
Problem Set #9 (9/17/2010)
Problem Set #10 (9/20/2010)
Problem Set #11 (10/01/2010)
Problem Set #12 (10/04/2010)
Problem Set #13 (10/06/2010)
Problem Set #14 (10/08/2010)
Problem Set #15 (10/11/2010)
Problem Set #16 (10/18/2010)
Problem Set #17 (10/20/2010)
Problem Set #18 (10/22/2010)
Problem Set #19 (10/25/2010)
Problem Set #20 (10/27/2010)
Problem Set #21 (10/29/2010)
Problem Set #22 (11/01/2010)
Problem Set #23 (11/03/2010)
Problem Set #24 (11/05/2010)
Problem Set #25 (11/17/2010)
Problem Set #26 (11/19/2010)
No Title
Aug 25, 2010
PHY 741 - Problem Set #1
Link to PDF version
Read Chapter 1 in Mahan; homework is due Monday, August 30, 2010.
For a system described by the probability amplitude ψ(x), we can define the square modulus of the variance of a Hermitian operator A as
| ∆A |2 ≡ 〈ψ| A2 | ψ〉−(〈ψ| A | ψ〉)2.
In class we showed that for the 3 Hermitian operators A, B, and C with the commutation relations
[A,B] = i C,
the variances satisfy the inequality
∆A ∆B ≥ 1
2
 〈ψ| C | ψ〉.
(1)
For this Homework, choose
A = x,      and       B = p ≡ −i ħ ∂
∂x
 .
  1. What is the operator C for this case?
  2. For each of the following probability amplitudes, evaluate the left and right hand sides of Eq. (1) and check the validity of the inequality.

    1. ψ(x) = 1

      √
      a

      √
      		 [ˉ(2 π)]
      		 e i k0 x − x2/(4 a2).
      In this expression a is a length parameter and k0 is a positive parameter with the dimensions of 1/length.

    2. ψ(x) = ⎧
      ⎪
      ⎪
      ⎨
      ⎪
      ⎪
      ⎩
        ⎛
      √
      630
      a9
       
      		 x2(a−x)2
      for      0 ≤ x ≤ a
      0
      otherwise.
      In this expression a is another length parameter.


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On 25 Aug 2010, 11:30.
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Sep 1, 2010
PHY 741 - Problem Set #2
PDF version
Finish reading Chapter 1 and start Chapter 2 in Mahan; homework is due Wednesday, September 1, 2010.
Find the current density vector J(r,t) for the following probability amplitudes. Note that in cartesian coordinates, the gradient operator can be written:
∇f(r,t) ≡
^
x
 
 ∂f(r,t)
∂x
 +
^
y
 
 ∂f(r,t)
∂y
 +
^
z
 
 ∂f(r,t)
∂z
 .
In spherical polar coordinates, the gradient operator can be written:
∇f(r,t) ≡
^
r
 
 ∂f(r,t)
∂r
 +
^
θ
 
 1
r
 ∂f(r,t)
∂θ
 +
^
ϕ
 
 1
r   sinθ
 ∂f(r,t)
∂ϕ
 .

  1. ψ(r,t) = 1

    √
    64 π
    		 ⎛
    ⎝     		Z
    a
    		⎞
    ⎠     		3/2

     
    		 Z r
    a
    		 e−Zr/(2a) sinθe−i ϕ e+i ωt,
    where Z and a are constants and ω = Z2 e2/(8a ħ).

  2. ψ(r,t) = ⎧
    ⎪
    ⎨
    ⎪
    ⎩
    (A eikx + B e−ikx) e−i ħ k2 t /(2m)
    for         x ≤ 0
    C eiqx e−i ħ k2 t /(2m)
    for         x > 0,
    where k and q are constants as are A, B, and C.


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On 1 Sep 2010, 18:12.
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Aug 31, 2010
PHY 741 - Problem Set #3
PDF version
Continue Chapter 2 in Mahan; homework is due Friday, September 3, 2010.
Consider a particle of mass m moving in one dimension in a finite potential well:
V(x) = ⎧
⎪
⎨
⎪
⎩
−V0
for       −L/2 ≤ x ≤ L/2
0
otherwise.
In terms of the length parameter L, the constant potential has the value
V0 = 16 ħ2
2 m L2
 .
Solve the Schrödinger equation to find at least one (extra credit for two) bound-state eigenfunction and eigenvalue for this particle. If you are using Maple to solve the equations, you might take advantage of "fsolve".


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On 31 Aug 2010, 15:24.

PHY 741 -- Assignment #4

September 3, 2010

Continue reading Chap. 2 in Mahan.

  1. Work problem 2.11
  2. Work problem 2.15
No Title
Sep 4, 2010
PHY 741 - Hint for Problem Set #4
PDF version
In order to solve problem 2.15, the following identities may prove useful. For an operator A, a function of A may be evaluated be using a Taylor's expansion. For example, for any constant s,
es A ≡ 1 + sA
1!
 + s2A2
2!
 + s3A3
3!
 + ….
A famous identity involving two operators A and B can be shown to be equivalent to a series of commutators (see, for example, Merzbacher's text):
es A B e−s A = B + s
1!
 [A,B ] + s2
2!
 [A,[A,B ]]+ s3
3!
 [A,[A,[A,B ]]] + ….


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On 4 Sep 2010, 13:11.
No Title
Sep 4, 2010
PHY 741 - Problem Set #5
PDF version
Continue Chapter 2 in Mahan; homework is due Wednesday, September 8, 2010.
Consider a particle of mass m moving in a one dimensional potential defined by Eq. (2.97) of your text. The electric field F is given by
F=v ħ2
2 m a3
 ,
where in our case, v=8 and a represents the bohr radius. Determine the first 3 eigenvalues En and eigenfunctions ψn(x) of this system. Plot the eigenfunctions for 0 ≤ x/a ≤ 10.


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On 4 Sep 2010, 14:57.
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Sep 7, 2010
PHY 741 - Problem Set #6
PDF version
Finish reading Chapter 2 in Mahan and start reading Chapter 3; homework is due Friday, September 10, 2010.
Consider a particle of mass m moving in a one dimensional potential:
V(x) = ⎧
⎪
⎨
⎪
⎩
V0
for         x ≤ 0
0
for         x ≥ 0,
where V0 > 0.
  1. Find the form (for both x ≤ 0 and x ≥ 0) of the continuous eigenfunction with eigenenergy E > V0.
  2. Calculate the current density J(x,t) and check whether or not it is continuous at x=0.


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On 7 Sep 2010, 13:51.

PHY 741 -- Assignment #7

September 10, 2010

Start reading Chap. 3 in Mahan.

  1. Work problem 3.9
  2. Work problem 3.10

PHY 741 -- Assignment #8

September 13, 2010

Continue reading Chap. 3 in Mahan. Homework is due Friday, September 17, 2010.

  1. Work problem 3.11
  2. Work problem 3.15
No Title
Sep 15, 2010
PHY 741 - Problem Set #9
PDF version
Start reading Chapter 4 in Mahan; homework is due Monday, September 20, 2010.
Consider a particle of mass m moving in a one dimensional potential:
V(x) = ⎧
⎪
⎨
⎪
⎩
− V0 sin(πx/a)
for         0 ≤ x ≤ a
∞
otherwise,
where V0 = 16 [(ħ2)/(2 m a2)].
  1. Write the Schrödinger equation in dimensionless units u=x/a and ϵ = E/(ħ2/(2ma2)) where E denotes the eigenstate energy.
  2. Using one of the numerical methods presented in the Lecture notes of 9/15/2010, estimate the lowest energy eigenvalue ϵ.
  3. Use a second approximation to check your answer. (Perhaps use a different number of grid points or use the variational method for example.)


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On 15 Sep 2010, 22:28.

PHY 741 -- Assignment #10

September 20, 2010

Continue reading Chap. 4 in Mahan. This assignment is due Wed. September 22, 2010.

  1. Construct the 4x4 matrix representations of the Mx, My, and Mz operators in the |jm> basis for j=3/2. Evaluate the matrices for (Mx)2, (My)2, and (Mz)2 and show that their sum is consistent with the expected form the total squared angular momentum operator M2.
No Title
Sep 30, 2010
PHY 741 - Problem Set #11
PDF version
Continue reading Chapter 5 in Mahan; homework is due Monday, October 4, 2010.
Consider an electron of mass M moving in 3 dimensions in a central potential V(r) representing the interaction of that electron with the nucleus and other electrons in an atom, presented by the form for r → 0:
V(r) r → 0
=
 
 − Z e2
r
and by the form for r → ∞:
V(r) r → ∞
=
 
 − q e2
r
 .
Here Z represents the nuclear charge and q is usually 1 or 2 depending on whether the atom is neutral or positively charged. It is convenient to write the Schrödinger equation in terms of the scaled variables:
ρ ≡ r
a
            where a is the Bohr radius        a ≡ ħ2
M e2
 .

ϵ ≡ − E
ERy
            where ERy is the Rydberg constant        ERy ≡ ħ2
2 M a2
 ≡ e2
2a
 .
  1. Show that the radial Schrödinger equation to determine the radial component of the wavefunction R(ρ) can be written
    ⎧
    ⎨
    ⎩     		d2
    d ρ2
    		 + 2
    ρ
    		 d
    d ρ
    		 − l(l+1)
    ρ2
    		 − 2 M a2
    ħ2
    		 V(ρ) − ϵ ⎫
    ⎬
    ⎭     		R(ρ) = 0.
  2. Consider the limit ρ→ ∞, where the dominant terms in the equation simplify to
    ⎧
    ⎨
    ⎩     		d2
    d ρ2
    		 + 2
    ρ
    		 d
    d ρ
    		 + 2 q
    ρ
    		 − ϵ ⎫
    ⎬
    ⎭     		R(ρ) = 0.
    In general, the value of ϵ needs to be determined numerically. Show that, if it were known, the radial wavefunction would take the form:
    R(ρ) = N ρq/√{ϵ}−1 e−√{ϵ} ρ ⎛
    ⎝     		1 +
    ∑
    n=1 
    		 Cn
    ρn
    		⎞
    ⎠     		,
    where N and {Cn} are constant coefficients.
  3. Now consider the limit ρ→ 0, where the dominant terms in the equation simplify to
    ⎧
    ⎨
    ⎩     		d2
    d ρ2
    		 + 2
    ρ
    		 d
    d ρ
    		 − l(l+1)
    ρ2
    		 + 2 Z
    ρ
    		⎫
    ⎬
    ⎭     		R(ρ) = 0.
    Find the leading power series for R(ρ) in this limit.


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On 30 Sep 2010, 09:42.

PHY 741 -- Assignment #12

October 4, 2010

Continue reading Chap. 5 in Mahan. Homework due October 6,2010.

  1. Work problem 5.5

PHY 741 -- Assignment #13

October 6, 2010

Finish reading Chap. 5 in Mahan. Homework due October 8,2010.

  1. Work problem 5.24

PHY 741 -- Assignment #14

October 8, 2010

Start reading Chapter 6 Mahan. Homework due October 11,2010.

  1. Verify the variational analysis of the He atom presented in your text in Sect. 5.7.3, working out several of the details to arrive at the estimate of the ground state energy given in Eq. 5.267.

PHY 741 -- Assignment #15

October 11, 2010

Start reading Chapter 6 Mahan. Homework due October 13,2010.

  1. Work Problem 6.3 in Mahan. Use both the "trick" method and also set up and diagonalize the 8x8 matrix.

PHY 741 -- Assignment #16

October 18, 2010

Finish reading Chapter 6 Mahan. Homework due October 20,2010.

  1. Work Problem 6.11 in Mahan. Compare your results with thos of Section 5.7.3 of your textbook.

PHY 741 -- Assignment #17

October 20, 2010

Finish reading Chapter 6 Mahan. Homework due October 22,2010.

  1. Work Problem 6.21 in Mahan. Note that there are 4 degenerate states that must be used to analyze the first order perturbation energy.

PHY 741 -- Assignment #18

October 22, 2010

Finish reading Chapter 6 Mahan. Homework due October 25, 2010.

  1. Consider a H atom in the state n l m ms = 2 1 1 1/2 in the presence of a uniform magnetic field along the z axis with magnitude B=1000 Gauss. Estimate the magnitude of the expectation values of the 3 contributions given in Eq. 6.297 of your text.

PHY 741 -- Assignment #19

October 25, 2010

Start reading Chapter 7 in Mahan. Homework due October 27, 2010.

  1. Consider a H atom in the states n l m ms with n=3 and l=2. Use degenerate perturbation theory to determine how the 10 degenerate states are split by a uniform magnetic field including the spin-orbit effects as well. Describe both the Zeeman and Paschen-Back limits.

PHY 741 -- Assignment #20

October 27, 2010

Continue reading Chapter 7 in Mahan. Homework due October 29, 2010.

  1. Work problem 7.3 in Mahan.
No Title
Oct 29, 2010
PHY 741 - Problem Set #21
PDF version
Continue reading Chap. 7 in Mahan; homework is due Monday, November 1, 2010.
Consider an electron in the ground state of H:
ϕ1s(r) = 1

√
πa03
 e−r/a0.
At t=0 a perturbing electric field of amplitude E0 (along the z-axis) is gradually turned on and off such that the perturbing Hamiltonian is given by
H1(r,t) = −e E0 r cosθ⎧
⎨
⎩ 		1
τ
√
π
 e−[(t−T)/τ]2 ⎫
⎬
⎭ 		.
Assume that T/τ >> 1.
  1. Find the general expression for the first order probability amplitudes for the electron to be in an excited state nlm for n > 1 and evaluate the expression for at least two excited states.
  2. Using convenient choices of T and τ, plot your results for the squared modulus of the amplitudes as a function of time.


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On 29 Oct 2010, 12:09.

PHY 741 -- Assignment #22

November 1, 2010

Start reading Chapter 10 in Mahan. Homework due November 3rd, 2010.

  1. Consider the same situation as described in Assignment 21 except that now assume that the z directed electric field has a frequency dependence of the form
    E0 cos(ωt)

    How does that effect the results compared to the case that &omega=0? How are your results related to the Fermi Golden Rule?
No Title
Nov 2, 2010
PHY 741 - Problem Set #23
Finish reading Chapter 7 and begin Chapter 10 in Mahan; homework is due Friday, November 5, 2010. PDF version
Consider a proton in an initial pure spin state
Ψ(0) = ⎛
⎜
⎜
⎝
1
0
⎞
⎟
⎟
⎠ 		.
At t=0, the proton enters a magnetic field of the form
B = B1 ( cos(Ωt)
^
x
 
 + sin(Ωt)
^
y
 
 ) + B0
^
z
 
 ,
(1)
where the rotation field strength is small (B1 << B0) compared to the constant field in the ∧z direction. The frequency Ω meets the resonance condition
Ω = Ω0 = −2 μp B0/ħ.
  1. Write a general expression for the proton spin wavefunction Ψ(t).
  2. Evaluate the time dependent expectation values
    1. 〈Ψ(t) | sz | Ψ(t) 〉.
    2. 〈Ψ(t) | sx | Ψ(t) 〉.
    3. 〈Ψ(t) | sy | Ψ(t) 〉.


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On 2 Nov 2010, 17:35.
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Nov 5, 2010
PHY 741 - Problem Set #24
PDF version
Continue reading Chapter 10 in Mahan; homework is due Monday, November 8, 2010.
Consider an electron scattering from a spherically symmetric potential well of the form
V(r) = ⎧
⎪
⎨
⎪
⎩
− ħ2
2ma2
 ν0
for    0 ≤ r ≤ a
0
for    r > a.
,
where ν0 is a positive constant.
  1. Find a general expression for the scattering phase shift δl(k), where k is related to the energy of the system, expressed in convenient units as:
    E = ħ2
    2ma2
    		 (ka)2.
  2. Use maple to plot the total scattering cross section of the system as a function of energy 1 ≤ (ka)2 ≤ 10, including contributions from at least 10 values of l and assuming the values ν0 = 100 and ν0 = 200.
Note: I have found the following procedure useful for defining functions that are derived from derivatives of functions. Assume that you have defined a function f(l,x) and you want to define
g(l,x) = ∂f(l,x)
∂x
 . 
f1:=diff(f(l,x),x);
g:=unapply(f1,l,x);


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On 5 Nov 2010, 12:12.
No Title
Nov 16, 2010
PHY 741 - Problem Set #25
PDF version
Continue reading Chapter 9 in Mahan; homework is due Friday, November 19, 2010.
Consider two identical particles, which have a spherically symmetric and spin-independent interaction, and which scatter in the center of mass frame of reference. In class, we showed that the differential scattering cross section in the center of mass frame takes the form:
d σc
d Ω
 = |fc(k,θ) ±fc(k,π− θ)|2,
where k denotes the magnitude of the wave vector in the center of mass frame and fc(k,θ) denotes the scattering amplitude which is expressed in terms of phase shifts δl(k).
fc(k,θ) = 1
k
 ∞
∑
l=0 
 (2l+1) ei δl(k) sinδl(k) Pl(cosθ).
The ± sign depends upon whether the particles obey Fermi or Bose statistics and on the spin configuration of each case c. Assume that the incident and target particles have uniform populations of all possible spin states and consider the form of the complete differential cross section
d σ
d Ω
 =
∑
c 
 Pc d σc
d Ω
 ,
where Pc denotes the probability of the particular case c.
Find the form of the cross section for each of these cases:
  1. The particles obey Bose statistics and each has spin 0.
  2. The particles obey Bose statistics and each has spin 1.
  3. The particles obey Fermi statistics and each has spin 1/2.
  4. The particles obey Fermi statistics and each has spin 3/2.


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On 16 Nov 2010, 12:28.

PHY 741 -- Assignment #26

November 19, 2010

Continue reading Chapter 9 in Mahan. Homework due November 23, 2010.

Consider a Ti+2 ion with the shell configuration 1s2 2s2 2p6 3s2 3p6 3d2

  1. How many degenerate states consistent with the Pauli exclusion principle are represented by the open shell configuration 3d2?
  2. List the corresponding terms in the notation
    2S+1L
    Note: you may wish to check your result on the NIST website     http://physics.nist.gov/PhysRefData/ASD/levels_form.html. (Type in Ti III into the form.)
  3. For the 1G states, the estimate of the Coulomb energy contribution for the 3d2 electrons can be written in the form
    E=E0 + A R2dd + B R4dd,
    where Rldd denotes the radial integrals defined in class. For one of the 1G states, find expressions for the constants A and B in terms of Gaunt coefficents. Use Maple or other software to evaluate the expressions. (You are welcome to use the Maple example linked on the lecturenote page.) Note: the correct answers are A=4/49 and B=1/441.

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