PH3203: Advanced Quantum Mechanics
Quiz 1
6th January 2023
Name.......................................................................... Roll No............................
Q.1) Suppose a quantum particle is moving in a 1-D Harmonic Oscillator
potential. Let us consider that the particle is in its ground state. Choose the
correct statement (s) from below:
A) ⟨x̂⟩ = 0, (⟨x̂2 ⟩ − ⟨x̂⟩2 ) = 0
B) ⟨x̂⟩ = 0, (⟨x̂2 ⟩ − ⟨x̂⟩2 ) ̸= 0
C) ⟨x̂⟩ =
̸ 0, (⟨x̂2 ⟩ − ⟨x̂⟩2 ) = 0
D) ⟨x̂⟩ =
̸ 0, (⟨x̂2 ⟩ − ⟨x̂⟩2 ) ̸= 0
Correct Option: B)
0
2
2
0
Q.2) What are the eigenvectors and eigenvalues of 2σx =
1
0
A) Eigenvector =
,
; Eigenvalues = {+1, −1}
0
1
1
1
B) Eigenvector =
,
; Eigenvalues = {+2, −2}
1
−1
1
1
C) Eigenvector =
,
; Eigenvalues = {+1, −1}
1
−1
1
0
D) Eigenvector =
,
; Eigenvalues = {+1, 0}
1
1
Correct Option: B)
Q.3) Find the correct statement/statements ?
A) Commutator of two hermitian operators is hermitian
B) Commutator of two anti-hermitian operators is anti-hermitian
C) Commutator of two hermitian operators is anti-hermitian
1
?
D) Commutator of two anti-hermitian operators operator is hermitian
Correct Options: B) and C)
Q. 4) A particle in 1-d (−∞ < x < ∞) is subjected to a potential V (x) = λx
(λ is constant). Now we replace this potential by V (x) = λ|x|. What changes
do we expect to see ?
A) There will be no bound state in either case, but the nature of wave functions
will change.
B) Now the system will have discrete bound state energies.
C) There will be discrete bound states in both cases, but the bound state
energies will change.
D) There will be no change at all.
Correct Option: B)
ˆ
ˆ x î + L
ˆ y ĵ + L
ˆ z k̂ represents the total angular momentum. The
Q.5) ⃗L = L
ˆ
ˆ
value of ⃗L × ⃗L is:
A) 0
ˆx
B) 2iℏL
ˆy
C) 2iℏL
ˆ
D) iℏ⃗L
Correct Option: D)
λσx
Q.6)
Compute e
λ
cosh λ − sinh λ
e
A)
B)
sinh λ cosh λ
0
cosh λ sinh λ
D)
sinh λ cosh λ
0
e−λ
C)
cos λ
sin λ
− sin λ
cos λ
Correct Option: D)
Q.7) A beam of spin- 21 particles goes through a series of Stern-Gerlach type
measurements as follows:
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I ) The 1st measurement accepts sz = + ℏ2 particles, and rejects sz = − ℏ2
particles.
II ) The 2nd measurement accepts sn = + ℏ2 particles and rejects sn = − ℏ2
⃗ with n̂ making an angle β in
particles. sn is the eigenvalue of the operator S.n̂
the x − z plane w.r.t z axis.
III ) The 3rd measurement accepts sz = − ℏ2 particles and rejects sz = + ℏ2
particles.
What is the intensity of the final beam sz = − ℏ2 when the intensity of the
is normalized
sz = + ℏ2 beam surviving the first measurement
to unity ?
0
1
0
−i
0
ℏ
ℏ
ℏ 1
⃗
Note: S = Sx î+Sy ĵ +Sz k̂ ; Sx = 2
; Sy = 2
; Sz = 2
1 0
i 0
0 −1
A) sin2 (β)
B)
1
4
sin2 (β)
C)
1
8
sin2 (β)
D)
1
4
cos2 (β)
Correct Option: B)
Q.8) Which of the following operators are physical measureables ? More
than one option can be correct here.
d
A) i dx
d
B) x dx
C) iσx
D)
d
dp
Correct Option: A)
Q.9) Let X̂ be Hermitian. Which of the following are always unitary ?
A) X̂
C) eX̂
B) iX̂
D) eiX̂
Correct Option: D)
Q.10) The wave function of a particle is given by |ψ⟩ = √12 |ϕ0 ⟩ + i|ϕ1 ⟩,
where |ϕ0 ⟩ and |ϕ1 ⟩ are orthonormal eigenstates of the Hamiltonian (governing
the particle’s dynamics) with eigen-energies E0 and E1 respectively. The expectation value of the Hamiltonian in the state |ψ⟩ is,
A)
E0
2
+ E1
B)
E0
2
− E1
C)
E0 +2E1
3
D)
E0 −2E1
3
Correct Option: C)
Q.11) |ψ⟩ is a quantum state of a two level system. Which of the following
set of statements is generally true for ρ = |ψ⟩⟨ψ| ?
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A) ρ = ρ† , rank = 2, T race(ρ) = 2
C) ρ = ρ† , rank = 2, T race(ρ) = 1
B) ρ2 = ρ, rank= 1, T race(ρ) = 1
D) ρ2 = ρ, rank = 1, T race(ρ) = 2
Correct Option: B)
Q.12) |ψ⟩ =
a|0⟩+b|1⟩+c|2⟩
√
a2 +b2 +c2
be a quantum state of a three level system in the or-
thonormal basis {|0⟩, |1⟩, |2⟩}. Consider the Hermitian operator Ĥ = x|x⟩⟨x| +
y|y⟩⟨y| + z|z⟩⟨z|, where |x⟩ = √16 (|0⟩ + 2|1⟩ + |2⟩), |y⟩ = √13 (|0⟩ − |1⟩ + |2⟩)
and |z⟩ = √12 (|0⟩ − |2⟩). What is the result of measuring Ĥ for particle in
quantum state |ψ⟩ ? The probability of measurement result l, m, n is given by
P (l), P (m), P (n).
A) P (l) =
B)P (l) =
(a−2b+c)2
6(a2 +b2 +c2 ) ,
(a+2b+c)2
6(a2 +b2 +c2 ) ,
C)P (l) = √
P (m) =
P (m) =
(a+b+c)2
3(a2 +b2 +c2 ) ,
(a+b−c)2
3(a2 +b2 +c2 ) ,
a−2b+c
,
6(a2 +b2 +c2 )
P (m) = √
a+2b+c
,
6(a2 +b2 +c2 )
P (m) = √
D)P (l) = √
P (n) =
P (n) =
(a−c)2
2(a2 +b2 +c2 )
(a−c)2
2(a2 +b2 +c2 )
a+b+c
,
3(a2 +b2 +c2 )
P (n) = √
a−c
2(a2 +b2 +c2 )
a+b−c
,
3(a2 +b2 +c2 )
P (n) = √
a−c
2(a2 +b2 +c2 )
Correct Option: A)
Q.13) The component along an arbitrary direction n̂(= nx î + ny ĵ + nz k̂) of the
spin of a spin- 21 particle is measured. The result of the measurement outcome is:
A) 0
B ) ± ℏ2
C ) ± √ℏ2
D ) ± ℏ2 (nx + ny + nz )
Correct Option: B)
Q.14) A spinless particle of mass m is confined in a 3-D cubic box poten2 2
tial of size a. What is the degeneracy of its energy level with energy 7 ℏmaπ2 (in
proper units):
A) 4,
B ) 6,
C ) 3,
D) 5
Correct Option: B)
Q.15) In a system describing 1-D harmonic oscillator with m = 1, ℏ = 1,
4
squeezed states |θ⟩ are constructed by acting on the eigenstates
q with the operi
1
θ F̂
2
†2
†
ator e , with F̂ = (â − â )/2 = 2 (x̂p̂ + p̂x̂), where x̂ =
2ω (â + â ) and
pω †
p̂ = i 2 (â − â). Choose the correct option from below:
A ) eθF̂ x̂e−θF̂ = e−θ x̂; eθF̂ p̂e−θF̂ = eθ p̂
B ) eθF̂ x̂e−θF̂ = e−θ x̂; eθF̂ p̂e−θF̂ = e−θ p̂
C ) eθF̂ x̂e−θF̂ = eθ x̂; eθF̂ p̂e−θF̂ = e−θ p̂
D ) eθF̂ x̂e−θF̂ = eθ x̂; eθF̂ p̂e−θF̂ = eθ p̂
Correct Option: C)
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