A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the average energy for the given quantum state?

A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the standard deviation in energy for the given quantum state?

A

σE = ((|n1|2E1 + |n2|2E2) – (|n1|2E1 + |n2|2E2)2)1/2

B

σE = ((E1 + E2) – (E1 + E2)2)1/2

C

σE = ((|n1|2E1 + |n2|2E2) – (|n1|2E1 + |n2|2E2)2)1/2 + ((|n1|2E1 + |n2|2E2))1/2

D

σE = (|n1|2E1 + |n2|2E2)2

A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the probability of measuring E2?

A

n1

B

|n1|2

C

|n2|2

D

n2

A particle of mass m is confined in the potential
Quantum Mechanics mcq question image

\[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x Let the wave function of the particle be given by $$\psi \left( x \right) = - \frac{1}{{\sqrt 5 }}{\psi _0} + \frac{2}{{\sqrt 5 }}{\psi _1}$$
where $${\psi _0}$$ and $${\psi _1}$$ are the eigen functions of the ground state and the first excited slate respectively. The expectation value of the energy is

A

$$\frac{{31}}{{10}}\hbar \omega $$

B

$$\frac{{25}}{{10}}\hbar \omega $$

C

$$\frac{{13}}{{10}}\hbar \omega $$

D

$$\frac{{11}}{{10}}\hbar \omega $$

A particle with energy E is in a time independent double well potential as shown in the figure. Which, of the following statements about the particle is not correct?
Quantum Mechanics mcq question image

A

The particle will always be in a bound state

B

The probability of finding the particle in one well will be time dependent

C

The particle will be confined to anyone of the wells

D

The particle can tunnel from one well to the other and back

Which of the following statements concerning the quantum numbers is correct? (i) The Angular quantum number determines the three-dimensional shape of the orbital. (ii) The Principal quantum number determines the orientation and energy of the orbital. (iii) The Magnetic quantum number determines the size of the orbital. (iv) Spin quantum number of an electron determines the orientation of the spin of the electron relative to the chosen axis.

A

ii and iv

B

i and iv

C

i and iii

D

ii and iii

A particle of mass m is confined in an infinite potential well \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{if }}0 It is subjected to a perturbing potential $${V_P}\left( x \right) = {V_0}\sin \left( {\frac{{2\pi x}}{L}} \right)$$ within the well. Let E⁽¹⁾ and E⁽²⁾ be the corrections to the ground state energy in the first and second order in V₀.
Quantum Mechanics mcq question image

Which of the following is correct?

A

[

A. E⁽¹⁾ = 0; E⁽²⁾

B. E⁽¹⁾ > 0; E⁽²⁾ = 0

B

E⁽¹⁾ = 0; E⁽²⁾ depends on the sign of V₀

C

E⁽¹⁾ (2)

A particle is in the normalized state $$\left| \psi \right\rangle $$ which is a superposition of the energy eigen states $$\left| {{E_0} = 10\,eV} \right\rangle $$ and $$\left| {{E_1} = 30\,eV} \right\rangle .$$ The average value of energy of the particle in the state $$\left| \psi \right\rangle $$ is 20 eV. The state $$\left| \psi \right\rangle $$ is given by

A

$$\frac{1}{2}\left| {{E_0} = 10\,eV} \right\rangle + \frac{{\sqrt 3 }}{4}\left| {{E_1} = 30\,eV} \right\rangle $$

B

$$\frac{1}{{\sqrt 3 }}\left| {{E_0} = 10\,eV} \right\rangle + \sqrt {\frac{2}{3}} \left| {{E_1} = 30\,eV} \right\rangle $$

C

$$\frac{1}{2}\left| {{E_0} = 10\,eV} \right\rangle - \frac{{\sqrt 3 }}{4}\left| {{E_1} = 30\,eV} \right\rangle $$

D

$$\frac{1}{{\sqrt 2 }}\left| {{E_0} = 10\,eV} \right\rangle - \frac{1}{{\sqrt 2 }}\left| {{E_1} = 30\,eV} \right\rangle $$

A particle is incident with a constant energy E on a one-dimensional potential barrier as shown in the figure.
Quantum Mechanics mcq question image

The wave functions in regions I and II are respectively

A

decaying, oscillatory

B

oscillatory, oscillatory

C

oscillatory, decaying

D

decaying, decaying

Questions below followed by three quantities. Find the relationship between them. Quantity I: In a class with a certain number of students if a teacher weighing 78 kg is added, then average weight of class is increased by 2 kg. If one more teacher weighing 66 kg is added, then the average weight of the class increases by 3.5 kg over the original average. What is the original average weight (in kg) of the class? Quantity II: The average age of 16 students and their teacher's age is 16 years. If the teacher's age is excluded, the average reduces by 2. What is the teacher's age? Quantity III: Average weight of three friends A, B and C is 46 kg. Another person D joins the group and now the average is 45 kg. If another person E whose weight is 6 kg more than D, joins the group replacing A, then average weight of B, C, D and E becomes 47 kg. What is the weight of A (in kg)?

A

Quantity I > Quantity II < Quantity III

B

Quantity I > Quantity II > Quantity III

C

Quantity I < Quantity II > Quantity III

D

Quantity I < Quantity II < Quantity III

E

Quantity I = Quantity II = Quantity III

In a company there are 252 engineers, in which the ratio of the number of electronics engineers and computer engineers is 2 ∶ 1. The ratio of the number of electronics engineers and computer engineers becomes 1 ∶ 1 after recruitment of some more computer engineers. The average age of all the engineers is now 22 years and the average age of the computer engineers is 2 years less than the average age of electronics engineers. (A) The average age (in years) of electronics engineers is 21 (B) The average age (in years) of computer engineers is 21 (C) The average age (in years) of electronics engineers is 23 (D) The average age (in years) of electronics engineers is 25 (E) The average age (in years) of computer engineers is 23 Choose the correct answer from the options given below:

A

(A) and (E) only

B

(B) and (C) only

C

(D) and (E) only

D

(E)only

A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the average energy for the given quantum state?

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