lOMoARcPSD|54878315
UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
PHY3702
Quantum Physics
Welcome to the PHY3702 examination
Hours: 4 hours
Total marks: 100
Examiner name: Dr M Ramantswana
Internal moderator name: Mr. BO Mnisi
External moderator name: Prof. R Duvenhage (University of Pretoria)
Instructions:
• This is a closed book examination.
• IRIS Invigilation App will be used for this exam.
• Remember to pledge the Honesty Declaration.
•
• Use of a non-programmable pocket calculator is permissible.
• Mark allocation for each question is indicated in brackets to the right.
• Information that you may find useful is given at the end of the paper.
• This examination paper consists of 4 pages and 6 questions.
• The exam is an online take-home on cset.myexams.unisa.ac.za.
Answer ALL questions.
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
2
PHY3702
October/November 2025
1. Calculate the de Broglie wavelength of
(a) An electron of kinetic energy 54 eV (5)
(b) A proton of kinetic energy 70 MeV (5)
(c) A 100 g bullet moving at 1200 ms-1 (5)
[15]
2. (a) Given the matrix representation of the operators A and B:
1 0 0 1 0 0
𝐴𝐴 = �0 −1 0 � , 𝐵𝐵 = �0 0 1�
0 0 −1 0 1 0
(i) Are A and B Hermitian operators? (2)
(ii) Show that [𝐴𝐴, 𝐵𝐵] = 0. (6)
(b) Given the matrix:
1 1 1
𝐴𝐴 = �1 1 1�
1 1 1
Find the eigenvalues and the normalized eigenvectors of 𝐴𝐴. (12)
[20]
3. (a) Prove the following relation:
Lˆz ,sin(2
= ϕ ) 2i sin 2 ϕ − cos 2 ϕ
( )
where ϕ is the azimuthal angle. (5)
(b) Find the expressions for the spherical harmonics Y30 (θ ,ϕ ) and Y3, ±1 (θ ,ϕ ) ,
(i) Y30 (θ ,ϕ )
= 7 16π (5cos3 θ − 3cosθ ) ,
2 ± iϕ
(ii) Y3, ±1 (θ ,ϕ ) 21 64π sin θ (5cos θ − 1)e
=
in terms of the Cartesian coordinates x, y, z. (10)
[15]
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
3
PHY3702
October/November 2025
4. (a) A particle is initially in its ground state in a one-dimensional harmonic oscillator
1 2
potential, Vˆ ( x) = kxˆ . If the spring constant is suddenly doubled, calculate the
2
probability of finding the particle in the ground state of the new potential. (10)
(b) Consider the dimensionless harmonic oscillator Hamiltonian
1 1 d
Hˆ = Pˆ 2 + Xˆ 2 , with Pˆ = −i .
2 2 dx
2 /2 2 /2
(i) Show the two wavefunctions ψ 0 ( x) = e− x and ψ 1 ( x) = xe− x are
1 3
eigenfunctions of Ĥ with eigenvalues and , respectively. (6)
2 2
(ii) Find the value of the coefficient α such that ψ 2 ( x)= (1 + α x 2 )e− x /2 is
2
orthogonal to ψ 0 ( x) and show that ψ 2 ( x) is an eigenfunction of Ĥ with
5
eigenvalue . (4)
2
[20]
5. A proton is confined to moving in a one-dimensional box of width 0.200 nm.
(a) Find the lowest possible energy of the proton. (5)
(b) What is the lowest possible energy of an electron confined to the same box? (3)
(c) How do you account for the large difference in your results for (a) and (b)? (2)
[10]
6. Consider a system of four non-interacting identical spin ½ particles (each of mass
m ) that are confined to move in a one-dimensional infinite potential well of length
a : V ( x) = 0 for 0 < x < a and V ( x) = ∞ for other values of x . Determine the energies
and wave functions of the ground state and the first three excited states. Draw a
figure showing how the particles are distributed among the levels. [20]
[TOTAL = 100]
______________________________________________________________________
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
4
PHY3702
October/November 2025
USEFUL INFORMATION
−34
Planck ’s constant=h 6.63 × 10 J.s
−10
=
Bohr radius a0 0.529 × 10 m
∞ n − ax n! a
∫ x e dx = ( n ≥ 0 and a > 0 ) Xˆ =
0 a n +1 n 2
a2 a2
x = r sin θ cosϕ Xˆ 2 = − 2 2
n 3 2n π
2π 2 2
y = r sin θ sin ϕ En = n for n = 1,2,3...
2ma 2
z = r cosθ
CONVERSION OF UNITS
1 fm =10-15 kT = [38.682]-1 eV
1 barn = 10-28 m2 1 eV = 1.6 𝚡𝚡 10-19 J
1 G = 10-4 T
2 2
∆r
= Rˆ 2 − Rˆ Pr
∆= Pˆr 2 − Pˆr
© Unisa 2025
[TURN OVER]
Downloaded by Stephen ()
UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
PHY3702
Quantum Physics
Welcome to the PHY3702 examination
Hours: 4 hours
Total marks: 100
Examiner name: Dr M Ramantswana
Internal moderator name: Mr. BO Mnisi
External moderator name: Prof. R Duvenhage (University of Pretoria)
Instructions:
• This is a closed book examination.
• IRIS Invigilation App will be used for this exam.
• Remember to pledge the Honesty Declaration.
•
• Use of a non-programmable pocket calculator is permissible.
• Mark allocation for each question is indicated in brackets to the right.
• Information that you may find useful is given at the end of the paper.
• This examination paper consists of 4 pages and 6 questions.
• The exam is an online take-home on cset.myexams.unisa.ac.za.
Answer ALL questions.
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
2
PHY3702
October/November 2025
1. Calculate the de Broglie wavelength of
(a) An electron of kinetic energy 54 eV (5)
(b) A proton of kinetic energy 70 MeV (5)
(c) A 100 g bullet moving at 1200 ms-1 (5)
[15]
2. (a) Given the matrix representation of the operators A and B:
1 0 0 1 0 0
𝐴𝐴 = �0 −1 0 � , 𝐵𝐵 = �0 0 1�
0 0 −1 0 1 0
(i) Are A and B Hermitian operators? (2)
(ii) Show that [𝐴𝐴, 𝐵𝐵] = 0. (6)
(b) Given the matrix:
1 1 1
𝐴𝐴 = �1 1 1�
1 1 1
Find the eigenvalues and the normalized eigenvectors of 𝐴𝐴. (12)
[20]
3. (a) Prove the following relation:
Lˆz ,sin(2
= ϕ ) 2i sin 2 ϕ − cos 2 ϕ
( )
where ϕ is the azimuthal angle. (5)
(b) Find the expressions for the spherical harmonics Y30 (θ ,ϕ ) and Y3, ±1 (θ ,ϕ ) ,
(i) Y30 (θ ,ϕ )
= 7 16π (5cos3 θ − 3cosθ ) ,
2 ± iϕ
(ii) Y3, ±1 (θ ,ϕ ) 21 64π sin θ (5cos θ − 1)e
=
in terms of the Cartesian coordinates x, y, z. (10)
[15]
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
3
PHY3702
October/November 2025
4. (a) A particle is initially in its ground state in a one-dimensional harmonic oscillator
1 2
potential, Vˆ ( x) = kxˆ . If the spring constant is suddenly doubled, calculate the
2
probability of finding the particle in the ground state of the new potential. (10)
(b) Consider the dimensionless harmonic oscillator Hamiltonian
1 1 d
Hˆ = Pˆ 2 + Xˆ 2 , with Pˆ = −i .
2 2 dx
2 /2 2 /2
(i) Show the two wavefunctions ψ 0 ( x) = e− x and ψ 1 ( x) = xe− x are
1 3
eigenfunctions of Ĥ with eigenvalues and , respectively. (6)
2 2
(ii) Find the value of the coefficient α such that ψ 2 ( x)= (1 + α x 2 )e− x /2 is
2
orthogonal to ψ 0 ( x) and show that ψ 2 ( x) is an eigenfunction of Ĥ with
5
eigenvalue . (4)
2
[20]
5. A proton is confined to moving in a one-dimensional box of width 0.200 nm.
(a) Find the lowest possible energy of the proton. (5)
(b) What is the lowest possible energy of an electron confined to the same box? (3)
(c) How do you account for the large difference in your results for (a) and (b)? (2)
[10]
6. Consider a system of four non-interacting identical spin ½ particles (each of mass
m ) that are confined to move in a one-dimensional infinite potential well of length
a : V ( x) = 0 for 0 < x < a and V ( x) = ∞ for other values of x . Determine the energies
and wave functions of the ground state and the first three excited states. Draw a
figure showing how the particles are distributed among the levels. [20]
[TOTAL = 100]
______________________________________________________________________
[TURN OVER]
Downloaded by Stephen ()
, lOMoARcPSD|54878315
4
PHY3702
October/November 2025
USEFUL INFORMATION
−34
Planck ’s constant=h 6.63 × 10 J.s
−10
=
Bohr radius a0 0.529 × 10 m
∞ n − ax n! a
∫ x e dx = ( n ≥ 0 and a > 0 ) Xˆ =
0 a n +1 n 2
a2 a2
x = r sin θ cosϕ Xˆ 2 = − 2 2
n 3 2n π
2π 2 2
y = r sin θ sin ϕ En = n for n = 1,2,3...
2ma 2
z = r cosθ
CONVERSION OF UNITS
1 fm =10-15 kT = [38.682]-1 eV
1 barn = 10-28 m2 1 eV = 1.6 𝚡𝚡 10-19 J
1 G = 10-4 T
2 2
∆r
= Rˆ 2 − Rˆ Pr
∆= Pˆr 2 − Pˆr
© Unisa 2025
[TURN OVER]
Downloaded by Stephen ()
