1. science
2. physics

1
What is modern physics?
2
Write a sentence that seems obviously true to a
classical physicist, but which is not completely
accurate due to quantum mechanics.
3
Given the bizarreness of quantum mechanics,
why would anyone believe it to be true?
4
What is a black body?
5
What is blackbody radiation?
6
What is the ultraviolet catastrophe?
7
Equation (1.1) in the text is P = σAT 4 .
Does it matter which units we wish to use for P?
for A?
for T ?
Why or why not?
8
On page 4, Scherrer refers to “the total energy
density ρ of radiation inside a blackbody cavity”.
In figure 1.2, Scherrer plots “the energy density
ρ(ν) of blackbody radiation as a function of
frequency ν at temperatures of T = 1000 K,
1500 K, and 2000 K”.
Do ρ and ρ(ν) mean the same thing?
Explain your answer.
9
Evaluate the denominator of equation (1.6), i.e.
Z ∞
P(E)dE
0
by using
e−E/kT
P(E) =
kT
Interpret your result.
10
Planck suggested a change to the classical
theory mean energy:
R∞
e−E/kT
0 P(E)EdE
R
E classical = ∞
,
P(E) =
kT
0 P(E)dE
Write down the expression for E Planck .
Evaluate E classical and E Planck .
In what limit does E classical → E Planck ?
11
What is a photon?
12
The photoelectric effect is described by
Emax = hν − EB
Carefully define each quantity in that equation.
13
Derive the Compton scattering formula,
λf − λi = λC (1 − cos θ),
and thereby determine the parameter λC in
terms of fundamental constants.
14
An isolated pion of mass m can spontaneously
decay into a pair of photons.
Calculate the momentum of each photon.
15
Can an isolated electron emit a photon?
Why or why not?
16
What is an electron?
17
Compare/contrast the Compton wavelength and
the de Broglie wavelength.
18
Calculate your de Broglie wavelength while
walking.
19
In Bohr’s model of an atom, is the electron
relativistic?
20
Is a virus a particle or a wave?
Is an insulin molecule a particle or a wave?
Reference:
www.nature.com/news/2011/110405/full/news.2011.210.html
21
True or false:
√
−1 = i.
Explain your answer.
22
True or false: Re(zw) = Re(z)Re(w).
Explain your answer.
23
Evaluate the following and express your result in
Cartesian form.
2 + 3i
4−i
24
Evaluate the following and express your result in
polar form.
2 + 3i
4−i
25
Evaluate the product of the following three
complex numbers.
z1 = 2eπi/2 ,
z2 = 3e−πi/6 ,
z3 = 4eπi/4 .
26
Evaluate
e7πi/4 − e−πi/4 .
Interpret your result.
27
Write down the complex conjugate of
(2 + 3i)4 eiπ
(1 − i)3
28
Is the operator defined by
C[f (x)] = f ∗ (x)
a linear operator?
Explain your answer.
If it is linear,
provide eigenfunctions and eigenvalues.
29
Is the operator defined by
A[f (x)] = |f (x)|
a linear operator?
Explain your answer.
If it is linear,
provide eigenfunctions and eigenvalues.
30
Is the operator defined by
D[f (x)] =
df (x)
dx
a linear operator?
Explain your answer.
If it is linear,
provide eigenfunctions and eigenvalues.
31
Is the operator defined by
Π[f (x)] = f (−x)
a linear operator?
Explain your answer.
If it is linear,
provide eigenfunctions and eigenvalues.
32
Our textbook says that a linear operator must
satisfy two properties.
(a) Write down those two properties.
(b) Can the two be replaced by a single
property?
(c) Why do you suppose the name is “linear”
operator?
33
Consider the function
Ψ(x, t) = Bei(kx−ωt)
where B is complex but k , x, ω and t are real.
Describe this function using basic wave
language.
¨
(No reference to the Schrodinger
equation is required.)
34
Is i~ ∂∂t a linear operator?
If so, find the eigenvalue corresponding to the
following eigenfunction:
Ψ(x, t) = Bei(kx−ωt)
35
∂
Is −i~ ∂x
a linear operator?
If so, find the eigenvalue corresponding to the
following eigenfunction:
Ψ(x, t) = Bei(kx−ωt)
36
Given the momentum operator
∂
∂x
and the total energy operator
P = −i~
E = i~
∂
∂t
(a) construct the kinetic energy operator for a
nonrelativistic object.
(b) write down an operator equation for
conservation of energy.
37
V=
8
V=
V=0
x=0
x=a
8
What forces are felt by an object in the following
potential well?
38
V=
8
V=
V=0
x=0
x=a
8
¨
Derive a Schrodinger
wave function for an
object in the following potential well.
39
Consider an object described by the wave
function

for x < 0,

0
−i~π 2 t
Ψ(x, t) =
2ma2
A sin πx
for 0 ≤ x ≤ a,
a e


0
for x > a
What is the probability of observing the object
within 0 < x < a?
40
Consider an object described by the wave
function

for x < 0,

0
q
−i~π 2 t
2
2ma2
Ψ(x, t) =
sin πx
for 0 ≤ x ≤ a,
a
a e


0
for x > a
What is the probability of observing the object
within 0 < x < a2 ?
41
Consider an object described by the wave
function

for x < 0,

0
q
−i~π 2 t
2
2ma2
Ψ(x, t) =
sin πx
for 0 ≤ x ≤ a,
a
a e


0
for x > a
Calculate hxi.
Z
Fact :
0
π
θ sin2 θ dθ =
π2
4
42
Consider an object described by the wave
function

for x < 0,

0
q
−i~π 2 t
2
2ma2
Ψ(x, t) =
sin πx
for 0 ≤ x ≤ a,
a
a e


0
for x > a
Calculate hpi.
43
Consider an object described by the wave
function

for x < 0,

0
q
−i~π 2 t
2
2ma2
Ψ(x, t) =
sin πx
for 0 ≤ x ≤ a,
a
a e


0
for x > a
Calculate hEi.
44
Explain the distinction between
¨
the time-dependent Schrodinger
equation
and
¨
the time-independent Schrodinger
equation.
45
What is a stationary state?
Why is the word “stationary” appropriate?
46
Derive the following equation.
d 2ψ
2m
=
[V (x) − E]ψ
dx 2
~2
List situations when it cannot be used.
47
Sketch ψ(x) for a particle in the ground state of
this potential.
V(x)
x
What does ψ(x) represent?
48
Sketch ψ(x) for a particle in the first excited
state of this potential.
V(x)
x
What does ψ(x) represent?
49
Sketch ψ(x) for a particle in the second excited
state of this potential.
V(x)
x
What does ψ(x) represent?
50
Can an unbound particle exist in this potential?
If so, sketch ψ(x).
V(x)
x
51
Sketch ψ(x) for an unbound particle in this
potential.
V(x)
x
Which energy values are available to an
unbound particle?
52
A
√ particle with energy E > V0 and momentum
2mE is incident from the left in the potential
V(x)
shown.
V=Vo
V=0
x=0
Calculate the probability of reflection.
x
53
A
√ particle with energy E < V0 and momentum
2mE is incident from the left in the potential
V(x)
shown.
V=Vo
V=0
x=0
Calculate the probability of reflection.
x
54
A
√ particle with energy E < V0 and momentum
2mE is incident from the left in the potential
V(x)
shown.
V=Vo
V=0
x=0
x=a
Sketch ψ(x).
Could this particle be transmitted?
Could this particle be reflected?
x
55
V=
8
V=
V=0
x=0
x=a
8
Derive all allowed energies for an object in the
following potential well.
56
Define zero-point energy.
57
In one or two sentences, explain whether the
¨
Schrodinger
equation can accommodate a
discontinuous wave function.
58
In one or two sentences, explain whether the
¨
Schrodinger
equation can accommodate a
discontinuous potential V (x).
59
In one or two sentences, explain whether the
¨
Schrodinger
equation can accommodate a kink
in the wave function (i.e. a discontinuous
dΨ/dx).
60
There are kinks (i.e. discontinuous first
derivatives) in the energy eigenfunctions, ψ(x),
at the walls of an infinite potential well.
Which of the following provides the best
explanation?
(a) V (x) is discontinuous at the walls.
(b) V (x) is infinite at and beyond the walls.
(c) Both (a) and (b) are required to produce the
kinks.
(d) Neither (a) nor (b) can produce the kinks.
61
Why is the harmonic oscillator potential so
valuable to physicists?
62
Are there unbound solutions to the harmonic
oscillator potential?
63
Prove that,
for any one-dimensional potential V (x),
each energy E has only one eigenfunction ψ(x)
(up to the usual unphysical phase).
64
Prove that
for any one-dimensional even potential,
i.e. V (x) = V (−x),
each eigenfunction ψ(x) will be even or odd.
65
Find the allowed energies for the harmonic
oscillator potential.
66
Sketch the wave function Ψ(x, t) and probability
density for the first excited state of a harmonic
oscillator potential.
Choose t = 0 and t 6= 0.
67
Given two observables a and b, is it possible for
a particle to be in a state of definite a and
definite b at the same time?
68
Which of the following operators can share a
common set of eigenfunctions with the
harmonic oscillator Hamiltonian?
position
momentum
kinetic energy
potential energy
total energy
parity
69
Suppose we have a particle in the first excited
state of a harmonic oscillator.
Now we measure the particle’s position.
Does the wave function change?
70
TRUE OR FALSE: The wave function of a
particle in an infinite square well is always one
of the energy eigenfunctions, ψn (x) for some n.
71
Write down the most general solution to the
¨
Schrodinger
equation for an infinite square well.
72
Given
~v = axˆ + byˆ + c zˆ
express the coefficients a, b and c in terms of
dot products.
73
Derive the coefficients An and Bn in the Fourier
series
∞ X
An
Bn
√ sin(nx) + √ cos(nx)
f (x) =
π
π
n=0
defined for 0 < x < 2π.
74
Let
Ψ(x, t) =
∞
X
cn ψn (x)e−iEn t/~
n=0
denote the most general solution to the
¨
Schrodinger
equation for some potential V (x).
ψn (x) are the normalized energy eigenfunctions
as usual.
Use the normalization condition to derive a
constraint on the values of the coefficients cn .
75
Let
Ψ(x, t) =
∞
X
cn ψn (x)e−iEn t/~
n=0
denote the most general solution to the
¨
Schrodinger
equation for some potential V (x).
ψn (x) are the normalized energy eigenfunctions
as usual.
Derive an expression for hEi and use your result
to interpret the physical meaning of each
coefficient cn .
76
Consider a particle in an infinite square well at
0 < x < a with wave function
r
πx 2
Ψ(x, t) ∝
sin
e−iE1 t/~
a
ra
2
2πx
e−iE2 t/~
sin
+
a
a
Derive the probability density.
77
Is the derivative operator, d/dx, Hermitian?
Prove your answer.
78
Is the momentum operator, −i~d/dx,
Hermitian?
Prove your answer.
79
Find the allowed energies for a electron trapped
in an infinite potential box of dimensions
L × L × L.
80
¨
Express the 3-dimensional Schrodinger
equation, for arbitrary V (~r , t), in terms of the
angular-momentum-squared operator L2 .
81
¨
Derive the radial Schrodinger
equation for a
time-independent central potential V (r ).
82
Calculate [Lx , Ly ] and [Ly , Lz ] and [Lz , Lx ].
What is the implication of your results?
83
Calculate [L2 , Lx ] and [L2 , Ly ] and [L2 , Lz ].
What is the implication of your results?
84
Calculate [H, L2 ] and [H, Lx ] and [H, Ly ] and
[H, Lz ].
What is the implication of your results?
85
¨
Solve the angular part of the Schrodinger
equation for any time-independent central
potential V (r ).
86
Show that Y`m is an eigenfunction of Lz .
What is the eigenvalue?
87
It is standard to combine Lx and Ly into new
operators called L± .
Then instead of Lx , Ly , and Lz we can use
L+ = Lx + iLy
L− = Lx − iLy
and Lz
Are these operators Hermitian?
What are the implications of your results?
88
Show that L+ Y`m and L− Y`m are eigenfunctions
of Lz .
What are the eigenvalues?
What are the conventional names for L+ and
L− ?
89
Beginning from
(L+ Y`m |L+ Y`m ) ≥ 0
and
(L− Y`m |L− Y`m ) ≥ 0
prove that −` ≤ m ≤ +`.
90
Consider an electron in a central potential
having 2~2 as the eigenvalue of L2 .
Suppose the component of angular momentum
along the z axis was measured to be ~.
What is the probability of finding the electron
within 10 degrees of the z axis?
What is the probability of finding the electron
within 10 degrees of the xy plane?
91
Sketch the potential energy for one proton plus
one electron.
On your sketch, label the bound-state region.
On your sketch, label the scattering region.
92
Write down the total energy of one proton plus
one electron.
Re-express that total energy in
center-of-momentum coordinates.
Based on your findings, propose a
center-of-momentum Hamiltonian for hydrogen.
93
¨
When solving the radial Schrodinger
equation,
Scherrer described one boundary condition as
follows (page 133):
“as usual, as r → ∞, we require u → 0”
But the radial wave function is not u. It is u/r .
Should the boundary condition really be this:
as usual, as r → ∞, we require u/r → 0 ?
Explain carefully.
94
Write down the most general solution,
¨
Ψ(r , θ, φ, t), to the hydrogen Schrodinger
equation in terms of Rn` (r ) and Y`m (θ, φ).
95
Consider two eigenfunctions, ψn`m and ψn0 `0 m0 , of
the hydrogen Hamiltonian.
Write down the mathematical definition of
orthonormality.
96
Orthogonal spherical harmonics?
R 2π R π ∗
(a) 0 0 Y`m
(θ, φ)Y`0 m0 (θ, φ)dθdφ = δ`,`0 δm,m0
(b)
R 2π R π
0
0
∗
(θ, φ)Y`0 m0 (θ, φ) sin θdθdφ = δ`,`0 δm,m0
Y`m
(c) Both of the above.
(d) None of the above.
97
Rn` (r ) denotes a hydrogen radial wave function.
Identify each of the following as true or false.
R∞
(a) 0 R10 (r )R20 (r )dr = 0
R∞
(b) 0 R10 (r )R21 (r )r 2 dr = 0
R∞
(c) 0 R20 (r )R21 (r )r 2 dr = 0
R∞
(d) 0 R31 (r )R31 (r )dr = 1
98
These curves are |rRn,` |2 for n ≤ 3. Label each.
•
••
••
•
|rR10 (r)|2
|rR20 (r)|2
|rR30 (r)|2
•
••
••
•
|rR21 (r)|2
|rR31 (r)|2
•
••
••
•
|rR32 (r)|2
•
•
•
l=0
n = 1,2,3
l=1
n = 2,3
•
•
l=2
n=3
0
5
•
10
15
20
99
The hydrogen ground state energy is
E1 = −13.6 eV.
How many hydrogen eigenfunctions have
energy E91 ?
Write down the quantum numbers for each.
100
Consider
√1
3
q
i
Is it normalized?
!
2
3
101
Consider
ψ1 =
√1
3
q
i
q !
!
2
3
Are they orthogonal?
and
ψ2 =
i 23
− √13
102
Consider
Is it Hermitian?
0 −i
i 0
103
Consider
ψ1 =
√1
3
q
i
q !
!
2
3
and
ψ2 =
Calculate (ψ1 |ψ2 ) and (ψ2 |ψ1 ).
i 23
− √13
104
Consider
ψ1 =
√1
3
q
i
q !
!
2
3
ψ2 =
and
and
Calculate (ψ2 |σ1 ψ1 ).
σ1 =
i 23
− √13
0 1
1 0
105
Consider
σ1 =
0 1
1 0
Calculate [σ1 , σ2 ].
and
σ2 =
0 −i
i 0
106
Find all eigenvalues of
0 −i
σ2 =
i 0
107
Find all normalized eigenvectors of
0 −i
σ2 =
i 0
What space is spanned by the eigenvectors?
Write down the most general vector in that
space.
108
Explicitly calculate
N
X
|vi ihvi |
i=1
where the |vi i are the normalized eigenvectors
of
0 −i
σ2 =
i 0
109
Provide explicit expressions for c1 and c2 in
|ψi = |v1 ic1 + |v2 ic2
If the |vi i are the normalized eigenvectors of
0 −i
σ2 =
i 0
then find c1 and c2 for
|ψi =
7
4
110
Suppose the charge density of an electron is
approximated by
ρ(~r ) = −eδ 3 (~r − ~r0 )
What are the appropriate physical units for ρ(~r )?
111
Suppose the charge density of an electron is
approximated by
ρ(~r ) = −eδ 3 (~r − ~r0 )
What is the total charge of the electron?
112
Suppose the charge density of an electron
along a wire is approximated by
ρ(x) = cδ(2x)
where c is some constant.
What is the total charge of the electron?
113
Suppose the charge density of an electron is
approximated by
ρ(x, y , z) = cδ(x)δ(y + z)δ(1 − 2z)e−(x
where c is some constant.
What is the total charge of the electron?
2
+y 2 +z 2 )
114
Given any electron,
can you increase its spin angular momentum
by rotating the electron faster?
Explain.
115
The operator L2 has eigenvalues `(` + 1)~2 .
The operator S 2 has eigenvalues s(s + 1)~2 .
Write down the allowed values for ` and s.
116
What is a Stern-Gerlach apparatus?
In 1927, Phipps and Taylor shot a beam of
hydrogen atoms, all in their ground state,
through a Stern-Gerlach apparatus.
What did they observe?
117
What effect does a Stern-Gerlach apparatus
have on a small bar magnet?
What effect does a Stern-Gerlach apparatus
have on a small current loop?
118
Electrons are called spin- 12 particles.
In terms of experimental observations, exactly
what does this mean?
119
The electron in a hydrogen atom is in an ` = 1
state. What are the possible values of j and mj ?
120
The electron in a hydrogen atom is in an ` = 5
state. What are the possible values of j and mj ?
121
Suppose the electron in a hydrogen atom is
replaced by a rho meson.
The rho meson is a spin-1 particle.
Let’s call this a “rho-onic atom”.
The rho meson in a rho-onic atom is in an ` = 5
state. What are the possible values of j and mj ?
122
1
Let
represent an electron that is spin-up
0
along the z axis.
0
represent an electron that is
Let
1
spin-down along the z axis.
(a) Write the standard eigenvalue equation for
each, for operator Sz .
(b) Derive the four elements in the matrix
representing Sz .
123
1
Let
represent an electron that is spin-up
0
along the z axis.
0
Let
represent an electron that is
1
spin-down along the z axis.
(a) Write the matrix representing the raising
operator S+ (with units of angular momentum).
(b) Write the matrix representing the lowering
operator S− (with units of angular momentum).
124
Given the familiar identities
S+ = Sx + iSy
S− = Sx − iSy
use the matrix expressions for S± to obtain
matrix expressions for Sx and Sy .
125
Express S+ S− in terms of S 2 and Sz .
Verify explicitly that the matrices really do satisfy
this relation.
126
Find the eigenvectors of Sx .
Can these eigenvectors be expressed as linear
combinations of the Sz eigenvectors? (If so,
then do it.)
127
Find the eigenvectors of Sy .
Can these eigenvectors be expressed as linear
combinations of the Sz eigenvectors? (If so,
then do it.)
128
Consider a constant magnetic field B zˆ
containing a stationary electron that is spin-up
along the z-axis at t = 0.
At time t = T /3, the z-component of the
electron’s spin is measured.
Definition: T =
π~
µB B
What are the possible values, and what is the
probability of each?
129
Consider a constant magnetic field B zˆ
containing a stationary electron that is spin-up
along the z-axis at t = 0.
At time t = T /4, the x-component of the
electron’s spin is measured.
Definition: T =
π~
µB B
What are the possible values, and what is the
probability of each?
130
Consider a constant magnetic field B zˆ
containing a stationary electron that is spin-up
along the x-axis at t = 0.
At time t = T /2, the x-component of the
electron’s spin is measured.
Definition: T =
π~
µB B
What are the possible values, and what is the
probability of each?
131
Consider a constant magnetic field B zˆ
containing a stationary electron that is spin-up
along the x-axis at t = 0.
At time t > 0, the y -component of the electron’s
spin is measured.
What are the possible values, and what is the
probability of each?
132
(a) Complete the list of possible wave functions
for a pair of neutrons. (Show your work.)
|1 1i
|1 0i
|1 − 1i
|0 0i
=
=
=
=
| ↑ ↑i
?
| ↓ ↓i
?
(b) Carefully and completely define the notation
used in the list above.
133
Consider two electrons at rest with Hamiltonian
~1 · S
~ 2.
H = λS
(a) Express H in terms of S12 , S22 and S 2 .
(b) Are the eigenstates of S 2 ,
i.e. |s ms i = |1 1i, |1 0i, |1, −1i, |0, 0i,
also eigenstates of H?
(c) List all possible energy levels of this
two-electron system.
134
Find the energy levels of a two-neutron system
with Hamiltonian
H=
µ
~1 · µ
~ 2 3(~µ1 · ~r )(~µ2 · ~r )
−
r3
r5
where ~r extends from one neutron to the other.
(This is example 8.5 on page 184 of Scherrer’s textbook.)
135
Consider a constant magnetic field B zˆ
containing a stationary electron that is spin-up
along the x-axis at t = 0.
Define T = µπ~
.
BB
In a few sentences, compare/contrast these two
scenarios:
(a) The system is undisturbed until t = T /2
when the x-component of the electron’s spin is
measured.
(b) The x-component of the spin is measured
frequently throughout 0 < t < T /2.
136
Consider a free stationary electron (not in any
magnetic field) that is spin-up along the x-axis
at t = 0.
Define any constant time T > 0.
In a few sentences, compare/contrast these two
scenarios:
(a) The system is undisturbed until t = T when
the x-component of the electron’s spin is
measured.
(b) The v -component of the spin is measured
frequently throughout 0 < t < T , where
vˆ = xˆ cos(πt/T ) + yˆ sin(πt/T ).
137
List (by name) the perturbations
for a hydrogen atom.
Explain the meaning of each.
138
Which states of hydrogen are affected by the
Lamb shift? Why?
139
Fill in the blanks:
unperturbed En is proportional to
(dEn )fine is proportional to
(dEn )hyperfine is proportional to
(dEn )Lamb is proportional to
Select your answers from:
me
2
2
4
2
5
2
α me c , α me c , α me c , m
α4 me c 2 .
p
140
Order these hydrogen correction terms
from largest to smallest:
• fine
• hyperfine
• Lamb
141
Consider the hydrogen energy level n`j = 5F7/2 .
(a) Which states are degenerate with 5F7/2
when all perturbations are neglected?
(b) Which states are degenerate with 5F7/2
when fine structure is the only perturbation
retained?
(c) In real life, which states are degenerate with
5F7/2 ?
142
What extra term appears in V (~r ) for hydrogen
but not for He+ ?
143
Write down V (~r ) for pionic He+
(bound state of an alpha particle and a pion).
Explain the physics of each term.
144
Calculate the Stark effect for a ground-state
hydrogen atom in a constant electric field E yˆ .
145
Spin-orbit and an external B field:
H = H0 +
α~ ~ ~ µB B
S·L+
(Lz + 2Sz )
2me2 cr 3
~
The Zeeman effect is obtained by treating the B
term as a perturbation.
~ · ~L term and write down the
First neglect the S
energy shifts E (1) .
~ · ~L term is not
Is your method valid if the S
neglected? Explain.
146
TRUE OR FALSE:
¨
Adding a constant to V (~r ) in the Schrodinger
equation should not change the observable
physics.
147
¨
Begin with the Schrodinger
equation
H0 |ψn i = En |ψn i.
Add a small constant V0 to the potential.
Use perturbation theory to calculate the new
energy levels (to second order) and the new
wave function (to first order).
148
The perturbation for hydrogen in a magnetic
field is H1 = µB~B (Lz + 2Sz ).
Write down the perturbation for helium in a
magnetic field.
Calculate the shift of the helium ground state
energy.
149
Given H|ψi i = Ei |ψi i, define |ψi =
P∞
i=0 ci |ψi i.
(a) Which functions |ψi can be expressed as a
sum of this form?
(b) Evaluate hψ|ψi.
150
Given H|ψi i = Ei |ψi i, define |ψi =
(a) Evaluate
P∞
i=0 ci |ψi i.
hψ|H|ψi
.
hψ|ψi
(b) Is this ratio larger than, smaller than, or
equal to the ground state energy E0 ?
151
Use the variational principle to estimate the
ground state energy of a neutron bouncing on a
tabletop due to gravity.
Hints:
(a) Choose the trial wave function to be
−αx
xe
for x ≥ 0
ψ(x) =
0
for x < 0
R∞
(b) A useful integral is 0 x n e−αx = αn!
n+1 .
152
In chapter 6, the ground state hydrogen wave
function was found to be
e−r /a0
ψ(~r ) = √ 3/2
πa0
Generalize this to a one-electron atom with any
nucleus.
153
A first guess for the helium Hamiltonian might be
Hguess
P12
P22
2e2
2e2
=
+
−
−
2me 2me 4π0 r1 4π0 r2
What important term has been omitted?
Show that the product of two one-electron wave
functions
ψ(~r1 )ψ(~r2 )
¨
satisfies the Schrodinger
equation with Hguess .
154
Write down the normalization condition for a
hydrogen wave function ψ(~r ).
Write down the normalization condition for a
helium wave function ψHe (~r1 , ~r2 ).
What are the dimensions of each wave
function?
155
The ground state wave function for a
one-electron atom is
Z 3/2 e−Zr /a0
ψ(~r ) = √ 3/2
πa0
Write down the standard trial wave function for
helium.
In one or two sentences, explain why this choice
is physically reasonable.
156
Scherrer’s Equation (10.8):
P12
2e2
hψ|H|ψi = 2hψ|
|ψi − 2hψ|
|ψi + . . .
2me
4π0 r1
Z 3 −Z (r1 +r2 )/a0
where ψ =
e
πa03
Scherrer (page 232) says: “The integrals in the
first two terms on the right-hand side of
Equation (10.8) are straightforward”.
Response options:
strongly agree
agree
disagree
strongly disagree
157
For helium,
P12
P22
2e2
2e2
e2
H =
+
−
−
+
2me 2me 4π0 r1 4π0 r2 4π0 |~r1 − ~r2 |
Z 3 −Z (r1 +r2 )/a0
e
ψ =
πa03
leads to
hψ|H|ψi
e2
=
hψ|ψi
4π0 a0
5
Z 2 − 4Z + Z
8
Estimate the ground-state energy of helium.
(The experimental value is -79.0 eV.)
158
H|ψm i=Em |ψm i and (H + H1 (t)) |ψ(t)i=i~ ∂∂t |ψ(t)i
P
−iEm t/~
|ψm i
and |ψ(t)i = ∞
m=0 cm (t)e
(a) Which functions |ψ(t)i can be expressed as
a sum of this form?
(b) Suppose you have a system described by
|ψ(t)i, and you measure to see whether it is in
the state |ψn i. What is the probability that you
will find it in that state?
159
H|ψm i=Em |ψm i and (H + H1 (t)) |ψ(t)i=i~ ∂∂t |ψ(t)i
P
−iEm t/~
and |ψ(t)i = ∞
|ψm i
m=0 cm (t)e
(a) Derive an expression for
dcn
dt .
(b) For initial condition ci (0) = 1 and cf (0) = 0
∀f 6= i, simplify your answer to (a) by assuming
H1 (t) is a small perturbation.
(c) From (b), derive an expression for cf (t)
∀f 6= i.
(d) What is the transition probability P(i → f )?
160
A hydrogen atom is in its ground state. A weak
uniform electric field E zˆ is turned on at time
t = 0 and left on. At some later time tf > 0, what
is the probability that the atom will be in each of
the following excited states?
(a) n=2, `=0, m` =0
(b) n=2, `=1, m` =-1
(c) n=2, `=1, m` =0
(Compare example 11.1 on page 245 of Scherrer’s textbook.)
161
The central conclusion of time-dependent
perturbation theory is
Z
2
1 tf
P(i → f ) = 2 hψf |H1 (t)|ψi iei(Ef −Ei )t/~ dt ~
ti
The leading contribution to atomic transitions is
from
H1 (t) = eE0 r cos θe−iωt
Which factor in P(i → f ) is the key to deriving
selection rules?
162
In one sentence, define the phrase
“forbidden transitions”.
In one sentence, define the phrase
“selection rules”.
163
Add arrows to Scherrer’s figure 9.7 to indicate
all allowed transitions.
164
Add the n = 4 states to Scherrer’s figure 9.7 and
then use arrows to indicate all allowed
transitions.
165
Can the 3S1/2 state of hydrogen decay to the
ground state? Explain.
166
Define cross section.
Define differential cross section.
167
Calculate the differential and total cross section
for classical (not quantum) scattering of tiny
projectiles from an impenetrable sphere of
radius a.
168
Using classical (not quantum) mechanics,
derive the differential cross section for
Rutherford scattering.
169
What is the Born approximation
and when would you use it?
What is the partial wave expansion
and when would you use it?
170
Work through Scherrer’s Example 12.2
to derive the equation for Figure 12.11.
171
Using quantum mechanics, derive the
differential cross section for Rutherford
scattering.
172
Calculate the differential and total cross section
for low-energy quantum scattering of tiny
projectiles (of definite momentum) from an
impenetrable sphere of radius a.
173
Show that the exchange operator E12 commutes
with the two-particle Hamiltonian if
V (~r1 , ~r2 ) = V (~r2 , ~r1 ).
174
Show that the eigenvalues of the exchange
operator, E12 , are +1 and -1.
175
Neglecting electron-electron interactions, write
down 2-particle spatial eigenstates in terms of
single-particle eigenstates.
(Be sure they are eigenstates of the exchange
operator.)
176
TRUE OR FALSE:
(a) When the exchange operator is applied to
two particles, it interchanges both their spatial
positions and their spins.
(b) For two electrons, |0 ms i is always
symmetric and |1 ms i is always anti-symmetric.
177
Neglecting electron-electron interactions, write
down the ground state wave function for helium.
178
Neglecting electron-electron interactions, write
down 3-particle spatial eigenstates in terms of
single-particle eigenstates.
(Be sure they are eigenstates of the exchange
operator.)
179
Define the following.
(a) bosons
(b) fermions
(c) Pauli exclusion principle
180
Define shell for a multi-electron atom.
Define subshell for a multi-electron atom.
181
Energy levels in a multi-electron atom depend
on n and `. They do not depend on m` and ms .
Why is this to be expected?
182
How many electrons can fit into one subshell?
How many electrons can fit into one shell?
183
Name that (ground-state) atom:
(a) 1s2 2s2 2p2
(b) 1s2 2s2 2p4
184
Label each element in the periodic table by its
highest-energy populated subshell 1s, 3d, . . .
185
MRI images the internal human body, and has
no harmful side-effects.
It relies on spin-1/2 quantum mechanics of
hydrogen nuclei.
Why are hydrogen nuclei the optimal choice?
186
You are inside an MRI machine with magnetic
field
~ = B0 zˆ + B1 cos(ωt)xˆ
B
Describe the physics of your hydrogen nuclei for
the case of B1 = 0.
187
The central conclusion of time-dependent
perturbation theory
Z tis
2
1 f
i(Ef −Ei )t/~ P(i → f ) = 2 hψf |H1 (t)|ψi ie
dt ~
ti
The perturbation for MRI is
0
1
~ 1 = µ p B1
H1 = −~µp · B
cos(ωt)
1 0
1
0
where
has energy Ei = −µp B0 and
0
1
has energy Ef = +µp B0 .
What is the probability of a hydrogen nucleus
flipping into the excited state?
188
The transition probability for a spin flip in MRI is
B12 µ2p sin2 [(ω − ω0 )t/2]
P(i → f ) =
4~2 [(ω − ω0 )/2]2
where ω0 = 2µp B0 /~.
What is an optimal choice for the phase ω of the
transverse magnetic field?
In one or two sentences, discuss the value in
having the magnetic field depend on position:
B0 (x, y , z), B1 (x, y, z) and ω(x, y , z).
189
How are proton spin flips
translated into an MRI image?
190
Consider a quantum computer made of two
spin- 12 particles (i.e. two qubits).
Using the notation



1
0
 0 
 1


| ↓↓i ⇔ 
 0  , | ↓↑i ⇔  0
0
0




0
0

 0 
 0
 , | ↑↓i ⇔ 



 1  , | ↑↑i ⇔  0
0
1
build operators: AND, OR, XOR, NOT,
√
Which of these operators are available to
classical computers and which are not?
NOT.


,

191
Suppose all of your money is in a bank in
Ukraine. You want to withdraw it immediately,
but remote withdrawals are not permitted. A
trusted friend in Ukraine has volunteered to go
to the bank for you. All your friend needs from
you is the 7-digit account number.
If the two of you shared a “symmetric key”,
explain how you could encrypt the account
number using XOR, and how your friend could
decrypt it.
192
Using the quantum key distribution protocol
named BB84, show how you and your friend
could
(a) create a symmetric key shared only by the
two of you.
or
(b) discover that an eavesdropper is trying to
acquire your symmetric key.
193
Put the energy and momentum operators into
E 2 = m2 c 4 + p 2 c 2
to derive the Klein-Gordon equation.
Are there solutions with negative energies?
194
Dirac suggested an equation first-order in ∂/∂t:
∂ψ 2
~
i~
= −i~c~
α · ∇ + βmc ψ
∂t
Use consistency with the Klein-Gordon equation
to obtain expressions for αx , αy , αz and β.
195
Solve the Dirac equation for a particle at rest.
Are there solutions with negative energies?
Discuss.
What physics is represented by the four degrees
of freedom?