Quantum Mechanics MCQs Question and Answer in Engineering Physics Page-4 section-1

31.
$${{\bf{\hat A}}}$$ 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

A. $${\bf{\hat A\hat B\hat A\hat B}} - {\bf{\hat B\hat A\hat B\hat A}}$$

B. $${\bf{\hat A}}\left( {{\bf{\hat A\hat B}} - {\bf{\hat B\hat A}}} \right) - {\bf{\hat B}}\left( {{\bf{\hat B\hat A}} - {\bf{\hat A\hat B}}} \right)$$

C. zero

D. $${\left( {\left[ {{\bf{\hat A}},\,{\bf{\hat B}}} \right]} \right)^2}$$

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32.
The parities of the wave functions (i) cos (kx) and (ii) tanh (kx) are

A. (i) odd, (ii) odd

B. (i) even, (ii) even

C. (i) odd, (ii) even

D. (i) even, (ii) odd

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33.
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

A. $$l\left( {l + 1} \right)\hbar $$

B. $$ - m\hbar $$

C. $$m\hbar $$

D. $$ + l\hbar $$

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34.
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. $$\frac{1}{{\sqrt 2 }}\left| {{E_0} = 10\,eV} \right\rangle - \frac{1}{{\sqrt 2 }}\left| {{E_1} = 30\,eV} \right\rangle $$

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35.
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

A. $$i\hbar \left( {L_x^2 + L_y^2} \right)$$

B. $$2i\hbar {L_z}$$

C. $$i\hbar \left( {L_x^2 - L_y^2} \right)$$

D. zero

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36.
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 E₀ and E₁. The wave function of the particle at a time t is

A. $$\frac{1}{{\sqrt 5 }}{e^{ - i\left( {{E_0}{E_1}} \right)t/2h}}\left[ {2{\psi _0}\left( x \right) - {\psi _1}\left( x \right)} \right]$$

B. $$\frac{1}{{\sqrt 5 }}{e^{ - i{E_0}t/h}}\left[ {2{\psi _0}\left( x \right) - {\psi _1}\left( x \right)} \right]$$

C. $$\frac{1}{{\sqrt 5 }}{e^{ - i{E_1}t/h}}\left[ {2{\psi _0}\left( x \right) - {\psi _1}\left( x \right)} \right]$$

D. $$\frac{1}{{\sqrt {15} }}\left[ {2{\psi _0}\left( x \right){e^{ - i{E_0}t/h}} - {\psi _1}\left( x \right){e^{ - i{E_1}t/h}}} \right]$$

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37.
An electron in a time independent potential is in a state which is the superposition of the ground state (E₀ = 11 eV) and the first excited state (E₁ = 1 eV). The wave function of the electron will repeat itself with a period of

A. 3.1 × 10^-18 s

B. 2.1 × 10^-15 s

C. 1.2 × 10^-12 s

D. 1.0 × 10^-9 s

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38.
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)?

A. \[I = 0,\, < \mathfrak{M} > = 0\]

B. \[I = \frac{1}{2},\, < \mathfrak{M} > = 1{\mu _B}\]

C. \[I = 1,\, < \mathfrak{M} > \approx 1{\mu _B}\]

D. \[I = 0,\, < \mathfrak{M} > = 2{\mu _B}\]

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39.
If [x, p] = $$i\hbar ,$$ the value of [x³, p] is

A. $$2i\hbar {x^2}$$

B. $$ - 2i\hbar {x^2}$$

C. $$3i\hbar {x^2}$$

D. $$ - 3i\hbar {x^2}$$

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40.
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

A. $$\left[ {0,\, \pm 1,\, \pm 2,\, \pm 3,\,...} \right]$$

B. $$\left\{ {\frac{1}{2},\,1,\,\frac{3}{2},\,2,\,\frac{5}{2},\,...} \right\}$$

C. $$\left\{ {0 \pm \frac{1}{2},\, \pm 1,\, \pm \frac{3}{2},\, \pm 2,\, \pm \frac{5}{2},\,...} \right\}$$

D. $$\left\{ { - \frac{1}{2},\, + \frac{1}{2}} \right\}$$

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Quantum Mechanics MCQs Question and Answer in Engineering Physics

Answer & Solution

32. The parities of the wave functions (i) cos (kx) and (ii) tanh (kx) are

Answer & Solution

33. The commutator $$\left[ {{L_z},\,{Y_{lm}}\left( {\theta ,\,\phi } \right)} \right],$$ where Lz is the z component of the orbital angular momentum and $${{Y_{lm}}\left( {\theta ,\,\phi } \right)}$$ is a spherical harmonic, is

Answer & Solution

Answer & Solution

35. If Lx, Ly and Lz are respectively the x, y and z components of angular momentum operator L, the commutator [Lx, Ly, Lz] is equal to

Answer & Solution

Answer & Solution

37. An electron in a time independent potential is in a state which is the superposition of the ground state (E0 = 11 eV) and the first excited state (E1 = 1 eV). The wave function of the electron will repeat itself with a period of

Answer & Solution

Answer & Solution

39. If [x, p] = $$i\hbar ,$$ the value of [x3, p] is

Answer & Solution

40. 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

Answer & Solution

32.
The parities of the wave functions (i) cos (kx) and (ii) tanh (kx) are

33.
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

35.
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

37.
An electron in a time independent potential is in a state which is the superposition of the ground state (E₀ = 11 eV) and the first excited state (E₁ = 1 eV). The wave function of the electron will repeat itself with a period of

39.
If [x, p] = $$i\hbar ,$$ the value of [x³, p] is

40.
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