Quantum Mechanics — Part 2 MCQs with Explanations

$$\left( {n + \frac{1}{2}} \right)\hbar \omega $$ is the correct answer to “The energy levels of a particle of mass m in a potential of the form \[\begin{gathered} V\left( x \right) = \infty ,\,\,\,\,\,\,\,\,\,\,\,\,x \leqslant 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}m{\omega ^2}{x^2},\,\,\,x > 0 \hfill \\ \end{gathered} \] are given, in terms of quantum number n = 0, 1, 2, 3, . . ., by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$ - i\hbar \frac{d}{{dx}}$$ is the correct answer to “The quantum mechanical operator for the momentum of a particle moving in one dimension is given by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$${\bf{\hat B}}$$ is Hermitian and $${\bf{\hat A}}$$ and $${\bf{\hat B}}$$ commute is the correct answer to “$${\bf{\hat A}}$$ and $${\bf{\hat B}}$$ represent two physical characteristics of a quantum system. If $${\bf{\hat A}}$$ is Hermitian, then for the product $${\bf{\hat A\hat B}}$$ to be Hermitian, it is sufficient that”. This question is standard in competitive examination syllabi for Quantum Mechanics.

E(1) = 0; E(2) depends on the sign of V0 is the correct answer to “A particle of mass m is confined in an infinite potential well \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{if }}0 < x < L} \\ {\infty ,}&{{\text{otherwise}}} \end{array}} \right.\] 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 V 0 . Which of the following is correct”. This question is standard in competitive examination syllabi for Quantum Mechanics.

oscillatory, decaying is the correct answer to “A particle is incident with a constant energy E on a one-dimensional potential barrier as shown in the figure. The wave functions in regions I and II are respectively”. This question is standard in competitive examination syllabi for Quantum Mechanics.

A0e-2x is the correct answer to “Which one of the functions given below represents the bound state eigen function of the operator $$ - \frac{{{d^2}}}{{d{x^2}}}$$ in the region, 0 ≤ x < $$\infty $$ , with the eigen value -4”. This question is standard in competitive examination syllabi for Quantum Mechanics.

6 is the correct answer to “The degeneracy of the n = 2 level for a three-dimensional isotropic oscillator is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

Infinitely many is the correct answer to “A parallel beam of infrared radiation of wavelength of 1.01 × 10 -6 m is incident normally on a screen with two slits 5 × 10 -6 m apart and the resulting interference pattern is observed on a distant screen. What is the largest number of maxima that can be observed on the screen”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{2}{L}\sin \left( {\frac{{m\pi x}}{L}} \right)\sin \left( {\frac{{n\pi y}}{L}} \right)$$ is the correct answer to “A quantum particle of mass m is confined to a square region in XOY-plane whose vertices are given by (0, 0), (L, 0), (L, L) and (0, L). Which of the following represents an admissible wave function of the particle (for I, m, n positive integers)”. This question is standard in competitive examination syllabi for Quantum Mechanics.

This is an eigen state of both energy and momentum is the correct answer to “A particle has the wave function $$\psi $$ (x, t) = A[exp (iωt) cos (kx)]. Which one of the following is correct”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\sigma = \frac{{4\pi }}{k} \times {\text{Imaginary part of }}{\left| {f\left( {\theta = 0} \right)} \right|^2}$$ is the correct answer to “If $$\sigma $$ is the total cross-section and f(θ), θ being the angle of scattering, is the scattering amplitude for a quantum mechanical elastic scattering by a spherically symmetric potential, then which of the following is true? Note that k is the magnitude of the wave vector along the $${{\bf{\hat z}}}$$ direction.”. This question is standard in competitive examination syllabi for Quantum Mechanics.

σxσy = -σyσx is the correct answer to “Which one of the following relations is true for Pauli matrices σ x , σ y and σ z”. This question is standard in competitive examination syllabi for Quantum Mechanics.

Det (σi) = 1 is the correct answer to “σ i (i = 1, 2, 3) represents the Pauli spin matrices. Which one of the following is not true”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\left\langle T \right\rangle = 2\left\langle V \right\rangle $$ is the correct answer to “A particle is moving in a spherically symmetric potential V(r) = αr 2 , where α is a positive constant. In a stationary state, the expectation value of the kinetic energy $$\left\langle T \right\rangle $$ of the particle is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$i\hbar z$$ is the correct answer to “The commutator [L x , y], where L x is the x-component of the angular momentum operator and y is the y-component of the position operator, is equal to”. This question is standard in competitive examination syllabi for Quantum Mechanics.

u2(x)v is the correct answer to “A beam of mono-energetic particles having speed v is described by the wave function $$\psi $$ (x) = u(x) exp(ikx), where u(x) is a real function. This corresponds to a current density”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}} < {V_0} < \frac{{8{\pi ^2}{\hbar ^2}}}{{m{a^2}}}$$ is the correct answer to “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 V 0 of the potential satisfies”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$${\left| {{c_1}} \right|^2} + {\left| {{c_2}} \right|^2} = 1$$ is the correct answer to “A system in a normalized state $$\left| \psi \right\rangle = {c_1}\left| {{\alpha _1}} \right\rangle + {c_2}\left| {{\alpha _2}} \right\rangle $$ with $$\left| {{\alpha _1}} \right\rangle $$ and $$\left| {{\alpha _2}} \right\rangle $$ representing two different eigen states of the system requires that the constants c 1 and c 2 must satisfy the condition”. This question is standard in competitive examination syllabi for Quantum Mechanics.

16 is the correct answer to “In a hydrogen atom, the accidental or Coulomb degeneracy for the n = 4 state is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$${\left| A \right|^2}\frac{{\hbar k}}{m}$$ is the correct answer to “A particle of mass m is represented by the wave function $$\psi \left( x \right) = A{e^{ikx}},$$ where k is the wave vector and A is a constant. The magnitude of the probability current density of the particle is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

px, py, Lz, E is the correct answer to “A classical particle is moving in an external potential field V(x, y, z) which is invariant under the following infinitesimal transformations \[\begin{array}{*{20}{c}} {x \to x'}& = &{x + \delta x} \\ {y \to y'}& = &{y + \delta y} \\ {\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] \to \left[ {\begin{array}{*{20}{c}} {x'} \\ {y'} \end{array}} \right]}& = &{{R_z}\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right]} \end{array}\] where, R z is the matrix corresponding to rotation about the Z-axis. The conserved quantities are (the symbols have their usual meaning)”. This question is standard in competitive examination syllabi for Quantum Mechanics.

10-6 is the correct answer to “An atom emits a photon of wavelength λ = 600 nm by transition from an excited state of life time 8 × 10 -9 s. If $$\Delta \nu $$ represents the minimum uncertainty in the frequency of the photon, the fractional width $$\frac{{\Delta \nu }}{\nu }$$ of the spectral line is of the order of”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{13}}{{10}}\hbar \omega $$ is the correct answer to “A particle of mass m is confined in the potential \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x < 0} \\ {\infty ,}&{{\text{for }}x \leqslant 0} \end{array}} \right.\] 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”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{2}{{{a^2}}}$$ is the correct answer to “The normalized ground state, wave function of a hydrogen atom is given $$\psi \left( r \right) = \frac{1}{{\sqrt {4\pi } }}\frac{2}{{{a^{\frac{3}{2}}}}} - {e^{ - \frac{r}{a}}},$$ where a is the Bohr radius and r is the distance of the electron from the nucleus located at the origin. The expectation value $$\left\langle {\frac{1}{{{r^2}}}} \right\rangle $$ is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{1}{{\sqrt L }}\sin \frac{p}{\hbar }x$$ is the correct answer to “A free particle is moving in +X direction with a linear momentum p. The wave function of the particle normalised in a length L is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\left( {\frac{3}{4}{E_0}} \right){10^{ - 4}}$$ is the correct answer to “The wave function of a one-dimensional harmonic oscillator is $${\psi _0} = A\exp \left( {\frac{{ - {\alpha ^2}{x^2}}}{2}} \right)$$ for the ground state $${E_0} = \frac{{\hbar \omega }}{2},$$ where $${\alpha ^2} = \frac{{m\omega }}{\hbar }$$ in the presence of a perturbing potential of $${E_0}{\left( {\frac{{\alpha x}}{{10}}} \right)^4},$$ the first order Change in the ground state energy is $$\left[ {{\text{Given, }}\Gamma \left( {x + 1} \right) = \int_0^\infty {{t^x}\exp \left( { - t} \right)dt} } \right]$$”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$ - \frac{{i{\hbar ^3}}}{8}$$ is the correct answer to “For a spin $$\frac{1}{2}$$ particle, the expectation value of s x s y s z , where s x , s y and s z are spin operators, is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{3}{4}$$ is the correct answer to “A free particle with energy E whose wave function is a plane wave with wavelength λ enters a region of constant potential V > 0, where the wavelength of the particle is 2λ. The ratio (V/E) is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{N!}}{{\left( {\frac{N}{2}} \right)!\left( {\frac{N}{2}} \right)!}}{\left( {\frac{1}{2}} \right)^N}$$ is the correct answer to “A one-dimensional random walker takes steps to left or right with equal probability. The probability that the random walker starting from origin is back to origin after N even number of steps is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

e-αx is the correct answer to “An electron with energy E is incident from left on a potential barrier, given by \[\begin{gathered} V\left( x \right) = 0{\text{ for }}x < 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {V_0}{\text{ for }}x > 0 \hfill \\ \end{gathered} \] as shown in the figure. For E < V 0 , the space part of the wave function for x > 0 is of the form”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{L\sqrt m }}{{\pi \hbar \sqrt {2E} }}$$ is the correct answer to “The energy density of states of an electron in a one-dimensional potential well of infinitely high walls is (the symbols have their usual meanings)”. This question is standard in competitive examination syllabi for Quantum Mechanics.

The particle will be confined to anyone of the wells is the correct answer to “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”. This question is standard in competitive examination syllabi for Quantum Mechanics.

zero in state $${\phi _{1s}}\left( r \right)$$ is the correct answer to “In hydrogenic states, the probability of finding an electron at r = 0 is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

λn2 is the correct answer to “A quantum harmonic oscillator is in the energy eigen state $$\left| n \right\rangle .$$ A time independent perturbation 1λ(a t a) 2 acts on the particle, where λ is a constant of suitable dimensions and a and a t are lowering and raising operators respectively. Then the first order energy shift is given by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{\hbar \left\{ {s\left( {s + 1} \right) - {m^2}} \right\}}}{2}$$ is the correct answer to “For a spin s particle, in the eigen basis of $$\overrightarrow {{{\bf{s}}^2}} ,$$ s z the expectation value $$\left\langle {sm\left| {\overrightarrow {{\bf{s}}_x^2} } \right|sm} \right\rangle $$ is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{m\omega }}{{2\hbar }}$$ is the correct answer to “For a particle of mass m in a one-dimensional harmonic oscillator potential of the form $${V_{\left( x \right)}} = \frac{1}{2}m{\omega ^2}{x^2},$$ the first excited energy eigen state is $$\psi \left( x \right) = x{e^{ - a{x^2}}}.$$ The value of a is”. This question is standard in competitive examination syllabi for Quantum Mechanics.