# User manual | The Handbook of Mathematics, Physics and Astronomy Data is provided ```The Handbook of Mathematics, Physics and
Astronomy Data is provided
KEELE UNIVERSITY
EXAMINATIONS, 2012/13
Level II
Thursday 17th January 2013, 16:00 – 18:00
PHYSICS/ASTROPHYSICS
PHY-20006
QUANTUM MECHANICS
Candidates should attempt ALL of PART A
and TWO questions from PART B.
PART A yields 40% of the marks, PART B yields 60%.
A sheet of useful formulae can be found on page 9.
NOT TO BE REMOVED FROM THE EXAMINATION HALL
PHY-20006
Page 1 of 9
PART A
A1
Answer all TEN questions
Give an expression for the probability of observing a particle in a
thin shell of radius r and infinitesimal width dr for a particle with
a spherically symmetric wave function Ψ(r, t).
A2

Sketch the following function and give two reasons why it cannot be
part of a realistic wave function (a > 0 is a real constant).



eax x < 0

ax x ≥ 0
f (x) = 

A3
Calculate the value of A for the following wave function and explain
your method.
Ψ(x, t) = A sech(x) e−iωt
You may use the following integral without proof in your answer.
Z
sech2 (x) dx = tanh(x) + C
N.B. tanh(x) → ±1 for x → ±∞.

/Cont’d
PHY-20006
Page 2 of 9
A4
State three differences between the predictions of classical physics
and the predictions of quantum mechanics for the properties of a
particle in the semi-infinite square well potential,






∞
x<0
V (x) =  −V0 0 ≤ x ≤ a



 0
x>a

A5
Sketch the energy eigenfunctions for the ground state and first excited state of a particle trapped in the region 0 < x < a by the
8
8
potential shown below assuming that the energy of the particle is
E < VB in both cases.
V(x)
VB
0
x
a
0

A6
Calculate the expectation value hEi for the energy of a particle with
the wavefunction
1
1
Ψ(x, t) = √ ψ1 (x)e−iE1 t/h̄ − √ ψ2 (x)e−iE2 t/h̄ ,
2
2
√
where En = n eV is the energy for the state ψn .

/Cont’d
PHY-20006
Page 3 of 9
A7
Show that the wave function Ψ(x, t) = e−i(kx+wt) is an eigenfunction
of the momentum operator, p̂, and state the eigenvalue.

A8
The wave function for 2 identical particles in a 1-dimensional potential is Ψ(x1 , x2 ). State and explain the possible values of Ψ(x2 , x1 ).

A9
List all possible values for the magnitude of the total angular momentum, J, for an electron in the 2p state of the hydrogen atom.
Give your answers in units of h̄.

A10
A particle has the wave function
3
a2
Ψ(x, t) = √ x e−ax/2 e−iωt ,
2
x > 0.
Calculate an approximate correction to the energy of the particle if
the potential is perturbed by a field V 0 (x) = ²x, where ² = 0.02 eV/nm
and a = 1 nm−1 . You may use the following integral without proof
in your answer.
Z ∞
n!
rn e−ar dr = n+1
0
a

/Cont’d
PHY-20006
Page 4 of 9
PART B
A particle of mass m is trapped in the following potential:

1


kx2


 2
V (x) =  0



 ∞
x<0
0≤x≤a
x>a
V(x)
8
B1
Answer TWO out of FOUR questions
0
a
x
For some values of k it is possible to write the solution of the time
independent Schrödinger equation as






ψ(x) = 




2
ψA = Ae−αx
ψB = B cos(βx) + C sin(βx)
0
x<0
0≤x≤a
x>a
(a) Apply the appropriate boundary condition at x = 0, and hence
show that B = A and C = 0.

(b) Apply the appropriate boundary condition at x = a and hence
derive the possible values of β.

(c) Show that ψB is a solution of the time independent Schrödinger
equation, and hence derive an expression for the energy, E, in
terms of a.

(d) Discuss the behaviour of
∂ψ
∂x
at x = a

(e) Describe and explain what the solutions of the time independent Schrödinger equation would look like in the alternative
case k → ∞.

/Cont’d
PHY-20006
Page 5 of 9
B2
The energy of a particle with mass m in a 2-dimensional harmonic
oscillator potential
1
1
V (x, y) = mω 2 (x2 + y 2 ) = kr2
2
2
is given by
Enx ,ny = (nx + ny + 1)h̄ω
nx = 0, 1, 2, . . .
ny = 0, 1, 2, . . .
The angular momentum operator for a 2-dimensional system is
L̂2 = x̂pˆy − ŷ pˆx ,
∂
, and similarly for pˆy .
where pˆx = −ih̄ ∂x
(a) Write down the energy in units of h̄ω for the first three energy
levels. State the degeneracy and the values of nx and ny for
each energy level.

(b) Show that Y (x, y) = y − ix is an eigenfunction of the operator L̂2 . State the values of the expectation value hL2 i and the
uncertainty ∆L2 for the particle in this state.

(c) Show that the commutator for L̂2 and x̂ has the value [L̂2 , x̂] =
ih̄y. State briefly what your result implies for the observed
values of L2 and x.

(d) Give a physical reason why eigenfunctions of the L2 operator can
be used to find a solution to the time independent Schrödinger
equation for this potential.

/Cont’d
PHY-20006
Page 6 of 9
B3
The energy of a particle with mass m in the harmonic oscillator
potential V (x) = 12 kx2 is given by
Ã
!
1
En = n +
h̄ω,
2
where ω =
n = 0, 1, 2, . . . ,
q
k/m. The energy eigenfunction for the first excited
state is
Ã
ψ1 (x) = A1
!
x −x2 /2a2
e
,
a
where a is a constant.
(a) Show that ψ1 has definite parity and state its value.

(b) Sketch the position probability distribution function for the
state ψ1 .

(c) The spectrum of H35 Cl shows spectral features due to changes
in vibration state spaced equally in frequency by 8.66×1013 Hz.
Calculate the bond strength, k, for this molecule. State any
assumptions you have made in your calculation.

(d) Describe how the energy eigenfunctions ψj , j = 0, 1, 2 . . ., for
this potential can be used to represent a time-dependent wave
function Ψ(x, t) given an initial state Ψ(x, 0).

(e) Discuss briefly whether the following two statements are consistent with each other.
• The momentum, p, of a particle with kinetic energy E1 is
given by p2 = 2mE1 .
• The expectation value of the momentum for the particle described by ψ1 (x) is hpi = 0.

/Cont’d
PHY-20006
Page 7 of 9
B4
The wave functions for an electron in a simple model of the hydrogen
atom have the form
Ψn,`,m` (r, θ, φ, t) =
u(r)
Y`,m` (θ, φ)e−iEt/h̄ .
r
(a) State the physical quantity most closely associated with each of
the quantum numbers n, ` and m` , and state the possible values
for each quantum number for an electron in a 3p state.

(b) The function u(r) is a solution of the radial Schrödinger equation
for the effective potential
`(` + 1)h̄2
e2
Ve (r) =
.
−
2me r2
4π²0 r
Explain the origin of the two terms in this equation for the
effective potential.

(c) With the aid of a labelled diagram, describe the main features
and results of the Stern-Gerlach experiment. Explain how this
experiment shows that the eigenfunctions Ψn,`,m` do not give
a complete description for the properties of an electron in a
hydrogen atom.

(d) The 3p state of hydrogen is split into two energy levels split by
about 0.05 meV. Explain the origin of this fine structure in the
energy spectrum of hydrogen.

/Cont’d
PHY-20006
Page 8 of 9
Quantum Mechanics formulae
Time independent Schrödinger equation


h̄2 d2 ψ
Ĥψ = −
+ V (x) ψ = Eψ
2m dx2
PHY-20006
Page 9 of 9
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