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 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

= |n1|2E1 + |n2|2E2

B

= |n1|2 + |n2|2

C

= $\frac{E1 + E2}{2}$

D

= E1 + E2

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 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 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

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

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

World Bank has employed major dimensions to construct measures of governance. These dimensions are:
1. Voice and Accountability - measuring political, civil and human rights
2. Instability an d violence - measuring the likelihood of violent threats to or changes in government, including terrorism
3. Government effectiveness - measuring the competence of the bureaucracy and the quality of public service delivery
4. Regulatory burden - measuring the incidence of market, unfriendly policies
5. Rule of law - measuring the quality of contract enforcement, the police, and the courts, as well as the likelihood of crime and violence
Select the correct answer:

A

1, 2 and 3

B

3, 4 and 5

C

1, 2, 3 and 5

D

1, 2, 3, 4 and 5

There are only three bound states for a particle of mass m in a one-dimensional potential well of the form shown in the figure. The depth V₀ of the potential satisfies
Quantum Mechanics mcq question image

A

[

A. $$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

B. $$\frac{{{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

B

$$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

C

$$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^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?

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