# 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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**These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers**

**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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**These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers**

**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 * e^{iwt}

c) (h)/2mw * e^{-iwt}

d) (4mw)/h * e^{iwt}

**Answer: c) (h)/2mw * e ^{-iwt}**

**Q6. Consider a HamiltonianH = 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) hw

_{0}when n = 0

b) 3hw

_{0}/2 when n = 0

c) hw

_{0}when n = 1

d) 3hw

_{0}/2 when n = 1

**Answer: a), d)**

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**These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers**

**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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**These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers**

**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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**These are Quantum Mechanics 1 Week 8 Assignment Nptel Answers**

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