Faculty of Science Final Examination Physics 457 ...
Physics 457: Honours Quantum Physics II
1
Faculty of Science Final Examination Physics 457 Honours Quantum Physics II
Examiner : Professor C. Gale Associate Examiner: Professor S. Jeon Date: Tuesday, April 17th 2012 Time: 9:00 – 12:00
Instructions: This examination comprises five (5) questions. Attempt all questions. No notes or texts allowed; write your answers (with intermediate steps: show your work) in the exam booklet(s). Some useful formulæ are given at the end of the exam and with some of the questions. Translation dictionaries are allowed. Calculators are not allowed. You may keep the exam.
1. The spin Hamiltonian for a spin- 12 particle in an external magnetic field is ˆ = H
~ µ ~ˆ · B
gq ˆ ~ The particle has mass m, charge q, and g is the “g-factor”. Take where µ ~ˆ = 2mc S. ~ = B0~k + B2~j, with B2 ⌧ B0 . Note that ~k and ~j are unit vectors in the z and y B
direction, respectively. (a) Write a matrix representation of the Hamiltonian, using eigenstates of the Sz operator. (b) Determine the energy eigenvalues exactly. (c) Use perturbation theory, to evaluate the energy eigenvalues to second order in B2 /B0 . Compare with the exact solutions.
2. A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state
Physics 457: Honours Quantum Physics II
2
is represented by the position eigenket |Ri(|Li), where spatial variations within each half of the box are neglected. The most general state vector can then be written as |↵i = |RihR|↵i + |LihL|↵i , where hR|↵i and hL|↵i can be regarded as “wave functions”. The particle can tunnel through the partition; this tunneling e↵ect is characterized by the Hamiltonian (|LihR| + |RihL|) ,
H = where
is a real number with the dimension of energy.
(a) Find the normalized energy eigenkets. What are the possible energy eigenvalues? (b) Suppose the system is represented by |↵i as given above at t = 0. Find the state vector |↵(t)i for t > 0.
(c) Suppose at t = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (d) Suppose that the Hamiltonian had been written wrongly as H =
(|LihR|).
By explicitly solving the most general time evolution problem with this Hamiltonian (or otherwise), show that probability conservation is violated.
3. The amplitude for the elastic scattering of a particle of mass m with an original wavevector ~k into a state ~k 0 is f (~k 0 , ~k) =
1 2m (2⇡)3 2 h~k 0 |V | i , 4⇡ h ¯
where V is the local scattering potential, and
is the eigenfunction of the free Hamil-
tonian plus the scattering potential. Consider the repulsive delta-shell potential V (~r) =
(r
R).
(a) Compute the scattering amplitude in the Born approximation.
Physics 457: Honours Quantum Physics II
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(b) What is the minimum energy necessary for zeroes to appear in the di↵erential cross section? (c) For energies above this minimum, at what angles will the di↵erential cross section vanish? (d) What is the total cross section in the low energy (kR ⌧ 1 ) limit? 4. Consider a particle of mass m being subjected to V (x): a one-dimensional potential well of length L, with infinite walls at x = 0 and x = L. This potential gets modified by a perturbation V1 (x): V (x) =
(
0, 0 < x < L 1, elsewhere
V1 (x) =
(
V0 , 0 x L/2 0, elsewhere
where V0 is small as compared to the other energies in this problem. (a) Using the WKB approximation, calculate En : the energy of level n. (b) Compare your answer in (a) with that obtained using first-order perturbation theory. If these two results are di↵erent, is there a limit where they agree? (c) Evaluate the ground state energy using the variational approximation (with a trial wave function of your choice), and compare with the appropriate results obtained in (a) and (b).
5. Only considering the interaction between the electrons and the nucleus (neglecting the spin-orbit interaction, the mutual interaction between the electrons, etc...) which has charge Z, find the energy levels and write down the two-body wave functions of the three lowest states for a two-electron atom. Use the fact that the energy levels for a Hydrogen-like atom are given by En =
µc2 Z 2 ↵2 2n2
where µ is the reduced mass, and the other symbols have their usual meaning. Take into account the fact that the electrons are identical spin- 12 particles, and just use the notation
x), n`m (~
for the Hydrogen wave function.
Physics 457: Honours Quantum Physics II
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Hints and Useful Formulæ: For non-commuting operators,A and B, such that [A, B] = C, and [A, [C, A]] [A, [C, B]] = [B, [C, B]] = 0, then exp(A + B) = eA eB e ˆ 2 = hAˆ2 i ( A)
C 2
e
[A,C] 6
ˆ2 hAi
q
J± |j, mi = (Jx ± iJy ) |j, mi = h ¯ j(j + 1) hj j 1 0|j ji =
s
.
j , hj j j+1
m(m ± 1)|j, m ± 1i
1 1 1|j ji =
s
1 j+1
(Note Clebsch-Gordan notation: hj1 j2 m1 m2 |JM i ) Sx =
where x0 =
h ¯ i¯h (|+ih | + | ih+|) , Sy = (|+ih | 2 2
x0 m!x0 x = p (a† + a) , p = i p (a† a) , 2 2 p p † a |ni = n + 1 |n + 1i , a|ni = n |n 1i ,
q
h ¯ /m!. ~ µ ~ˆ · B
ˆ = H
f (k 0 , k) = Z Z 1
1
| ih+|) .
dx e
q
dx 1/x
ax2
=
r
[xi , G(p)] = i¯h
1 2m (2⇡)3 2 hk 0 |V | i 4⇡ h ¯
q
1 = x 1/x ⇡ Z1 , dx x e a 0
1
ax2
@G , [pi , F (x)] = @pi
1 arctan 2 1 Z = , 2a i¯h
q
( 1/x
1 1
2(x
dx(1
1)(2x
1)
1) x2 )3/2 = 3⇡/8
@F ✏ , lim 2 = ⇡ (x) @xi ✏!0 x + ✏2
=
1
Faculty of Science Final Examination Physics 457 Honours Quantum Physics II
Examiner : Professor C. Gale Associate Examiner: Professor S. Jeon Date: Tuesday, April 17th 2012 Time: 9:00 – 12:00
Instructions: This examination comprises five (5) questions. Attempt all questions. No notes or texts allowed; write your answers (with intermediate steps: show your work) in the exam booklet(s). Some useful formulæ are given at the end of the exam and with some of the questions. Translation dictionaries are allowed. Calculators are not allowed. You may keep the exam.
1. The spin Hamiltonian for a spin- 12 particle in an external magnetic field is ˆ = H
~ µ ~ˆ · B
gq ˆ ~ The particle has mass m, charge q, and g is the “g-factor”. Take where µ ~ˆ = 2mc S. ~ = B0~k + B2~j, with B2 ⌧ B0 . Note that ~k and ~j are unit vectors in the z and y B
direction, respectively. (a) Write a matrix representation of the Hamiltonian, using eigenstates of the Sz operator. (b) Determine the energy eigenvalues exactly. (c) Use perturbation theory, to evaluate the energy eigenvalues to second order in B2 /B0 . Compare with the exact solutions.
2. A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state
Physics 457: Honours Quantum Physics II
2
is represented by the position eigenket |Ri(|Li), where spatial variations within each half of the box are neglected. The most general state vector can then be written as |↵i = |RihR|↵i + |LihL|↵i , where hR|↵i and hL|↵i can be regarded as “wave functions”. The particle can tunnel through the partition; this tunneling e↵ect is characterized by the Hamiltonian (|LihR| + |RihL|) ,
H = where
is a real number with the dimension of energy.
(a) Find the normalized energy eigenkets. What are the possible energy eigenvalues? (b) Suppose the system is represented by |↵i as given above at t = 0. Find the state vector |↵(t)i for t > 0.
(c) Suppose at t = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (d) Suppose that the Hamiltonian had been written wrongly as H =
(|LihR|).
By explicitly solving the most general time evolution problem with this Hamiltonian (or otherwise), show that probability conservation is violated.
3. The amplitude for the elastic scattering of a particle of mass m with an original wavevector ~k into a state ~k 0 is f (~k 0 , ~k) =
1 2m (2⇡)3 2 h~k 0 |V | i , 4⇡ h ¯
where V is the local scattering potential, and
is the eigenfunction of the free Hamil-
tonian plus the scattering potential. Consider the repulsive delta-shell potential V (~r) =
(r
R).
(a) Compute the scattering amplitude in the Born approximation.
Physics 457: Honours Quantum Physics II
3
(b) What is the minimum energy necessary for zeroes to appear in the di↵erential cross section? (c) For energies above this minimum, at what angles will the di↵erential cross section vanish? (d) What is the total cross section in the low energy (kR ⌧ 1 ) limit? 4. Consider a particle of mass m being subjected to V (x): a one-dimensional potential well of length L, with infinite walls at x = 0 and x = L. This potential gets modified by a perturbation V1 (x): V (x) =
(
0, 0 < x < L 1, elsewhere
V1 (x) =
(
V0 , 0 x L/2 0, elsewhere
where V0 is small as compared to the other energies in this problem. (a) Using the WKB approximation, calculate En : the energy of level n. (b) Compare your answer in (a) with that obtained using first-order perturbation theory. If these two results are di↵erent, is there a limit where they agree? (c) Evaluate the ground state energy using the variational approximation (with a trial wave function of your choice), and compare with the appropriate results obtained in (a) and (b).
5. Only considering the interaction between the electrons and the nucleus (neglecting the spin-orbit interaction, the mutual interaction between the electrons, etc...) which has charge Z, find the energy levels and write down the two-body wave functions of the three lowest states for a two-electron atom. Use the fact that the energy levels for a Hydrogen-like atom are given by En =
µc2 Z 2 ↵2 2n2
where µ is the reduced mass, and the other symbols have their usual meaning. Take into account the fact that the electrons are identical spin- 12 particles, and just use the notation
x), n`m (~
for the Hydrogen wave function.
Physics 457: Honours Quantum Physics II
4
Hints and Useful Formulæ: For non-commuting operators,A and B, such that [A, B] = C, and [A, [C, A]] [A, [C, B]] = [B, [C, B]] = 0, then exp(A + B) = eA eB e ˆ 2 = hAˆ2 i ( A)
C 2
e
[A,C] 6
ˆ2 hAi
q
J± |j, mi = (Jx ± iJy ) |j, mi = h ¯ j(j + 1) hj j 1 0|j ji =
s
.
j , hj j j+1
m(m ± 1)|j, m ± 1i
1 1 1|j ji =
s
1 j+1
(Note Clebsch-Gordan notation: hj1 j2 m1 m2 |JM i ) Sx =
where x0 =
h ¯ i¯h (|+ih | + | ih+|) , Sy = (|+ih | 2 2
x0 m!x0 x = p (a† + a) , p = i p (a† a) , 2 2 p p † a |ni = n + 1 |n + 1i , a|ni = n |n 1i ,
q
h ¯ /m!. ~ µ ~ˆ · B
ˆ = H
f (k 0 , k) = Z Z 1
1
| ih+|) .
dx e
q
dx 1/x
ax2
=
r
[xi , G(p)] = i¯h
1 2m (2⇡)3 2 hk 0 |V | i 4⇡ h ¯
q
1 = x 1/x ⇡ Z1 , dx x e a 0
1
ax2
@G , [pi , F (x)] = @pi
1 arctan 2 1 Z = , 2a i¯h
q
( 1/x
1 1
2(x
dx(1
1)(2x
1)
1) x2 )3/2 = 3⇡/8
@F ✏ , lim 2 = ⇡ (x) @xi ✏!0 x + ✏2
=
