Quantum Mechanics — Part 1 MCQs with Explanations

For an arbitrary state $$\left| \psi \right\rangle ,$$ the probability of finding the particle in the left half of the well is half is the correct answer to “A particle is placed in a one-dimensional box of size L along the X-axis, (0 < x < L). Which of the following is true”. This question is standard in competitive examination syllabi for Quantum Mechanics.

(e-ax1 - e-ax2) is the correct answer to “If the probability that x lies between x and x + dx is P(x) dx = ae -ax dx, where 0 < x < $$\infty $$ , a > 0, then the probability that x lies between x 1 and x 2 (x 2 > x 1 ) is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

550 nm is the correct answer to “A Michelson interferometer is illuminated with monochromatic light. When one of the mirrors is moved through a distance of 25.3 μm, 92 fringes pass through the cross-wire. The wavelength of the monochromatic light is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

0.25 is the correct answer to “An electron is in a statewith spin wavefunction \[{\phi _s} = \left[ {\begin{array}{*{20}{c}} {\frac{{\sqrt 3 }}{2}} \\ {\frac{1}{2}} \end{array}} \right]\] in the s z representation. What is the probability of, finding the z-component of its spin along the $$ - {\bf{\hat Z}}$$ direction”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{L}{4}$$ and $$\frac{{3L}}{4}$$ is the correct answer to “If the wave function of a particle trapped in space between x = 0 and x = L is given by $$\psi \left( x \right) = A\sin \left( {\frac{{2\pi x}}{L}} \right),$$ where A is a constant, for which value(s) of x will the probability of finding the particle be the maximum”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{{2{V_0}}}{{3\pi }}$$ is the correct answer to “The normalized eigen states of a particle in a one-dimensional potential well \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{if }}0 \leqslant x \leqslant a} \\ {\infty ,}&{{\text{otherwise}}} \end{array}} \right.\] are given by $${\psi _n}\left( x \right) = \sqrt {\frac{2}{a}} \sin \left( {\frac{{n\pi x}}{a}} \right)$$ where, n = 1, 2, 3, . . . The particle is subjected to a perturbation $$\eqalign{ & V'x = {V_0}\cos \left( {\frac{{\pi x}}{a}} \right),\,{\text{for }}0 \leqslant x \leqslant \frac{a}{2} \cr & \,\,\,\,\,\,\,\,\,\,\, = 0,\,\,\,{\text{otherwise}} \cr} $$ The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

decreasing the momentum of the electrons is the correct answer to “A parallel beam of electrons of a given momentum pass through a screen S 1 containing a slit and then produces a diffraction pattern on a screen S 2 placed behind it. The width of the central maximum observed on the screen S 2 can be increased by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

Pr(E = E0) = 0 is the correct answer to “An exact measurement of the position of a Simple Harmonic Oscillator (SHO) is made with the result x = x 0 , [The SHO has energy levels E n (n = 0, 1, 2, . . .) and associated normalized wave functions $${\psi _{\text{n}}}$$ ]. Subsequently, an exact measurement of energy E is made, using the general notation Pr(E = E') denoting the probability that a result E' is obtained for this measurement, the following statements are written. Which one of the following statements is correct”. This question is standard in competitive examination syllabi for Quantum Mechanics.

l = 1, m = 1, n = 2 is the correct answer to “An atomic state of hydrogen is represented by following wave function $$\psi \left( {r,\,\theta ,\,\phi } \right) = \frac{1}{{\sqrt 2 }}{\left( {\frac{1}{{{a_0}}}} \right)^{\frac{3}{2}}}\left( {1 - \frac{r}{{2{a_0}}}} \right){e^{\frac{{ - r}}{{2{a_0}}}}}\cos \theta $$ where, a 0 is a constant. The quantum numbers of the state are”. This question is standard in competitive examination syllabi for Quantum Mechanics.

The kinetic energy and the potential energy of the particle cannot simultaneously have sharp values is the correct answer to “A spinless particle moves in a central potential V(r).”. This question is standard in competitive examination syllabi for Quantum Mechanics.

zero is the correct answer to “The expectation value of the z coordinate, (z), in the ground state of the hydrogen atom (wave function: $${\psi _{100}}\left( r \right) = A{e^{ - \frac{r}{{{a_0}}}}},$$ where A is the normalization constant and $${a_0}$$ is the Bohr radius), is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$${\tau _\rho } < {\tau _\omega } < {\tau _\phi }$$ is the correct answer to “The resonance widths $$\Gamma $$ of $$\rho ,\,\omega $$ and $$\phi $$ particle resonances satisfy the relation $${\Gamma _\rho } > {\Gamma _\omega } > {\Gamma _\phi }$$ . Their lifetimes r satisfy the relation”. This question is standard in competitive examination syllabi for Quantum Mechanics.

\[\frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} i \\ 0 \end{array}} \right]\] is the correct answer to “The spin' function of a free particle, in the basis in which s z is diagonal can be written as \[\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]\] with eigen values \[ + \frac{\hbar }{2}\] and \[ - \frac{\hbar }{2}\] respectively. In the given basis, the normalized eigen function of s y with eigen value \[ - \frac{\hbar }{2}\] is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{9}{{14}}$$ is the correct answer to “A one-dimensional harmonic oscillator is in the state $$\psi \left( x \right) = \frac{1}{{\sqrt {14} }}\left[ {3{\psi _0}\left( x \right) - 2{\psi _1}\left( x \right) + {\psi _2}\left( x \right)} \right],$$ where, $${\psi _0}\left( x \right),\,{\psi _1}\left( x \right)$$ and $${\psi _2}\left( x \right)$$ are the ground, first excited and second excited states, respectively. The probability of finding the oscillator in the ground state is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

The kinetic energy acquired by the electron is the largest when the incident and scattered photons move in opposite directions is the correct answer to “Which one of the following statements concerning the Compton effect is not correct”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$l{\hbar ^2}$$ is the correct answer to “Let $$\overrightarrow {\bf{L}} $$ = (L x , L y , L z ) denotes the orbital angular momentum operators of a particle and let L + = L x + i L y and L - = L x - i L y . The particle is in aneigen state of L 2 and L z eigen values $${\hbar ^2}\left( {l + 1} \right)$$ and $$\hbar l$$ respectively. The expectation value of L + L - in this state is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$ - \frac{{{\hbar ^2}}}{{2m}}\frac{{{d^2}}}{{d{x^2}}} + \frac{1}{2}k{x^2} - qEx$$ is the correct answer to “A one-dimensional harmonic oscillator carrying a charge -q is placed in a uniform electric field $$\overrightarrow {\bf{E}} $$ along the positive X-axis. The corresponding Hamiltonian operator is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$0,\,\frac{{{\hbar ^2}}}{4},\,0,\,\frac{{{\hbar ^2}}}{4}$$ is the correct answer to “A spin half particle is in the state $${S_z} = \frac{{{\hbar ^2}}}{2}.$$ The expectation values of $${S_x},\,S_x^2,\,{S_y},\,S_y^2$$ are given by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\left[ {{L_x},\,{L_y}} \right]\left| {{\psi _0}} \right\rangle = 0$$ is the correct answer to “Let $$\left| {{\psi _0}} \right\rangle $$ denotes the ground state of the hydrogen atom. Choose the correct statement from those given below.”. This question is standard in competitive examination syllabi for Quantum Mechanics.

7.322 x 105 m/s is the correct answer to “An electron propagating along the X-axis passes through a slit of width ∆y = 1 nm. The uncertainty in the y-component of its velocity after passing through the slit is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

vp = vg is the correct answer to “For a perfect free electron gas in a metal the magnitudes of phase velocity (v p ) and group velocity (v g ) are such that”. This question is standard in competitive examination syllabi for Quantum Mechanics.

total energy of the system is the correct answer to “The time independent Schrodinger equation of a system represents the conservation of the”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\phi \left( x \right) = Ex\exp \left( { - F{x^2}} \right);E,F > 0$$ is the correct answer to “Which of the following functions represents acceptable wave function of a particle in the range $$ - \infty \leqslant x \leqslant \infty ?$$”. This question is standard in competitive examination syllabi for Quantum Mechanics.

V(x) = 0 is the correct answer to “The wave function of a particle moving in a one-dimensional time independent potential V(x) is given by $$\psi \left( x \right) = {e^{ - iax + b}},$$ where a and b are constants. This means that the potential V(x) is of the form”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$2i\hbar {p_x}$$ is the correct answer to “The commutator [x, p 2 ], where x and p are position and momentum operators respectively, is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\lambda = \frac{h}{{\sqrt {3m{k_B}T} }}$$ is the correct answer to “The de-Broglie wavelength of particles of mass m with average momentum p at a temperature T in three-dimensions is given by”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$${L_z}$$ is the correct answer to “The Hamiltonian of a particle is given by $$H = \frac{{{p^2}}}{{2m}} + V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right) + \phi \left( { + \left| {\overrightarrow {\bf{r}} } \right|} \right)\overrightarrow {\bf{L}} .\overrightarrow {\bf{S}} ,$$ where $$\overrightarrow {\bf{S}} $$ is the spin, $$V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ and $$\phi \left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ are potential functions and $$\overrightarrow {\bf{L}} \left( { = \overrightarrow {\bf{r}} \times \overrightarrow {\bf{p}} } \right)$$ is the angular momentum. The Hamiltonian does not commute with”. This question is standard in competitive examination syllabi for Quantum Mechanics.

\[\psi = \left\{ {\begin{array}{*{20}{c}} {{\text{non - zero constant}}}&{{\text{if, }}r < R} \\ {0,}&{{\text{if, }}r > R} \end{array}} \right.\] is the correct answer to “Which of the following is an allowed wave function for a particle in a bound state? N is a constant and α, β > 0.”. This question is standard in competitive examination syllabi for Quantum Mechanics.

10-8 m is the correct answer to “The de-Broglie wavelength λ for an electron of energy 150 eV is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

a0 is the correct answer to “The radial wave function of the electrons in the state of n = 1 and 1 = 0 in hydrogen atom is \[{R_{10}} = \frac{2}{{{\text{a}}_0^{\frac{3}{2}}}}\exp \left( { - \frac{r}{{{{\text{a}}_0}}}} \right),{\text{ }}{{\text{a}}_0}\] is the Bohr radius. The most probable value of r for an electron is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

zero is the correct answer to “$${{\bf{\hat A}}}$$ and $${{\bf{\hat B}}}$$ are two quantum mechanical operators. If $$\left[ {{\bf{\hat A}},\,{\bf{\hat B}}} \right]$$ stands for the commutator of $${{\bf{\hat A}}}$$ and $${{\bf{\hat B}}}$$ then $$\left[ {\left[ {{\bf{\hat A}},\,{\bf{\hat B}}} \right],\,\left[ {{\bf{\hat B}},\,{\bf{\hat A}}} \right]} \right]$$ is equal to”. This question is standard in competitive examination syllabi for Quantum Mechanics.

(i) even, (ii) odd is the correct answer to “The parities of the wave functions (i) cos (kx) and (ii) tanh (kx) are”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$m\hbar $$ is the correct answer to “The commutator $$\left[ {{L_z},\,{Y_{lm}}\left( {\theta ,\,\phi } \right)} \right],$$ where L z is the z component of the orbital angular momentum and $${{Y_{lm}}\left( {\theta ,\,\phi } \right)}$$ is a spherical harmonic, is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{1}{{\sqrt 2 }}\left| {{E_0} = 10\,eV} \right\rangle - \frac{1}{{\sqrt 2 }}\left| {{E_1} = 30\,eV} \right\rangle $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$i\hbar \left( {L_x^2 - L_y^2} \right)$$ is the correct answer to “If L x , L y and L z are respectively the x, y and z components of angular momentum operator L, the commutator [L x , L y , L z ] is equal to”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\frac{1}{{\sqrt {15} }}\left[ {2{\psi _0}\left( x \right){e^{ - i{E_0}t/h}} - {\psi _1}\left( x \right){e^{ - i{E_1}t/h}}} \right]$$ is the correct answer to “The wave function of a particle in a one-dimensional potential at time t = 0 is $$\psi \left( {x,\,t = 0} \right) = \frac{1}{{\sqrt {15} }}\left[ {2{\psi _0}\left( x \right) - {\psi _1}\left( x \right)} \right]$$ where, $${\psi _0}\left( x \right)$$ and $${\psi _1}\left( x \right)$$ are the ground arid the first excited states of the particle with corresponding energies E 0 and E 1 . The wave function of the particle at a time t is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

2.1 × 10-15 s is the correct answer to “An electron in a time independent potential is in a state which is the superposition of the ground state (E 0 = 11 eV) and the first excited state (E 1 = 1 eV). The wave function of the electron will repeat itself with a period of”. This question is standard in competitive examination syllabi for Quantum Mechanics.

\[I = 0,\, < \mathfrak{M} > = 0\] is the correct answer to “Consider the combined system of proton and electron in the hydrogen atom in its (electronic) ground state. Let $$I$$ denotes the quantum number associated with the total angular momentum and let \[ < \mathfrak{M} > \] denote the magnitude of the expectation value of the net magnetic moment in the state. Which of the following pairs represents a possible state of the system (\[{\mu _B}\] is Bohr magneton)”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$3i\hbar {x^2}$$ is the correct answer to “If [x, p] = $$i\hbar ,$$ the value of [x 3 , p] is”. This question is standard in competitive examination syllabi for Quantum Mechanics.

$$\left\{ { - \frac{1}{2},\, + \frac{1}{2}} \right\}$$ is the correct answer to “Three operators X, Y and Z satisfy the commutation relations, $$\left[ {X,\,Y} \right] = i\hbar Z,\,\left[ {Y,\,Z} \right] = i\hbar X$$ and $$\left[ {Z,\,X} \right] = i\hbar Y.$$ The set of all possible eigen values of the operator Z, in units of $$\hbar $$ is”. This question is standard in competitive examination syllabi for Quantum Mechanics.