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 σy 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. Eg ≤ En
b. Eg ≥ En
c. Eg = En
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: ωfi = 0 is maximum when
a. Ef = Ei
b. Ef > Ei
c. Ef < Ei
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 ωfi is equal to
a. + ω
b. ω
c. $\pm\mathbf{\omega}$
d. ω = 0
49: Probability at constant perturbation is appreciable at
a. ωfi = 0
b. ωfi = 1
c. ωfi = 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 (px) 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. ms
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 (ml) 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. Jx
b. Jy
c. Jz
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
