Quantum Mechanics-II - Quanta: BS/MSc Physics Books

1: The wave functions of electron, proton and neutron are

a. Symmetric

b. Anti-symmetric

c. mixed.

d. None of these.

2: The wave functions of α particle, photon and deuteron are

a. Symmetric

b. Phonons

c. Anti- symmetric

d. None of these

3: $\mathbf{\psi_1\left({\vec{r}}_1,\ {\vec{r}}_2\right)=\left({\vec{r}}_1-{\vec{r}}_2\right)^2}$

a. Symmetric

b. Phonons

c. Anti- symmetric

d. None of these

4: Every single particle state can be occupied by at most one fermion is called

a. Uncertainty principle

b. Pauli principle

c. Hund’s rule

d. Symmetry principle

5: The particles with integral spin, $S=0,1\hbar,\ 2\hbar,\ 3\hbar,\cdots$ are called

a. Superconductor

b. Neon

c. Fermions

d. Bosons

6: The exchange operator apply on

a. single particle

b. two particles

c. Both

d. None of these

7: A state of N fermions must be………. under every possible exchange operator:

a. Anti- symmetric

b. Symmetric

c. Both

d. None of these

8: The expression which describes the wave function of a multi-fermionic system that satisfies anti-symmetry conditions?

a. Fock space

b. Slater determinant

c. Vacuum space

d. None of these

9: Quarks are … particles.

a. Fermions

b. Bosons

c. Elementary

d. All

10: Quantum Field Theory describes the interaction among

a. Fermions

b. Bosons

c. Elementary particles

d. All

11: The magnetic moment of an electron circulating in an orbit with an orbital angular momentum L is given by

a. $-\frac{\mathbf{eL}}{\mathbf{2mc}}$

b. $-\frac{ep}{2mc}$

c. $-\frac{er}{2mc}$

d. None of above

12: Eigen values of square of spin angular momentum are:

a. $\mathbf{s}{\left(\mathbf{S}+\mathbf{1}\right)\hbar}^\mathbf{2}$

b. $s\left(2s+1\right)\hbar^2$

c. $s\left(s+1\right)\hbar$

d. $s\left(2s+1\right)\hbar$

13: Spin angular momentum S in term of Pauli spin matrices can be written as

a. $-\frac{eL}{2mc}$

b. $s\left(s+1\right)\hbar^2$

c. $\mathbf{\frac{h}{2}\vec{\sigma}}$

d. $\frac{\hbar}{2}S_z$

14: Eigen values of square of total or general angular momentum

a. $\mathbf{j}\left(\mathbf{j}+\mathbf{1}\right)\hbar^\mathbf{2}$

b. $j\left(2j+1\right)\hbar^2$

c. $j\left(j+1\right)\hbar$

d. $j\left(2j+1\right)\hbar$

15: In case of general angular momentum, the values of magnetic quantum number are

a. $-\mathbf{j}\le\mathbf{m}\le\mathbf{j}$

b. $m\le j$

c. m=2j

d. None of above

16: Y-component of spin angular momentum is matrix form has complex entities, its eigen values are

a. Complex

b. Real

c. Imaginary

d. All of above

17: Eigen values of none of Pauli spin matrix σ_y are real, what can you say about its eigen vectors?

a. Real

b. Complex

c. Both a & b

d. None of above

18: Value of square of any Pauli spin matrix is … matrix

a. Null

b. Diagonal

c. Identity

d. scalar

19: Components of spin angular momentum

a. Commute

b. Are orthogonal

c. Anti-commute

d. None of above

20: Trace of any of Pauli spin matrix is

a. Unity

b. Zero

c. 2i

d. None of above

21: The electron proton and neutron are

a. Fermion

b. Boson

c. Photons

d. None of above

22: The α particle, photon and deuteron are

a. Fermion

b. Boson

c. Phonons

d. None of above

23: The wave functions of electron, proton and neutron are

a. Symmetric

b. Anti-symmetric

c. Photons

d. None of above

24: The wave function of α particle, photon and deuteron are

a. Symmetric

b. Anti-symmetric

c. Phonons

d. None of above

25: $\mathbf{\psi_1\left(r_1,\ r_2\right)=\left(r_1-r_2\right)^2}$ is

a. Symmetric

b. Anti-symmetric

c. Phonons

d. None of above

26: If $\mathbf{\psi_3\left(r_1,\ r_2\right)=\frac{5(r_1-r_2)}{(r_1-r_2)}}$ then $\boldsymbol{{\hat{P}}_{12}\psi_3(r_1,\ r_2)}$

a. -ψ₃(r₁, r₂)

b. $\psi_3(r_1,\ r_2)$

c. $-\psi_3(r_2-r_1)$

d. None of above

27: The particles with integral spin, $S_i=0,\ 1\hbar,\ 2\hbar,\ 3\hbar\cdots$

a. Superconductor

b. Neon

c. Fermions

d. Bosons

28: The particles with half-odd-integral spin, $\mathbf{S_i=\frac{\hbar}{2},\frac{3\hbar}{2},\frac{5\hbar}{2}\cdots}$

a. Superconductor

b. Neon

c. Fermions

d. Bosons

29: Every single particle state can be occupied by at most one fermion. This is called

a. Uncertainty Principle

b. Pauli Exclusion Principle

c. Hund’s rule

d. Symmetry Principle

30: For S=0 multiplicity is

a. 1

b. 3

c. 2

d. 4

31: For S=1, multiplicity is

a. 1

b. 3

c. 2

d. 4

32: The spin multiplicity is defined as

a. 2S+ 2

b. 2S+3

c. 2S+1

d. 2S+4

33: The z-component of the spin angular momentum S can have only two eigenvalues

a. $\frac{+\hbar}{2}$ and $\frac{+\hbar}{2}$

b. $\frac{+\hbar}{2}$ and $\frac{-\hbar}{2}$

c. $\frac{-\hbar}{2}$ and $\frac{-\hbar}{2}$

d. None of above

34: In non-degenerate time-independent perturbation theory the λ varies from

a. 0 to 1

b. 1 to 2

c. 0 to n

d. None of these

35: When the operator applying on a state and it gives the different eigen values

a. Symmetric

b. Degenerate

c. Anti- symmetric

d. Non-degenerate

36: The operator changes in time while the Hamiltonian remains fixed in time

a. Dirac picture

b. Heisenberg picture

c. Schr¨odinger picture

d. None of these

37: WKB method can be viewed as … approximation

a. Quantum mechanical

b. Classical

c. Semi-classical

d. All of these

38: Both basis and operators carry time dependence in

a. Heisenberg picture

b. Dirac picture

c. Schrodinger picture

d. None of these

39: Potential V (r) slightly varies in

a. WKB

b. Perturbation

c. Variational

d. All of these

40: Variational principle states that

a. E_g ≤ E_n

b. E_g ≥ E_n

c. E_g = E_n

d. None of these

41: WKB method is a technique for obtaining solutions in

a. One dimension

b. Two dimensions

c. Three dimensions

d. All of these

42: Variational method is used to find the approximation to the state

a. Excited state

b. Ground state

c. Intermediate state

d. None of these

**43: The wave function ……. becomes in WKB approximation when E = V (r)**

a. Real

b. Imaginary

c. Infinite

d. None of these

44: Height of central peak in probability is proportional to

a. $\frac{1}{t}$

b. $t^2$

c. t

d. None

**45: ω_fi = 0 is maximum when**

a. E_f = E_i

b. E_f > E_i

c. E_f < E_i

d. $Ef\neq Ei$

46: The transition probability exhibits a pattern like

a. Scattering

b. Interference

c. Straight

d. None of these

47: The transition rate exits for constant perturbation when the energy is

a. Conserved

b. Non-conserved

c. Zero

d. None

48: Transition probability of harmonic perturbation will be maximum when ω_fi is equal to

a. + ω

b. ω

c. $\pm\mathbf{\omega}$

d. ω = 0

49: Probability at constant perturbation is appreciable at

a. ω_fi = 0

b. ω_fi = 1

c. ω_fi = 1/2

d. None of above

50: We attain delta function when time is

a. t = 0

b. t > 0

c. t < 0

d. t → ∞

51: Quantization of radiation is achieved by which operators

c. Raising

b. Lowering

c. Ladder

d. Hamiltonian

52: No electronic transition will occur unless

a. ∆m = 0, ±1

b. ∆m = 0

c. ∆m = ±1

d. None of these

53: The spin helicity of a photon is

a. $\hbar$

b. $\pm\hbar$

c. Both (a) and (b)

d. None of these

54: The perpendicular distance between the path of incident particle and center of a potential field created by target nucleus is

a. Impact parameter

b. Impact scattering

c. Impact element

d. None of these

55: What kind of wave we get when plane wave strikes with target in Born approximation

a. Plane wave

b. Spherical wave

c. Light wave

d. Wave packet

56: In quantum scattering, we take incident particle in the form of

a. Solid particle

b. Solid field

c. Field form

d. Liquid form

57: In quantum scattering, motion of incident particle is in

a. One dimension

b. Two dimensions

c. Three dimensions

d. Zero dimension

58: dσ is directly proportional to

a. dΩ

b. dV

c. D(θ)

d. dA

59: If Kr>>1, the region will be called as

a. Radiation Zone

b. Intermediate region

c. Scattering region

d. Closed region

**60: V (r) can be spherically — — — — — — —— potential**

a. Symmetric

b. Anti-symmetric

c. Non-zero

d. All

61: For large values of r, potential V (r) approaches to

a. 0

b. 1

c. ∞

d.) None

62: Born approximation is valid for incident particle with

a. Large energy

b. Small energy

c. Strong potential

d.None

63: In the scattering of Born approximation, particles have

a. Loss of energy

b. Energy Conserved

c. K=K’

d. None of these

**64: The position (x) and momentum (p_x) operators are always**

a. Commutative

b. Anti Commutative

c. Normalized

d. Orthogonal

65: Expression for z-component of angular momentum (L) is

a. $-\mathbf{i}\hbar\frac{\partial}{\partial\phi}$

b. $-i\hbar\frac{\partial}{\partial\theta}$

c. $i\hbar\frac{\partial}{\partial\phi}$

d. $i\hbar\frac{\partial}{\partial\phi}$

66: Components of angular momentum () are

a. Orthogonal

b. Normal

c. Commute

d. All of aboe

67: Value of [L,L] is

a. 0

b. 1

c. $i\hbar L$

d. All of above

68: The angular momentum of an isolated system is

a. Conserved

b. Non-Conserved

c. Variable

d. None

69: Spin does not depend on

a. S

b. m_s

c. Spatial degrees of freedom

d. All of above

70: For Hydrogen atom that is in ground state, the orbital angular momentum will be

a. 1

b. 2

c. 0

d. Unknown

71: If azimuthal quantum number (l) is 2, the number of values of the magnetic quantum number (m_l) will be

a. 2

b. 3

c. 4

d. 5

72: Square of angular momentum is

a. Hermitian

b. Anti-Hermitian

c. Linear

d. Projection Operator

73: The square of angular momentum J² commutes with

a. J_x

b. J_y

c. J_z

d. All of these

74: Application of Barrier Tunneling are

a. Radioactive Decay

b. Semiconductor Devices

c. Both a and b

d. None

75: The energy of free particle in 3D is

a. Triply degenerate

b. Non degenerate

c. Infinitely Degenerate

d. None

76: The Centrifugal potential depends on …. quantum number

a. Orbital

b. Magnetic

c. Principle

d. All of above

77: Three dimensional problems often exhibit degeneracy, which occur whenever ….

is symmetric

a. Eigen value

b. Wavefunction

c. Potential

d. None

**78: Value of square of any Pauli spin matrix is — — —— matrix**

a. Null

b. Diagonal

c. Identity

d. None

79: As the particle angular momentum increases, the particle becomes less and less

a. Repulsive

b. Unbound

c. Bound

d. Both a and b

80: Eigen values of square of spin angular momentum are

a. $\mathbf{s}\left(\mathbf{s}+\mathbf{1}\right)\mathbf{2}\hbar^\mathbf{2}$

b. $s\left(2s+1\right)\hbar^2$

c. $s\left(s+1\right)\hbar^2$

d. $s\left(2s+1\right)2\hbar^2$

**81: Radial equation for central potential depends on — — — — — — quantum number**

a. Azimuthal

b. Magnetic

c. Spin

d. None

82: The Hamiltonian of rigid rotator

a. $\hat{H}=\frac{{\hat{L}}^2}{4I}$

b. $\hat{H}=\frac{\hat{L}}{4I}$

c. $\hat{\mathbf{H}}=\frac{{\hat{\mathbf{L}}}^\mathbf{2}}{\mathbf{2I}}$

d. $\hat{H}=\frac{\hat{L}}{8I}$

83: The quantized energy of rigid rotator is

a. $\mathbf{\frac{\hbar^2l(l+1)}{2I}}$

b. $\frac{\hbar l(l+1)}{2I}$

c. $\frac{\hbar^2l(l+1)}{4i}$

d. $\frac{\hbar^2l(l+1)}{6I}$

84: Consider a hydrogen atom that is placed in an external magnetic field. The effect of an external magnetic field on atom is to cause a shift of its energy levels, this is called

a. Stark effect

b. Zeeman effect

c. Paschen back effect

d. None of above

85: The strong field Zeeman effect is also called

a. Stark effect

b. Coulomb effect

c. Paschen back effect

d. None of above

86: Both basis and operators carry time dependence in

a. Dirac picture

b. Heisenberg picture

c. Schrodinger picture

d. None of above

87: The operators stay fixed while the Schrodinger equation changes the basis with time in

a. Dirac picture

b. Heisenberg picture

c. Schrodinger picture

d. None of above

88: The operator changes in time while the basis of the basis of the space remains fixed in

a. Dirac picture

b. Heisenberg picture

c. Schrodinger picture

d. None of above

89: The magnetic moment of an electron circulating in an orbit with an orbital angular momentum is given by:

a. $-\frac{\mathbf{eL}}{\mathbf{2mc}}$

b. $-\frac{ep}{2mc}$

c. $-\frac{er}{2mc}$

d. None of these

90: Eigen values of square of spin angular momentum are:

a. $\mathbf{s}\left(\mathbf{s}+\mathbf{1}\right)\hbar^\mathbf{2}$

b. $s\left(2s+1\right)\hbar^2$

c. $s\left(s+1\right)\hbar$

d. $s(2s+1)\hbar$

91: Spin angular momentum in terms of Pauli spin matrices can be written as:

a. $-\frac{eL}{2mc}$

b. $s\left(s+1\right)\hbar^2$

c. $\frac{\hbar}{\mathbf{2}}\mathbf{\sigma}$

d. $\frac{\hbar}{2}S_z$

92: Eigen values of square of total or general angular momentum are:

a. $\mathbit{j}\left(\mathbit{j}+\mathbf{1}\right)\hbar^\mathbf{2}$

b. $j\left(2j+1\right)\hbar^2$

c. $j\left(j+1\right)\hbar$

d. $j(2j+1)\hbar$

93: In case of general angular momentum the values of magnetic quantum number “m” are:

a. $-\mathbf{j}\le\mathbf{m}\le\mathbf{j}$

b. $m\le j$

c. m=2j

d. None of these

94: Y-component of spin angular momentum in matrix form has complex entities, its Eigen values are:

a. Complex

b. Real

c. Imaginary

d. All of these

95: Eigen values of one of Pauli spin matrix $\sigma_y$ are real what can you say about its Eigen vectors:

a. Real

b. Complex

c. a & b

d. None of these

96: Value of square of any Pauli spin matrix is…….. matrix:

a. Null

b. Diagonal

c. Identity

d. Scalar

97: Components of spin angular momentum are:

a. Commute

b. Orthogonal

c. Anti-commute

d. None of these

98: Trace of any Pauli spin matrix is:

a. Unity

b. Zero

c. -2i

d. None of these

99: WKB method is a technique for obtaining approximate solutions to the time independent Schrodinger equation in:

a. One dimension

b. Two dimension

c. Three dimension

d. All of these

100: Condition for validity of WKB approximation is:

a. $\left|\frac{\mathbf{d\lambda}}{\mathbf{dx}}\right|\ll\mathbf{1}$

b. $\left|\frac{d\lambda}{dx}\right|\gg1$

c. $\left|\frac{d\lambda}{dx}\right|\ll h$

d. $\left|\frac{d\lambda}{dx}\right|\gg h$

101: Consider a hydrogen atom that is placed in an external magnetic field. The effect of an external magnetic field on atom is to cause a shift of its energy levels this is called:

a. Stark effect

b. Zeeman effect

c. Paschen Back effect

d. None of these

102: The strong field Zeeman effect is also called:

a. Stark effect

b. Coulomb effect

c. Paschen Back effect

d. None of these

103: WKB method can be viewed as………………. Approximation:

a. Quantum mechanical

b. Classical

c. Semi classical

d. All of these

104: For one dimensional harmonic oscillator, Bohr Sommerfeld quantization rule is:

a. $\int\mathbf{pdq}=\left(\mathbf{n}+\frac{\mathbf{1}}{\mathbf{2}}\right)\mathbf{h}$

b. $\int p d q=\left(n-\frac{1}{2}\right)h$

c. $\int p d q=\left(n+\frac{3}{2}\right)h$

d. All of these

105: The angular momentum operator acts on a state $\psi\left(r,\theta,\varphi\right)$ as:

a. $\mathbf{L}^\mathbf{2}\mathbf{\psi}=\mathbf{l}\left(\mathbf{l}+\mathbf{1}\right)\hbar^\mathbf{2}\mathbf{\psi}$

b. $L\psi=l\left(l+1\right)\hbar^2\psi$

c. $L\psi=l\left(l+1\right)\hbar\psi a$

d. None of these

106: The angular part of wave function for hydrogen atom is written by using:

a. Spherical harmonics

b. Hankel functions

c. Bessel functions

d. Neumann functions

107: The momentum operator is multiple dimensions can be written as:

a. $-\hbar^2\nabla^2$

b. $-\mathbit{i}\hbar\mathbf{\nabla}$

c. $-i\hbar\frac{\partial}{\partial x}$

d. None of these

108: Ground state wave function for hydrogen atom is:

a. $\mathbf{R}{\mathbf{10}}=\left(\frac{\mathbf{2}}{\mathbf{a}\mathbf{o}^{\frac{\mathbf{3}}{\mathbf{2}}}}\right)\mathbf{e}^{-\mathbf{r}/\mathbf{a}_\mathbf{o}}$

b. $R_{10}=\left(\frac{2}{a_o^{\frac{3}{2}}}\right)e^{-2r/a_o}$

c. $R_{10}=\left(\frac{2}{a_o^{\frac{3}{2}}}\right)e^{-r/2a_o}$

d. None of these

108: Suppose n=2, l=0 then average value of orbit of electron is:

a. $4a_o$

b. $5a_o$

c. $\mathbf{6}\mathbf{a}_\mathbf{o}$

d. None of these

109: For hydrogen atom, the radial part of wave function R(r) is normalized according to formula:

a. $\int_{0}^{\infty}{\left|R\left(r\right)\right|^2dr=1}$

b. $\int_{\mathbf{0}}^{\infty}{\left|\mathbf{R}\left(\mathbf{r}\right)\right|^\mathbf{2}\mathbf{r}^\mathbf{2}\mathbf{dr}=\mathbf{1}}$

c. $\int_{-\infty}^{+\infty}{\left|R\left(r\right)\right|^2dr=1}$

d. All of these

110: Reduced mass of hydrogen atom is:

a. $\mathbf{\mu}=\frac{\mathbf{m}\mathbf{e}\mathbf{m}\mathbf{p}}{\mathbf{m}\mathbf{e}+\mathbf{m}\mathbf{p}}$

b. $\mu=\frac{m_em_p}{m_e-m_p}$

c. $\mu=\frac{m_e+m_p}{m_em_p}$

d. none of these

111: In one dimension the time independent Schrodinger wave equation is:

a. $-\frac{\hbar^\mathbf{2}}{\mathbf{2m}}\frac{\mathbf{d}^\mathbf{2}\mathbf{\psi}\left(\mathbf{x}\right)}{\mathbf{d}\mathbf{x}^\mathbf{2}}+\mathbf{V}\left(\mathbf{x}\right)\mathbf{\psi}\left(\mathbf{x}\right)=\mathbf{E\psi}(\mathbf{x})$

b. $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V\left(x\right)=-i\hbar\frac{\partial\psi}{\partial t}$

c. $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V\left(x\right)=-i\hbar\frac{\partial\psi}{\partial t}$

d. None of these

112: The Hermitian conjugate of an operator can be written as:

a. ${\left\langle\psi\middle|\hat{A}\middle|\varphi\right\rangle}^\ast=\left\langle\psi\middle|{\hat{A}}^\dag\middle|\varphi\right\rangle$

b. ${\left\langle\mathbf{\psi}\middle|\hat{\mathbf{A}}\middle|\mathbf{\varphi}\right\rangle}^\ast=\left\langle\mathbf{\varphi}\middle|{\hat{\mathbf{A}}}^\dag\middle|\mathbf{\psi}\right\rangle$

c. ${\left\langle\psi\middle|\hat{A}\middle|\varphi\right\rangle}^\ast=\left\langle\varphi\middle|\hat{A}\middle|\psi\right\rangle$

d. None of these

113: The photoelectric current depends upon:

a. Frequency

b. Intensity

c. Nature of metal

d. All of these

114: The time energy uncertainty relation is:

a. ∆E∆t≤h

b. ∆E∆t<ℏ

c. ∆E∆t≥ℏ

d. All of these

115: If we denote position space operator by \hat{x}, which of the following best describes its action on the base kets:

a. $\left\langle\mathbf{x}\middle|\hat{\mathbf{x}}\middle|\mathbf{x}^\prime\right\rangle=\mathbf{x}^\prime\mathbf{\delta}(\mathbf{x}-\mathbf{x}^\prime)$

b. $\left\langle x\middle|\hat{x}\middle| x^\prime\right\rangle={x^\ast}^\prime\delta(x-x^\prime)$

c. $\left\langle x\middle|\hat{x}\middle| x^\prime\right\rangle=x^\prime$

d. None of these

116: The matrix mechanics was developed by:

a. Heisenberg

b. Schrodinger

c. Max Plank

d. Dirac

117: The expectation value of an operator can be written as:

a. $\int_{-\infty}^{+\infty}{\psi^\ast\left(x\right)\hat{A}\psi\left(x\right)dx}$

b. $\left\langle\psi\left(x\right)\middle|\hat{A}\middle|\psi\left(x\right)\right\rangle$

c. $\int_{-\infty}^{+\infty}{\psi^\ast\left(x\right)\hat{A}\psi\left(x\right)dp}$

d. a & b

118: The reflection coefficient R is given by:

a. $\frac{\mathbf{J}{\mathbf{ref}}}{\mathbf{J}{\mathbf{inc}}}$

b. $\frac{J_{inc}}{J_{ref}}$

c. $\frac{J_{inc}}{J_{trans}}$

d. $\frac{J_{trans}}{J_{ref}}$

119: A wave function is expanded in terms of orthogonal basis as:

$\left|\left.\psi\right\rangle\right.=\frac{1}{\sqrt5}\left|\left.\psi_1\right\rangle\right.+\sqrt{\frac{3}{5}}\left|\left.\psi_2\right\rangle\right.+A\left|\left.\psi_3\right\rangle\right.$

The normalization constant A must be:

a. $\frac{1}{5}$

b. $\frac{\mathbf{1}}{\sqrt\mathbf{5}}$

c. $\frac{2}{5}$

d. $\frac{1}{2}$

120: In case of E>V, if $V\ll\ll E$ then:

a. $R=0,\ T=1$

b. $\mathbf{R}=\mathbf{1},\ \mathbf{T}=\mathbf{0}$

c. R+T=0

d. R.T=1

121: Kinetic energy is ………… intensity of radiation in photoelectric effect:

a. Dependent of

b. Independent of

c. Proportional to

d. Inversely proportional to

122: A dispersion relation is a fundamental relationship that:

a. Relates energy and momentum

b. Defines group velocity in terms of particle states

c. Defines phase velocity

d. Relates frequency and wave number

123: In case of E<V the wave function to the right of step potential:

a. Zero

b. Decrease

c. Increase

d. Maximum

124: If two operators have simultaneous Eigen functions then they are:

a. Orthogonal

b. Normalize

c. Anti-commute

d. Commute

125: If $\int\psi_1^\ast\left(x\right)\psi_2\left(x\right)dx=0 then \psi_1\left(x\right)$ & $\psi_2\left(x\right)$ are:

a. Normalized

b. Orthogonal

c. Orthonormal

d. Commute

126: Consider hydrogen atom, the angular probability distribution is given by:

a. $Y_l^m\left(\theta,\varphi\right)$

b. $\left|\mathbf{\theta}\mathbf{Y}_\mathbf{l}^\mathbf{m}\left(\mathbf{\theta},\mathbf{\varphi}\right)\right|^\mathbf{2}\mathbf{d\Omega}$

c. $\left|Y_l^m\left(\theta,\varphi\right)\right|^2$

d. $\left|cos\theta Y_l^m\left(\theta,\varphi\right)\right|^2d\Omega$

127: In coordinate representation the Eigen states of one dimensional harmonic oscillator can be written in form:

a. Hankel functions

b. Hermite polynomials

c. Schrodinger polynomials

d. Legendre polynomials

128: A wave function is expanded in a set of basis states $\psi=\sum_{i} a_i\varphi_i$ , the coefficients of expansion must satisfy:

a. $\sum_{\mathbit{i}}\left|\mathbit{c}_\mathbit{i}\right|^\mathbf{2}=\mathbf{1}$

b. $\sum_{i}\left|c_i\right|^2=0$

c. $\sum_{i}\left|c_i\right|=1$

d. None of these

129: The Wien displacement law is defined by equation:

a. $\lambda_mT=1$

b. $\mathbf{\lambda}_\mathbf{m}\mathbf{T}=\mathbf{constant}$

c. $\frac{\lambda_m}{T}=constant$

d. $\lambda_mT=0$

130: ${\left\langle a\middle| b\right\rangle}^\ast$ =……………

a. $\left\langle b\middle| a\right\rangle$

b. $\left|\left.b\right\rangle\right.\left.\left\langle a\right.\right|$

c. ${\left\langle\mathbf{b}\middle|\mathbf{a}\right\rangle}^\ast$

d. $\left|\left.a\right\rangle\right.\left.\left\langle b\right.\right|$

131: The Compton shift in wavelength \Delta\lambda is given by:

a. $\frac{\mathbit{h}}{\mathbit{m}_\mathbit{o}\mathbit{C}}\left(\mathbf{1}-\mathbf{cos}{\mathbit{\phi}}\right)$

b. $\frac{h}{m_oC}\left(\cos{\phi}-1\right)$

c. $\frac{m_oC}{h}\left(1-\cos{\phi}\right)$

d. $\frac{m_oC}{h}\left(\cos{\phi}-1\right)$

132: The eigen states of a particle in one dimensional box are described by:

a. $\mathbit{\psi}_\mathbit{n}=\sqrt{\frac{\mathbf{2}}{\mathbit{l}}}\mathbf{sin}{\frac{\mathbit{n\pi x}}{\mathbit{l}}}$

b. $\psi_n=\frac{\sqrt2}{l}\sin{\frac{nlx}{\pi}}$

c. $\psi_n=\frac{l}{\sqrt2}\sin{\frac{n\pi l}{x}}$

d. $\psi_n=\frac{l}{\sqrt2}\sin{\frac{n\pi x}{l}}$

133: In one dimension the time dependent Schrodinger wave equation is:

a. $-\frac{\hbar^2}{2m}\frac{d^2\psi\left(x\right)}{dx^2}+V\left(x\right)\psi\left(x\right)=E\psi(x)$

b. $-\frac{\hbar^\mathbf{2}}{\mathbf{2m}}\frac{\mathbf{d}^\mathbf{2}\mathbf{\psi}}{\mathbf{d}\mathbf{x}^\mathbf{2}}+\mathbf{V}\left(\mathbf{x}\right)\mathbf{\psi}=\mathbf{i}\hbar\frac{\partial\psi}{\partial t}$

c. $-\frac{\hbar^2}{2m}\nabla^2\psi\left(x\right)+V\left(x\right)\psi\left(x\right)=E\psi(x)$

d. None of these

134: A wave function is written in terms of orthonormal basis as:

$\left|\left.\psi\right\rangle\right.=\frac{1}{\sqrt3}\left|\left.\psi_1\right\rangle\right.+\frac{1}{\sqrt6}\left|\left.\psi_2\right\rangle\right.+\sqrt{\frac{5}{6}}\left|\left.\psi_3\right\rangle\right$

The probability that the system is found in state $\left|\left.\psi_2\right\rangle\right$ . is:

a. $\frac{5}{6}$

b. $\mathbf{0}.\mathbf{17}$

c. 0.13

d. $\frac{1}{3}$

135: The x-component of operator for kinetic energy $\left(K.E.\right)$ is:

a. $-\frac{\hbar^\mathbf{2}}{\mathbf{2}\mathbit{m}}\frac{\partial^\mathbf{2}}{\partial\mathbit{x}^\mathbf{2}}$

b. $-\frac{\hbar}{2m}\frac{\partial}{\partial x}$

c. $-\frac{2\hbar}{m}\frac{\partial}{\partial x}$

d. $-\frac{\hbar}{m}\frac{\partial}{\partial x}$

136: If $\psi$ is a state function and A is an operator then in Heisenberg picture:

a. $A\left(t\right),\ \psi\left(0\right)$

b. $A\left(0\right),\ \psi\left(t\right)$

c. $\mathbf{A}\left(\mathbf{t}\right), \mathbf{\psi}\left(\mathbf{t}\right)$

d. $A\left(0\right), \psi\left(0\right)$

137: If $\int\psi^\ast\left(x\right)\psi\left(x\right)dx=1$ then $\psi\left(x\right)$ is:

a. Normalized

b. Orthogonal

c. Orthonormal

d. Commute

138: For a given surface there is a certain …………… of incident radiation below which no photoelectric current flows:

a. Velocity

b. Frequency

c. Intensity

d. K.E.

139: The momentum and total energy operators are said to:

a. Orthogonal

b. Normalized

c. Anti-commute

d. Commute

140: In black body radiation the energy of the oscillators and temperatures are related by …………… constant:

a. $\frac{E}{T^5}$

b. $\frac{\mathbf{E}}{\mathbf{T}^\mathbf{4}}$

c. $\frac{T^4}{\mathbit{E}}$

d. $\frac{T^5}{\mathbit{E}}$

141: The equation of continuity is described by:

a. $\frac{\partial J}{\partial t}-div\rho=0$

b. $div\rho-\frac{\partial J}{\partial t}=0$

c. $div\rho-\frac{\partial J}{\partial t}=0$

d. $\mathbf{divJ}-\frac{\partial\rho}{\partial t}=\mathbf{0}$

142: The energy levels of one dimensional harmonic oscillator are:

a. 2-fold degenerate

b. Equally spaced

c. Non-degenerate

d. a & c

143: The base kets of position space satisfy:

a. $\sum_{i}\left.\left|x\right.\right\rangle\left\langle\left.x\right|\right.$

b. $\int{\left.\left|\mathbf{x}\right.\right\rangle\left\langle\left.\mathbf{x}\right|\right.\mathbf{dx}=\mathbf{1}}$

c. $\int\left.\left|x\right.\right\rangle\left\langle\left.x\right|\right.\left.p\right\rangle dx=1$

d. All of these

144: The lowest possible energy E_o of simple harmonic oscillator is given by:

a. $\frac{2}{\hbar\omega}$

b. $\frac{1}{\hbar\omega}$

c. $\frac{\hbar\mathbf{\omega}}{\mathbf{2}}$

d. $\frac{3\hbar\omega}{2}$

145: Relativistic energy of an electron after collision is :

a. $\left(\mathbit{p}^\mathbf{2}\mathbit{c}^\mathbf{2}+\mathbit{m}_\mathbit{o}^\mathbf{2}\mathbit{c}^\mathbf{4}\right)^\frac{\mathbf{1}}{\mathbf{2}}$

b. $\left(p^2c^2+m_o^2c^2\right)^\frac{1}{2}$

c. $\left(p^2c^2+m_o^4c^4\right)^\frac{1}{2}$

d. $\left(pc+m_oc^2\right)^\frac{1}{2}$

146: The energy of simple harmonic oscillator can be described quantum mechanically as:

a. $E_n=h\left(n+\frac{1}{2}\right)$

b. $\mathbf{E}_\mathbf{n}=\mathbf{hv}\left(\mathbf{n}+\frac{\mathbf{1}}{\mathbf{2}}\right)$

c. $E_n=h\left(n-\frac{1}{2}\right)$

d. $E_n=hv\left(n-\frac{1}{2}\right)$

147: If two operators have simultaneous Eigen functions then they are:

a. Orthogonal

b. Normalized

c. Anti-commute

d. Commute

148: The results of classical quantum mechanics must agree with classical mechanics according to

a. Dirac

b. Ehrenfest

c. Schrodinger

d. Heisenberg

148: The operator $\left(\frac{d}{dx}+ax\right)$ ……………….. the state of a particle:

a. Lowers to one step

b. Raises one step

c. Lowers to ground

d. Doesn’t change

149: The coefficients of expansion for some basis can be written as:

a. $a_i=\left(\sum_{i}{|\left.\varphi_i\right\rangle}\right)|\left.\psi\right\rangle$

b. $\mathbf{a}\mathbf{i}=\left\langle\mathbf{\varphi}\mathbf{i}\middle|\mathbf{\psi}\right\rangle=\int\mathbf{\varphi}_\mathbf{i}^\ast\mathbf{\psi dx}$

c. φi*ψdx

d. All of these

150: The degeneracy w.r.t. an energy eigen state can be best described by saying:

a. Two or more particles states have zero momentum

b. Two or more different particles states have same energy

c. Two or more different particles states have different energy

d. Heisenberg uncertainty principle

MCQ's with Answers

1: The wave functions of electron, proton and neutron are

2: The wave functions of α particle, photon and deuteron are

4: Every single particle state can be occupied by at most one fermion is called

6: The exchange operator apply on

7: A state of N fermions must be………. under every possible exchange operator:

8: The expression which describes the wave function of a multi-fermionic system that satisfies anti-symmetry conditions?

9: Quarks are … particles.

10: Quantum Field Theory describes the interaction among

11: The magnetic moment of an electron circulating in an orbit with an orbital angular momentum L is given by

12: Eigen values of square of spin angular momentum are:

13: Spin angular momentum S in term of Pauli spin matrices can be written as

14: Eigen values of square of total or general angular momentum

15: In case of general angular momentum, the values of magnetic quantum number are

16: Y-component of spin angular momentum is matrix form has complex entities, its eigen values are

17: Eigen values of none of Pauli spin matrix σy are real, what can you say about its eigen vectors?

18: Value of square of any Pauli spin matrix is … matrix

19: Components of spin angular momentum

20: Trace of any of Pauli spin matrix is

21: The electron proton and neutron are

22: The α particle, photon and deuteron are

23: The wave functions of electron, proton and neutron are

24: The wave function of α particle, photon and deuteron are

29: Every single particle state can be occupied by at most one fermion. This is called

30: For S=0 multiplicity is

31: For S=1, multiplicity is

32: The spin multiplicity is defined as

33: The z-component of the spin angular momentum S can have only two eigenvalues

34: In non-degenerate time-independent perturbation theory the λ varies from

35: When the operator applying on a state and it gives the different eigen values

36: The operator changes in time while the Hamiltonian remains fixed in time

37: WKB method can be viewed as … approximation

38: Both basis and operators carry time dependence in

39: Potential V (r) slightly varies in

40: Variational principle states that

41: WKB method is a technique for obtaining solutions in

42: Variational method is used to find the approximation to the state

43: The wave function ……. becomes in WKB approximation when E = V (r)

44: Height of central peak in probability is proportional to

45: ωfi = 0 is maximum when

46: The transition probability exhibits a pattern like

47: The transition rate exits for constant perturbation when the energy is

48: Transition probability of harmonic perturbation will be maximum when ωfi is equal to

49: Probability at constant perturbation is appreciable at

50: We attain delta function when time is

51: Quantization of radiation is achieved by which operators

52: No electronic transition will occur unless

53: The spin helicity of a photon is

54: The perpendicular distance between the path of incident particle and center of a potential field created by target nucleus is

55: What kind of wave we get when plane wave strikes with target in Born approximation

56: In quantum scattering, we take incident particle in the form of

57: In quantum scattering, motion of incident particle is in

58: dσ is directly proportional to

59: If Kr>>1, the region will be called as

60: V (r) can be spherically — — — — — — —— potential

61: For large values of r, potential V (r) approaches to

62: Born approximation is valid for incident particle with

63: In the scattering of Born approximation, particles have

64: The position (x) and momentum (px) operators are always

65: Expression for z-component of angular momentum (L) is

66: Components of angular momentum () are

67: Value of [L,L] is

68: The angular momentum of an isolated system is

69: Spin does not depend on

70: For Hydrogen atom that is in ground state, the orbital angular momentum will be

71: If azimuthal quantum number (l) is 2, the number of values of the magnetic quantum number (ml) will be

72: Square of angular momentum is

73: The square of angular momentum J2 commutes with

74: Application of Barrier Tunneling are

75: The energy of free particle in 3D is

76: The Centrifugal potential depends on …. quantum number

77: Three dimensional problems often exhibit degeneracy, which occur whenever ….

78: Value of square of any Pauli spin matrix is — — —— matrix

79: As the particle angular momentum increases, the particle becomes less and less

80: Eigen values of square of spin angular momentum are

81: Radial equation for central potential depends on — — — — — — quantum number

82: The Hamiltonian of rigid rotator

83: The quantized energy of rigid rotator is

84: Consider a hydrogen atom that is placed in an external magnetic field. The effect of an external magnetic field on atom is to cause a shift of its energy levels, this is called

85: The strong field Zeeman effect is also called

17: Eigen values of none of Pauli spin matrix σ_y are real, what can you say about its eigen vectors?

**43: The wave function ……. becomes in WKB approximation when E = V (r)**

**45: ω_fi = 0 is maximum when**

48: Transition probability of harmonic perturbation will be maximum when ω_fi is equal to

**60: V (r) can be spherically — — — — — — —— potential**

**64: The position (x) and momentum (p_x) operators are always**

71: If azimuthal quantum number (l) is 2, the number of values of the magnetic quantum number (m_l) will be

73: The square of angular momentum J² commutes with

**78: Value of square of any Pauli spin matrix is — — —— matrix**

**81: Radial equation for central potential depends on — — — — — — quantum number**