, Question 1
A proton is accelerated to a kinetic energy equal to its rest energy. What is the
ratio of its relativistic momentum to its nonrelativistic momentum calculated
using the classical formula p = mv, where v is the relativistic speed?
A. 1
B. 2
C. 3
D. 2
Correct Answer: C - 3
RATIONALE
For K = mc², = 2, so v = (3/2)c. Relativistic momentum p = mv =
2m(3/2 c) = 3 mc. Classical momentum using the same v is mv =
m(3/2 c). Ratio = (3 mc) / (3/2 mc) = 2. Wait: compute carefully: p_rel
= m v = 2 m (3/2 c) = 3 m c. p_class = m v = m (3/2 c). Ratio = (3 m
c)/(3/2 m c) = 2. Actually ratio = 2, not 3. Re-evaluate: = 2, v/c = (1 -
1/²) = (1 - 1/4) = (3/4) = 3/2. p_rel = m v = 2 m (3/2 c) = 3 m c.
p_class = m v = m (3/2 c). Ratio = (3)/(3/2) = 2. So correct answer is
D (2). Correction: The ratio is 2, so correct option is D. Explanation:
For K = mc², = 2, = 3/2. Relativistic momentum p = mv = 3 mc;
classical momentum with same v is mv = (3/2)mc; ratio = 2. Therefore
D is correct; A, B, C are incorrect.
Question 2
An electron is confined in a one-dimensional infinite square well of width L. If
the well width is suddenly doubled (L -> 2L) while the electron remains in the
ground state of the original well, what is the probability that the electron is
found in the ground state of the new well?
A. 1/2
B. 8/(3)²
C. 16/(3)²
D. 1/4
Page 2
,Correct Answer: B - 8/(3)²
RATIONALE
The overlap integral between the original ground state (x) = (2/L)
sin(x/L) on [0,L] and the new ground state (x) = (1/L) sin(x/(2L)) on
[0,2L] yields coefficient magnitude squared = 8/(3)². This is a
standard sudden approximation result; the other options arise from
incorrect normalization or integration limits.
Question 3
Which of the following transitions in a hydrogen atom is forbidden by the
electric dipole selection rules?
A. 3d -> 2p
B. 4s -> 3p
C. 3p -> 1s
D. 2s -> 1s
Correct Answer: D - 2s -> 1s
RATIONALE
Electric dipole transitions require l = ±1 and m_l = 0, ±1. The 2s -> 1s
transition has l = 0, so it is forbidden (though it can occur via
two-photon emission). The other transitions satisfy l = ±1 and are
allowed.
Question 4
The binding energy per nucleon curve peaks near A 56 (iron). Which
statement best explains why fusion of light nuclei and fission of heavy nuclei
both release energy?
A. Both processes increase the total number of nucleons, releasing energy
via mass defect.
B. Both processes move the system toward the maximum binding energy
per nucleon, increasing total binding energy and reducing rest mass.
Page 3
, C. Both processes decrease the total binding energy, converting it into
kinetic energy of fragments.
D. Both processes involve the weak interaction, which converts neutrons
to protons and releases energy.
Correct Answer: B - Both processes move the system toward the
maximum binding energy per nucleon, increasing total binding
energy and reducing rest mass.
RATIONALE
Energy release in nuclear reactions occurs when the total binding
energy of products exceeds that of reactants, corresponding to a
decrease in rest mass (E = m c²). Both fusion of light nuclei and
fission of heavy nuclei move toward the iron peak, increasing binding
energy per nucleon. The other options misstate the mechanism or
incorrectly invoke the weak interaction.
Question 5
For a system of N non-interacting spin-1/2 fermions in a three-dimensional box
at T = 0 K, the Fermi energy is _F. If the volume is adiabatically compressed to
half its original value, what is the new Fermi energy?
A. _F / 2
B. _F (2)^(1/3)
C. _F (2)^(2/3)
D. 2 _F
Correct Answer: C - _F (2)^(2/3)
RATIONALE
Fermi energy scales as _F n^(2/3) where n = N/V is the number
density. Halving the volume doubles n, so _F' = _F (2)^(2/3). The
other options arise from incorrect scaling exponents or assuming
linear dependence on volume.
Page 4
A proton is accelerated to a kinetic energy equal to its rest energy. What is the
ratio of its relativistic momentum to its nonrelativistic momentum calculated
using the classical formula p = mv, where v is the relativistic speed?
A. 1
B. 2
C. 3
D. 2
Correct Answer: C - 3
RATIONALE
For K = mc², = 2, so v = (3/2)c. Relativistic momentum p = mv =
2m(3/2 c) = 3 mc. Classical momentum using the same v is mv =
m(3/2 c). Ratio = (3 mc) / (3/2 mc) = 2. Wait: compute carefully: p_rel
= m v = 2 m (3/2 c) = 3 m c. p_class = m v = m (3/2 c). Ratio = (3 m
c)/(3/2 m c) = 2. Actually ratio = 2, not 3. Re-evaluate: = 2, v/c = (1 -
1/²) = (1 - 1/4) = (3/4) = 3/2. p_rel = m v = 2 m (3/2 c) = 3 m c.
p_class = m v = m (3/2 c). Ratio = (3)/(3/2) = 2. So correct answer is
D (2). Correction: The ratio is 2, so correct option is D. Explanation:
For K = mc², = 2, = 3/2. Relativistic momentum p = mv = 3 mc;
classical momentum with same v is mv = (3/2)mc; ratio = 2. Therefore
D is correct; A, B, C are incorrect.
Question 2
An electron is confined in a one-dimensional infinite square well of width L. If
the well width is suddenly doubled (L -> 2L) while the electron remains in the
ground state of the original well, what is the probability that the electron is
found in the ground state of the new well?
A. 1/2
B. 8/(3)²
C. 16/(3)²
D. 1/4
Page 2
,Correct Answer: B - 8/(3)²
RATIONALE
The overlap integral between the original ground state (x) = (2/L)
sin(x/L) on [0,L] and the new ground state (x) = (1/L) sin(x/(2L)) on
[0,2L] yields coefficient magnitude squared = 8/(3)². This is a
standard sudden approximation result; the other options arise from
incorrect normalization or integration limits.
Question 3
Which of the following transitions in a hydrogen atom is forbidden by the
electric dipole selection rules?
A. 3d -> 2p
B. 4s -> 3p
C. 3p -> 1s
D. 2s -> 1s
Correct Answer: D - 2s -> 1s
RATIONALE
Electric dipole transitions require l = ±1 and m_l = 0, ±1. The 2s -> 1s
transition has l = 0, so it is forbidden (though it can occur via
two-photon emission). The other transitions satisfy l = ±1 and are
allowed.
Question 4
The binding energy per nucleon curve peaks near A 56 (iron). Which
statement best explains why fusion of light nuclei and fission of heavy nuclei
both release energy?
A. Both processes increase the total number of nucleons, releasing energy
via mass defect.
B. Both processes move the system toward the maximum binding energy
per nucleon, increasing total binding energy and reducing rest mass.
Page 3
, C. Both processes decrease the total binding energy, converting it into
kinetic energy of fragments.
D. Both processes involve the weak interaction, which converts neutrons
to protons and releases energy.
Correct Answer: B - Both processes move the system toward the
maximum binding energy per nucleon, increasing total binding
energy and reducing rest mass.
RATIONALE
Energy release in nuclear reactions occurs when the total binding
energy of products exceeds that of reactants, corresponding to a
decrease in rest mass (E = m c²). Both fusion of light nuclei and
fission of heavy nuclei move toward the iron peak, increasing binding
energy per nucleon. The other options misstate the mechanism or
incorrectly invoke the weak interaction.
Question 5
For a system of N non-interacting spin-1/2 fermions in a three-dimensional box
at T = 0 K, the Fermi energy is _F. If the volume is adiabatically compressed to
half its original value, what is the new Fermi energy?
A. _F / 2
B. _F (2)^(1/3)
C. _F (2)^(2/3)
D. 2 _F
Correct Answer: C - _F (2)^(2/3)
RATIONALE
Fermi energy scales as _F n^(2/3) where n = N/V is the number
density. Halving the volume doubles n, so _F' = _F (2)^(2/3). The
other options arise from incorrect scaling exponents or assuming
linear dependence on volume.
Page 4
