# 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
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
[4]
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) = 
[4]
A3
Calculate the value of A for the following wave function and explain
Ψ(x, t) = A sech(x) e−iωt
Z
sech2 (x) dx = tanh(x) + C
N.B. tanh(x) → ±1 for x → ±∞.
[4]
/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
[4]
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
[4]
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 .
[4]
/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.
[4]
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 ).
[4]
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.
[4]
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
Z ∞
n!
rn e−ar dr = n+1
0
a
[4]
/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.
[6]
(b) Apply the appropriate boundary condition at x = a and hence
derive the possible values of β.
[5]
(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.
[8]
(d) Discuss the behaviour of
∂ψ
∂x
at x = a
[5]
(e) Describe and explain what the solutions of the time independent Schrödinger equation would look like in the alternative
case k → ∞.
[6]
/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.
[6]
(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.
[8]
(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.
[10]
(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.
[6]
/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.
[4]
(b) Sketch the position probability distribution function for the
state ψ1 .
[4]
(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
[8]
(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).
[8]
(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.
[6]
/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.
[6]
(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.
[4]
(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.
[12]
(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.
[8]
/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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