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Quantum Mechanics PHYS09053 (SCQF Level 9)

Answer ALL of the questions in Section A,
ONE question from Section B and ONE question from Section C

Printed: Wednesday 15th April, 2020

PHYS09053

School of Physics & Astronomy

Quantum Mechanics PHYS09053 (SCQF Level 9)

Monday 4th May, 2020 13:00 16:00 BST (May Diet)

Please read full instructions before commencing writing. Examination Paper Information

Special Instructions

? This is an open-book examination.
? Electronic Calculators may be used during this examination.
? A sheet of physical constants is supplied for use in this examination.

Special Items

? School supplied Constant Sheets

Chairman of Examiners: Prof J Dunlop External Examiner: Prof Ian Ford

Anonymity of the candidate will be maintained during the marking of this examination.

Quantum Mechanics (PHYS09053)
Section A: Answer ALL of the questions in this Section

  1. A.1 ?Given a linear operator S and kets |χf? = S|χi?, show that ?χf|χf? = tr[ρS?S], where
    ρ ≡ |χi??χi|. (Recall the definition of the trace of a linear operator - tr(S) ??i?i|S|i?,
    where
    {|i?} is an arbitrary basis). [5]

  2. A.2 ?Write down the Hamiltonian operator for a harmonic oscillator, H, defined in terms of
    ladder operators
    a and a? and ??ω. Show that if |ψ? is an eigenstate of H with energy eigenvalue E ?, then a?|ψ? is also an eigenstate but with eigenvalue E ? + ??ω. Describe in
    one sentence physically what the operator
    a? does. [5]

  3. A.3 ?Using the definitions for the angular momentum operators J±, J2, and Jz, and using the
    known commutation relations for
    Jx,y,z show the following identity. [5]

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A.4 (a)

|J±|j, m?| = ????j(j + 1) ? m(m ± 1)
What is the ground state wavefunction of a particle of mass
m in a 1-dimensional

square well potential
V(x) = 0

[2]

0 0 2mL2
of the ground state energy induced by this potential. [3]

  1. A.5 ?For the hydrogen atom the spin-orbit interaction for the angular coordinates is written H?SO = ξn,l?s.?l with ?s and ?l operators for the spin and orbital angular momentum.

    1. (a) ?Explain in one or two sentences the physical origin of the spin-orbit interaction. [2]

    2. (b) ?Showthat[H?SO,?j]=0where?jisthethetotalangularmomentumoperator. [2]

    3. (c) ?Why is it advantageous to work with eigenstates of ?jz and ?j2 rather than those of
      ?lz and s?z when deriving the energy-level shifts due to the spin-orbit interaction? [1]

  2. A.6 ?A manganese atom has electronic configuration [Ar]4s23d5.
    (a) What is the term symbol for its ground state?
    [4]

    (b) Explain in words (one or two sentences) why the 4s level is filled before the 3d level.

=
(b) An additional potential V δ(x) is now applied with V ? ??2 . Estimate the change

?L/2<x<L/2 otherwise.

[1]

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Quantum Mechanics (PHYS09053)
Section B: Answer ONE question from this Section

B.1 Consider a potential that has the form,

V (r) = ? β , β > 0. r2

(a) Derive an approximate form for the Schrodinger Equation at large r and show it has

a solution of the form ψ e?ar. Note that the radial part of the Laplacian can be written

as 1 ? r2 ? . [10] r2?r ?r

(b) Show that for small r, the radial part of the Schrodinger Equation takes the form
given below and find an expression for
γ. [4]

′′ RR
R
+2r +γr2 =0,

(c) Seek out solutions R rs to show that there are two possible values for s. [4] (d) Use this result to show that the orbital angular momentum (l(l + 1)) is bounded from

below. [5] (e) How is this result different from the Hydrogen atom where the potential 1r . [2]

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Quantum Mechanics (PHYS09053)
B.2 (a)GiventwolinearoperatorsAandB,usingDiracnotationshowthattr(AB)=tr(BA).

Consider a particle with angular momentum j = 1/2.
(
b) Write down the two basis states which are eigenvectors of Jz in both Dirac notation

[3]

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and as column vectors. [1] (c) Derive the matrices for the operators J2, J+, J?, Jx, Jy, and Jz. Which of these

operators are Observables? [12] (d) The matrices σx, σy, and σz are defined in terms of the operators Ji by Jx,y,z = ??2σx,y,z.

Show that the trace of σx, σy, and σz is zero. [1] (e) By computing the products of σi (where i = x, y, z), show the following anticommu-

tationrelation{σij}≡σiσj +σjσi =2δij.
(
f) Using this relation to show that tr(σiσj) = 2δij.

[5] [3]

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Quantum Mechanics (PHYS09053)
Section C: Answer ONE question from this Section

C.1 This question is on degenerate perturbation theory for a particle in an anisotropic har- monic potential and the energy of two identical particles in the presence of an interparticle interaction.

The raising and lowering operators for a one dimensional harmonic oscillator of mass mandfrequencyωarea?±=??x??i??1 p?.

(a) Prove that the ground state wavefunction for a particle in a one dimensional har- monic potential is

? mωx2 e 2?? .

dx e?x2/2σ2 = 2πσ.]
Now consider a single particle in a two dimensional anisotropic harmonic potential with

Hamiltonian

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?x|0?=
[You may use the mathematical identity
??

2?? 2m??ω

?? ?? 14 π??

222
p?x +p?y ?? 2 2??

?
H 0 = 2 m + 2 x? + 4 y? .

?∞

[4]

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  1. (b) ?Explain why eigenstates of H?0 can be written |nx,ny? ≡ |nx?|ny? i.e. as a product
    of eigenstates of two one dimensional harmonic potentials.
    [2]

  2. (c) ?The second excited level comprises states |2,0? and |0,1? with energy 27??ω. Write
    down the eigenstates and energies for the ground state, first and third excited energy
    levels.
    [4]

Now consider an additional perturbing potential H?1 = λx?2y? where λ is small.
(d) Show that this perturbation splits the second excited level into two distinct levels

and calculate the resulting energy splitting to first order in λ. [7] Now consider two identical mass m, non-interacting, spin zero particles, confined to two

2 2 2 dimensions in the unperturbed potential 2 (x? + 4y? ).

The basis states can be expressed in terms of symmetrised kets |(n1x, n1y), (n2x, n2y)?. (e) Write down the energies of the lowest energy and first 2 excited energy levels and

all the corresponding eigenstates (kets). [4]

Now consider a very weak interaction between the particles H?I = V0δ(x1 ? x2)δ(y1 ? y2).

(f) What is energy shift of the ground state energy due to this interaction to lowest
order in
V0? [4]

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Quantum Mechanics (PHYS09053)

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C.2 This question is on the Stark effect in hydrogen, the mass-velocity relativistic correction to the energy levels of the hydrogen atom and linear Stark effect including relativistic corrections.

The Stark effect considers the change in energy of atomic states in an applied electric field. An electric field of strength E applied along the z-direction creates a perturbation H?1 = eEz?. The unperturbed atomic states are conventionally described by quantum numbers |n, l, m? with energy ??n. We consider atomic hydrogen in the following questions.

  1. (a) ?Explain why there is no shift of the ground state energy linear in E. [3]

  2. (b) ?Show, starting from second order perturbation theory, that a lower estimate of the

    energy shift of the ground state is given by the expression

2
? E 0 , 0 , 0 = E 0 , 0 , 0 ? ?? 0 e 2 E 2 ? 0 , 0 , 0 | z? | 0 , 0 , 0 ? ,

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??0 ? ??1
with ??1 the energy of the first excited state. [5]

(c) Explain briefly why the argument made in part (a) does not apply when calculating
the energy shift for the state
|n, l, m? = |1, 0, 0?. [3]

  1. (d) ?Starting from the relativistic expression for the kinetic energy E = ??p2c2 + m2c4 show that the lowest order relativistic correction to the kinetic energy (in the absence of E) gives rise to the perturbation

    4
    H? 1 = ? p? .

    8m3 c2

  2. (e) ?Hence show that the change of energy of a general state |n, l, m? due to this pertur-

[2]

[2]

[5]

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bation is

?Ekin=??n,l,m| 1 (H?0?V(r?))2|n,l,m?. 2mc2

H?0 is the non-relativistic hydrogen atom Hamiltonian. (f) Hence show that

α2??3 N?? ?Ekin=???nN2 4?l+1/2

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where α = e2 = ?? is the fine structure constant, ?? 4πε0??c ma0c

n

= ? 1 8πε0a0N2

with a

0

the

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Bohr radius and N = n + 1 (ε0 is the permittivity of free space). The following identities can be used:

??1?? = 1 and ??1?? = 1 . r N,l a0N2 r2 N,l a20N3(l+1/2)

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(g) Including all relativistic corrections (except the QED Lamb shift) the eight states in

the N = 2 shell are split into two sets of 4 degenerate levels. The first set comprise

the 2s1 and 2p1 levels and the second 2p3 . Explain which (if any) of these sets 222

would be split linearly by an applied electric field E and give the degeneracy of the resulting levels. [5]

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