Quantum Mechanics 1 | Week 8
Session: JAN-APR 2024
Course name: Quantum Mechanics I
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These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers
Q1. The eigenvalues of the Hamiltonian operator H = (σ.B) will be
a) 0, i|B|
b) 0, |B|
c) +|B|, -|B|
d) i|B|, -i|B|
Here B denotes the magnitude of magnetic field B and o’s are Pauli matrices.
Answer: c) +|B|, -|B|
Q2. The Hamiltonian of a system is H = εδ., where e is a constant having the dimensions of energy, n is an arbitrary unit vector, and στ, σy and are the Pauli matrices. The energy eigen- values of H will be
(a) ±€
(b) 0, €
(c) ±i€
(d) ±€/2
Answer: (a) ±€
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Q3. If a system is in the state (0, 0) = 1,-1(0,0)+1,0 (0,0) – √1,1(0,4), and L₂ is measured then the probabilities of getting the measured values with m = 1 will be
a) 4/7
b) √(2/7)
c) 2/7
d) 1/√7
Answer: c) 2/7
Q4. The number state (0) is the ground-state of the harmonic oscillator which is annihilated by a √(+) operator. The position space wavefunction (2) will be
a) Ae-(x2/B2)
b) Ae-(B2*x2)
c) Ae-(x2/B)
d) Ae-(B*x2)
where A is normalization factor and 3 is a dimensionful constant.
Answer: a) Ae-(x2/B2)
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Q5. The correlation function F(t) = (0x(t)(0)|0) with (t) as the position operator in Heisenberg picture and (0) as ground state of the one-dimensional harmonic oscillator will be
a) (2mw)/h * e-iwt
b) (h)/2mw * eiwt
c) (h)/2mw * e-iwt
d) (4mw)/h * eiwt
Answer: c) (h)/2mw * e-iwt
Q6. Consider a Hamiltonian
H = ħwo (c+c+1/2)
such that operator ĉ is defined by the following relations: ê² = 0, {ê, ĉ+} = I. If the states In) is the eigenstates of H, then the possible energy eigenvalues of this operator will be
a) hw0 when n = 0
b) 3hw0/2 when n = 0
c) hw0 when n = 1
d) 3hw0/2 when n = 1
Answer: a), d)
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Q7. Recall the definition of the spherical harmonics: Yem (0, 0) = (ñl me), where ñ is a unit vector giving the orientation i.e and . If Y1,-1 3e- sin 0, then Y1,0 = 8π 8π (√a cos 0), where a = _________. (Answer should be an integer)
Answer: 2
Q8.
Answer: d)
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Q9. D(a) = elaût-aa) is the displacement operator, where a and at are lowering and raising opera- tors respectively. The quantity D⁺(a) ât D(a) will be
a) â + a
b) â + a*
c) ât – a*
d) â⁺ + a*
Answer: d) â⁺ + a*
Q10. Consider a one-dimensional simple harmonic oscillator. Using the number basis construct a linear combination of (0) and (1) such that (2) is as large as possible.
a) 1/2 (|0> + |1>)
b) 1/√2 (|0> + |1>)
c) 1/√2 (|0> + i|1>)
d) 1/2 (|0> – i|1>)
Answer: b) 1/√2 (|0> + |1>)
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