This set of Engineering Physics Multiple Choice Questions & Answers (MCQs) focuses on “Quantum Nature of Elastic Waves”.

1. Elastic waves in crystals are made up of _______________

a) Photons

b) Nano particles

c) Atoms

d) Phonons

View Answer

Explanation: In a crystal, when energy is provided, the lattice absorbs energy and gets excited to a higher state. When it de-excites to ground state, it releases radiation in sound-wave region, known as phonons. These waves are quantized.

2. The energy of elastic waves is given by _____________

a) nω

b) n ħω

c) nħω/2

d) (n + ^{1}⁄_{2})ħω

View Answer

Explanation: The energy of a lattice vibration is quantized which is given by the expression: E = (n + ½)ħω, where w is the angular frequency of the waves.

3. Waves in the wave packets are such that they interfere constructively over a large region of space.

a) True

b) False

View Answer

Explanation: A wave packet consists of a group of waves, each of which have phases and amplitudes such that they interfere constructively over a small region of space where the particle can be located. Outside that region, they interfere destructively.

4. Which of the following is the correct expression for the group velocity?

a) υλ

b) dω/dυ

c) dE/dk

d) dE/ħdk

View Answer

Explanation: The velocity with which the wave packet moves is called the group velocity. It is equal to dω/dk, which can be further simplified to dE/ħdk, as E =ħω.

5. Which of the following is the correct relation between the group velocity and the phase velocity?

a) v_{g} = v_{p} + λdv_{p}/dλ

b) v_{p} = v_{g} + λdv_{p}/dλ

c) v_{p} = v_{g} – λdv_{p}/dλ

d) v_{g} = v_{p} – λdv_{p}/dλ

View Answer

Explanation: The relation between the phase velocity and the group velocity is given by: v

_{g}= v

_{p}– λdv

_{p}/dλ. Thus, when dv

_{p}/dλ = 0, v

_{g}= v

_{p}.

6. The motion of a wave packet is similar to _____________

a) Photons

b) Waves

c) Classical Particle

d) Quantum Particle

View Answer

Explanation: The motion of a material/classical particle is equivalent to a group of waves or a wave packet. Such an expression can be derived by using the basic principles of motion.

7. Which of the following is not a characteristic of wave function?

a) Continuous

b) Single valued

c) Differentiable

d) Physically Significant

View Answer

Explanation: The wave function has no physical significance. It merely helps in determining the state of a particle. It is the square of the wave function that has a physical significance.

8. There is an inextensible string of linear density µ. If it is given that ω = \(k\sqrt{\frac{T}{μ}}\), then what is the group velocity of the string?

a) \(\sqrt{\frac{T}{μ}}\)

b) ħ\(\sqrt{\frac{T}{μ}}\)

c) \(\frac{\hbar}{2}\sqrt{\frac{T}{μ}}\)

d) \(\frac{1}{\hbar}\sqrt{\frac{T}{μ}}\)

View Answer

Explanation: As we know, v

_{g}= dω/dk

Therefore, v

_{g}= d( k\(\sqrt{\frac{T}{μ}}\))/dk

= \(\sqrt{\frac{T}{μ}}\).

9. Ψ must be normalizable.

a) True

b) False

View Answer

Explanation: Ψ must fo to 0 as x-> infinity, y-> infinity and z -> infinity in order that ∫|Ψ|

^{2}dv all over space be a finite constant. Thus, Ψ must be normalizable.

10. Calculate the minimum uncertainty in the momentum of a 4He atom confined to 0.40 nm.

a) 2.02 X 10^{-25} kg m/s

b) 2.53 X 10^{-25} kg m/s

c) 2.64 X 10^{-25} kg m/s

d) 2.89 X 10^{-25} kg m/s

View Answer

Explanation: We know that 4He atom is somewhere in the 0.40 nm region, therefore, Δx = 0.40 nm.

Using, Δp

_{x}≥ \(\frac{\hbar}{\Delta x}\)

For minimum uncertainty, Δp

_{x}= 6.626 X 10

^{-34}Js/2π X 0.40 X 10

^{-9}

= 2.64 X 10

^{-25}kg m/s.

11. The following is a possible wave function.

a) True

b) False

View Answer

Explanation: The following cannot be the graph of a wave function as the graph is discontinuous. A wave function should be continuous and single valued.

12. The uncertainty in the location of a particle moving with velocity 7.28 X 10^{7}m /s is two times its de-Broglie wavelength. What is the uncertainty in measuring the velocity?

a) 5.79 X 10^{6} m/s

b) 6.12 X 10^{6} m/s

c) 7.63 X 10^{6} m/s

d) 8.45 X 10^{6} m/s

View Answer

Explanation: De-Broglie wavelength, λ = h/mv

Given: Δx = 2λ = 2h/mv

Now, Δx.m.Δv = ħ

Δv = v/4π

= 5.79 X 10

^{6}m/s.

13. Energy of a wave divided by its momentum gives _____________

a) Group velocity

b) Classical Velocity

c) Phase Velocity

d) Wave velocity

View Answer

Explanation: We know, E = ħω and P = ħk. Therefore, E/P gives us w/k which is the phase velocity of the wave. Hence, Energy of a wave divided by its momentum gives us phase velocity.

14. Which of the following can be a wave function?

a) tan x

b) sin x

c) cot x

d) sec x

View Answer

Explanation: Out of all the given options, sin x is the only function, that is continuous and single-valued. All the rest of the functions are either discontinuous or double-valued.

15. At what condition, v_{p} = v_{g}?

a) dv_{p}/dk = 0

b) dv_{p}/dω = 0

c) dv_{p}/dλ = 0

d) dv_{p}/dμ= 0

View Answer

Explanation: As we know, the relation between the phase and group velocity is v

_{g}= v

_{p}– λdv

_{p}/dλ. Thus, as dv

_{p}/dλ = 0, v

_{p}= v

_{g}.

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