# 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, 2014/15
Level II (FHEQ Level 5)
Wednesday 14th January 2015, 09:15 – 11:15
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
A particle in a 2-dimensional potential, V (r, θ), has a radially symmetric wave function, Ψ(r, t). Give an expression for the probability
of observing the particle in an annulus centred on the origin with
an inner radius R1 and an outer radius R2 .
A2
[4]
Explain why the wave function for a particle
(a) must be single-valued;
(b) must have a continuous derivative
∂ψ
∂x .
[2×2]
A3
Normalize the following solution of the time-independent Schrödinger
equation.






0
x ≤ −a
√
ψ(x) =  A 1 − (x/a)4




0
−a < x < a
x≥a
[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 a harmonic oscillator potential, V (x) = 21 kx2 .
A5
[4]
Sketch the energy eigenfunctions for the ground state and first excited state of a particle trapped in the region 0 < x < a by the
potential shown below assuming that the energy of the particle is
8
V(x)
8
E < 0 in both cases.
0
a
x
[4]
A6
Calculate the expectation value ⟨E⟩ for the energy of a particle with
the wavefunction
1
1
1
Ψ(x, t) = √ ψ1 (x)e−iE1 t/h̄ − √ ψ2 (x)e−iE2 t/h̄ + √ ψ3 (x)e−iE3 t/h̄ ,
3
3
3
√
where En = n is the energy in eV 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+ωt) is an eigenfunction
of the momentum operator, p̂, and state the eigenvalue.
A8
[4]
Explain the concept of definite exchange symmetry for 2 identical
particles that occupy the same potential and how this relates to the
spin-statistics theorem.
A9
[4]
List all possible values for the magnitude of the total angular momentum, J, for an electron in the 3d state of the hydrogen atom.
Give your answers in units of h̄. What eﬀect do these diﬀerent values
of J have on the energy of the electron?
A10
[4]
The energy of a 3-dimensional quantum harmonic oscillator is given
by
)
(
3
h̄ω.
E = nx + ny + nz +
2
• Write down the energy (in units of h̄ω) and the degeneracy of
the ground state.
• List all the states with energy E = 92 h̄ω and state the degeneracy
of this energy level.
[4]
/Cont’d
PHY-20006
Page 4 of 9
PART B
B1
Answer TWO out of FOUR questions
A particle of mass m with energy E < 0 is trapped in the following
potential:






∞
V (x) =  −V0




0
x≤0
(Region I)
0 < x < a (Region II)
x≥a
(Region III)
(a) Explain why the solution to the time-independent Schrödinger
equation in Region I must be ψI = 0.
[4]
(b) Show that ψII = B cos(k0 x)+C sin(k0 x) is a solution of the timeindependent Schrödinger equation in Region II and so derive an
expression for k0 in terms of V0 and E.
(c) Expain why B = 0 for a particle in this potential.
[8]
[3]
(d) The general solution of the time-independent Schrödinger equation in Region III is ψIII = A e−αx + A′ eαx , where α is a real,
positive constant. Explain why A′ = 0 for a particle in this
potential.
[4]
(e) Describe how the boundary conditions at x = a and the value
of V0 determine the allowable values of E.
(f) Discuss the behaviour of
∂ψI
∂x
and
∂ψII
∂x
at x = 0.
[6]
[5]
/Cont’d
PHY-20006
Page 5 of 9
B2
Consider a particle with mass m and energy E incident on a barrier
with height VB and width a, such that E < VB .
VB
E
0
x
a
The solution of the time-independent Schrödinger equation in the
region x < 0 is
ψI = AI eikx + AR e−ikx ,
and in the region x > a is
ψIII = AT eikx .
(a) What is the main diﬀerence between the predictions of classical
physics and quantum mechanics for the behaviour of the particle
in this case?
[4].
(b) What does the quantity T =
|AT |2
|AI |2
measure? Explain how the
value of T relates to the value of R =
|AR |2
|AI |2 .
[6]
(c) For “wide” barriers (aβ ≫ 1),


16E(VB − E)  −2βa
T ≈
e
,
VB2
√
where β = 2m(VB − E)/h̄.
i. Estimate the value of T for the case of two pieces of metal
separated by one atom’s width at room temperature.
[10]
ii. Explain how the dependence of T on a leads to the feasibility
of building a scanning tunneling electron microscope that
can make an image of surface showing individual atoms. [10]
/Cont’d
PHY-20006
Page 6 of 9
B3
The energy of a particle with mass m in a 1-dimensional harmonic
oscillator potential V (x) = 12 kx2 is given by
(
)
1
En = n +
h̄ω,
2
n = 0, 1, 2, . . . ,
√
where ω = k/m. The energy eigenfunction for the ground state is

1
1  2 −x2 /2a2

√
,
ψ0 (x) =
e
a π
where a is a constant.
(a) Show that ψ0 has definite parity and state its value.
[4]
(b) Sketch the position probability distribution function, P (x)dx,
for the state ψ0 .
[3]
(c) Explain why the expectation value for the momentum is ⟨p⟩ = 0
for a particle in the state ψ0 .
[3]
(d) Calculate the uncertainty on the momentum, ∆p, for a particle
in the state ψ0 .
[12]
(e) The spectrum of H79 Br shows spectral features spaced equally
in frequency by 7.66×1013 Hz due to transitions between molecular vibrational state. Calculate the bond strength, k, for this
molecule. State any assumptions you have made in your calculation.
[8]
You may use the following integrals without proof in your answers.
e−x
/a2
√
=a π
x2 e−x
/a2
1 √
= a3 π
2
∫ ∞
−∞
∫ ∞
−∞
2
2
/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,ℓ,mℓ (r, θ, ϕ, t) =
u(r)
Yℓ,mℓ (θ, ϕ)e−iEt/h̄ .
r
For the ground state (1s) of the hydrogen atom, u(r) =
where a0 =
4πϵ0 h̄2
e2 m e
√2
a0
(
r
a0
)
e−r/a0 ,
is the Bohr radius.
(a) What are the values of ℓ and mℓ for the electron in the 1s
state?
[2]
(b) Show that Yℓ,mℓ (θ, ϕ) has the constant value
√1
4π
for the 1s
state.
[6]
(c) Show that the Born interpretation applied to the wave function
Ψn,ℓ,mℓ (r, θ, ϕ, t) reduces to P (r)dr = u2 dr for the 1s state.
[6]
(d) Use the result in part (c) to calculate the expectation value ⟨r⟩
for an electron in the 1s state.
[8]
(e) Muons have the same charge and spin as an electron but are
about 200 times more massive. What do you think is the size of
a muonic hydrogen atom formed by replacing the electron with
a muon in a normal hydrogen atom? Justify your answer.
[8]
/Cont’d
PHY-20006
Page 8 of 9
Quantum Mechanics formulae
Time independent Schrödinger equation


2
h̄
d2
+ V (x) ψ = Eψ
Ĥψ = −
2m dx2
PHY-20006
Page 9 of 9
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