PHY4604–Introduction to Quantum Mechanics
Fall 2004
Practice Exam
Dec. 9, 2004
No materials allowed. If you can’t remember a formula, ask and I might help. If you
can’t do one part of a problem, solve subsequent parts in terms of unknown answer–
define clearly. All parts 10 pts., max=120. Problem 1 required, attempt 2 of remaining
3 problems; circle which ones you want graded.
Possibly helpful formulae and constants
q
L̂+ ψ`m = h̄ `(` + 1) − m(m + 1)ψ`m+1
q
L̂− ψ`m = h̄ `(` + 1) − m(m − 1)ψ`m−1
En =
me4 1
2h̄2µn2
¶
1
2
L̂+ L̂− + L̂2z − h̄L̂z
L̂− L̂+ + L̂2z + h̄L̂z
Eψ
∂ψ
ih̄
∂t
h̄2 2
−
∇ + V (r)
2m
En = h̄ω n +
L̂2 =
=
Hψ =
Hψ =
H =
Z
dx xn e−x = n!
Z a
0
a2 − cos(a2 ) sin(a2 )
2a
³ ´2
− 12 xx
1
0
ψ0 (x) = q
e
1/2
π x0
sin2 ax dx =
ψ1 (x) = q
1
2π 1/2 x0
ψ2 (x) = q
1
µ
à µ
x
4
x0
8π 1/2 x0
à µ
1
ψ3 (x) = q
¶
2x − 12
e
x0
48π 1/2 x0
1
³
x
x0
!
¶2
x
8
x0
´2
³
− 12
−2 e
¶3
!
x
x0
´2
−1
x
− 12
e 2
x0
³
x
x0
´2
1. Short answer. Must attempt (only) 4 of 6.
(a) Explain what is meant by a 2p state of an atomic electron.
(b) What is the degeneracy of the 1st excited state (E = (5/2)h̄ω) of the
isotropic 3D simple harmonic oscillator?
(c) Sketch the first 3 eigenfunctions of the 1D infinite square well with V = 0
for −a ≤ x ≤ a and V = ∞ otherwise. Label them according to their
parity.
2
(d) State whether the following operators are self-adjoint, anti-self-adjoint,
unitary, or “none of the above” and why.
i. L̂x
ii. xp̂x
iii.
d
dx
iv. a + a†
(a, a† are raising & lowering operators for the 1D simple harmonic
oscillator.)
(e) If a particle is in the state |ψi, and |ni is the nth eigenvector of Q̂ corresponding to eigenvalue qn , what is the probability of measuring q3 ?
(f) Identify:
i.
ii.
iii.
iv.
Photoelectric effect
Davisson-Germer effect
Ehrenfest theorem
Stefan-Boltzmann law
3
2. Hydrogenic orbitals. An electron moving in the Coulomb field of a proton is
in a state described by the wave function (ignoring spin)
1
Ψ(r, θ, φ) = √ [ψ100 (r) + 3ψ311 (r, θ, φ)]
10
(1)
(a) What is the expectation value of the energy?
(b) What is the expectation value of L̂2 ?
(c) Is the wavefunction an eigenstate of parity? Yes or no? Explain either
answer.
(d) What is the expectation value of the operator
4
∂
∂φ
in this state?
3. Electron in hydrogenic state. The electron in a hydrogen atom occupies a
state
s
1 0
ψ = R21 (r) 
Y +
3 1
s

2 1
Y
3 1
(2)
where
s
R21 (r) =
s
Y10 =
3
cos θ
4π
µ
1 1
3 2a0
¶3/2 µ
Y11 = −
,
s
¶
r
e−r/2a0
a0
3
sin θeiφ
8π
,
(3)
(4)
(a) What value(s) could a measurement of the z-component of the orbital
angular momentum, L̂z yield, and what is the probability of each? What
is the expectation value of L̂z in this state?
(b) Calculate the average distance of the electron from the nucleus in this
state.
(c) What is the expectation value of L̂x in this state?
(d) If you measured the z-component of the angular momentum and the distance of the electron from the origin r simultaneously, what is the probability density for finding L̂z with eigenvalue zero at a distance r?
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4. Scattering potential. For a 1D potential as shown in the figure and E < V0 ,
assuming the particle is incident from the left,
(a) write down the Schrödinger equation and its general solution in the regions
I,II, and III.
(b) Write down the matching conditions at the boundaries x = a, b.
(c) Do not solve for all coefficients, but reduce the problem to a single equation
determining the eigenvalues.
(d) Sketch the probability of finding the electron in all three regions.
6