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

1: The principle that all microscopic physical entities have both wave and particle properties is called the wave-particle

a. Singularity

b. Duality

c. Triality

d. Infinality

2: The full Schrodinger’s equation in compact form is

a. $\mathbf{H\mathrm{\Psi}=i\hbar\frac{\partial\Psi\ \ }{\partial t}}$

b. $H\mathrm{\Psi}\ =\hbar\frac{\partial\Psi}{\partial t}$

c. $H\mathrm{\Psi}\ =\ i\frac{\partial\Psi}{\partial t}$

d. $H\mathrm{\Psi}\ =\ i\hbar\frac{\partial\Psi}{\partial x}$

3: In the probabilistic interpretation of wave function , the quantity is

a. Probability density

b. Probability amplitude

c. Negative probability

d. 1

4: The probability of finding a particle in differential region dx is

a. $\mathrm{\Psi}(x,\ t)dx$

b. $\mathrm{\Psi}\left(x,t\right)^\ast dx$

c. $[\mathrm{\Psi}\left(x,t\right)^\ast/\mathrm{\Psi}(x,t)] dx$

d. $\mathbf{\Psi\left(x,t\right)^\ast\Psi(x,\ t)dx}$

5: A physical requirement on wavefunctions is that they should be

a. Reliable

b. Square integrable

c. Zero

d. Discrete

6: The charge on an electron is represented by e. Which of the following charges can exist

a. 2.0 e

b. 2.5 e

c. 3.6 e

d. 5.2 e

7: The energy of a photon depends on its

a. Amplitude

b. Speed

c. Temperature

d. Pressure

8: When an electron jumps from an orbit where n = 1 to n = 4, its energy in terms of the energy of the ground level E₁ is

a. $E_{1/9\ }$

b. $2E_1$

c. $\mathbf{4E_1}$

d. $\mathbf{4E_1}$

9: How much of the universe is comprised of matter and energy that is explained by current Physics theory

a. 95 %

b. 75 %

c. 50 %

d. 5 %

10: The kinetic energy of photoelectrons depends on the

a. Speed of light

b. Intensity of the light

c. Number of incident photons

d. Photon frequency

11: The dynamical variables are real quantities that are measurable and so they are represented by

a. Hermitian operator

b. Anti-Hermitian operator

c. Linear operator

d. Non-linear operator

12: The maximum number of linearly independent vectors of a vector space is called − − − − − of vector space

a. Dimension

b. Basis

c. Modulus

d. None of these

13: A set of vectors in a vector space is called

a. Dimension

b. Basis

c. Modulus

d. Probability

14: Heisenberg uncertainty principle is a direct consequence of − − − − − of operators

a. Commutivity

b. Non commutivity

c. Probability

d. Conductivity

15: The eigen values of Apnti Hermitian operator are always

a. Complex

b. Real

c. Imaginary

d. All of above

16: The eigen values of Hermitian operator are always

a. Complex

b. Real

c. Imaginary

d. All of above

17: The eigen values of operators corresponding to dynamical variables are

a. Complex

b. Real

c. Imaginary

d. All of above

18: The bra space is − − − − − − − to kept space

a. Hilbert space

b. Vector space

c. Dual space

d. All of above

19: Which one of the following is not a Hermitian operator

a. Position

b. Momentum

c. Projection

d. Unitary

20: If two vectors are normalized and orthogonal, we called them − − − − − − − vectors

a. Normalized

b. Orthogonal

c. Orthonormal

d. None of these

21: If two operators are commute with each other, they possess a −−−−−−− eigenstate

a. Degenerate

b. Non-degenerate

c. Simultaneous

d. Orthogonal

22: Equation $\hat{A}\psi=\lambda\psi$ is called −−−−−−− equation

a. Expectation value

b. Operator

c. Wave function

d. Eigen value

23: Expectation value of an observable is always

a. Imaginary

b. Complex

c. Real

d. Commutative

24: In Schrodinger Picture, the wavefunctions are time dependent while − − − − − − − are time independent

a. Operators

b. Wave functions

c. Eigen values

d. None

25: In Heisenberg Picture, the operators are time dependent while −−−−−−− are time independent

a. Wave functions

b. Operators

c. Eigen values

d. None

26: In Interaction Picture, both − − − − − − − are time dependent

a. Eigen values, Wave functions

b. Operators, Wave functions

c. Eigen values, Operators

d. None

27: The solution of time dependent Schrodinger equation for a time independent potential is − − − − − − − state

a. Bound

b. Unbound

c. Stationary

d. All of above

28: The states corresponding to discrete spectra are called − − − − − − − states

a. Bound

b. Unbound

c. Stationary

d. All of above

29: The states corresponding to Continuous spectra are called − − − − − − − states

a. Bound

b. Unbound

c. Stationary

d. All of above

30: If transformation from one orthonormal set into another is unitary, such transformation is

a. Linear

b. Non linear

c. Unitary

d. Similarity

31: In one dimensional problem the energy levels of a bound state system are discrete and

a. Degenerate

b. Non degenerate

c. Complete

d. None

32: In one dimensional problem the energy levels of unbound state system are continuous and

a. Degenerate

b. Non degenerate

c. Complete

d. None

33: Unbound states occur in those cases where the motion of system is not confined example is

a. Step potential

b. Potential barrier

c. Free Particle

d. All of above

34: In one dimensional problem the energy levels of unbound state system (V₁ < E < V₂) are continuous and

a. Degenerate

b. Non degenerate

c. Complete

d. None

35: Bound states occur in those cases where the motion of system is confined; example is

a. Particle in a box

b. Infinite Square Well

c. Free Particle

d. Both a and b

36: When V (x) is even, the corresponding Hamiltonian is

a. Odd

b. Even

c. Degenerate

d. Non degenerate

37: The Number operator N is − − − − − − − operator

a. Parity

b. Projection

c. Identity

d. Hermitian

38: The sum of Reflection (R) and Transmission (T) Coefficients is equal to

a. 0

b. 1

c. Both a and b

d. None

39: The action of a on |n > generates a new state |n-1 > with value

a. n

b. n-1

c. $\mathbf{\sqrt n}$

d. $\sqrt{n-1}$

40: The action of a on |n> generates a new state |n + 1> with value

a. n

b. n+1

c. $\sqrt n$

d. $\sqrt{n+1}$

41: The Position and Momentum operators are always

a. Commutative

b. Anti Commutative

c. Normalized

d. Orthogonal

42: Expression for z-component of angular momentum is

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

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

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

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

43: Components of angular momentum are

a. Orthogonal

b. Normal

c. Commute

d. Anti Commute

44: Value of L X L is

a. 0

b. 1

c. $i\hbar L$

d. All of above

45: The angular momentum of an isolated system is

a. Conserved

b. Non-Conserved

c. Variable

d. None

46: Spin does not depend on

a. S

b. $m_s$

c. spatial degrees of freedom

d. All of above

47: For Hydrogen atom that are in ground state, the orbital angular momentum will be

a. 1

b. 2

c. 0

d. Unknown

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

a. 2

b. 3

c. 4

d. 5

49: Square of angular momentum is

a. Hermitian

b. Anti Hermitian

c. Linear

d. Projection Operator

50: The square of angular momentum (J²) commutes with

a. $J_x$

b. $J_y$

c. $J_z$

d. All of these

51: Application of Barrier Tunneling are

a. Radioactive Decay

b. Semiconductor Devices

c. Both a and b

d. none

52: The energy of free particle in 3D is

a. Triply degenerate

b. Non degenerate

c. Infinitely Degenerate

d. None

53: The Centrifugal potential depends on − − −− quantum number

a. Orbital

b. Magnetic

c. Principle

d. All of above

54: Three dimensional problems often exhibit degeneracy, which occur whenever − − −− is symmetric

a. Eigen value

b. Wavefunction

c. Potential

d. None

55: Value of square of any Pauli spin matrix is − − −− matrix

a. Null

b. Diagonal

c. Identity

d. None

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

a. Repulsive

b. unbound

c. bound

d. Both a and b

57: Eigen values of square of spin angular momentum are

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

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

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

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

58: Radial equation for central potential depends on − − − − − − − quantum number

a. Azimuthal

b. Magnetic

c. Spin

d. None

59: The Hamiltonian of rigid rotator

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

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

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

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

60: The quantized energy of rigid rotator is

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

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

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

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

61: The principle that all microscopic physical entities have both wave and particle properties is called the wave-particle:

a. Singularity

b. duality

c. Triality

d. Infinality

62: The full Schrodinger’s equation in compact form is:

a. $H\Psi=i\hbar\frac{\partial\Psi}{\partial t}$

b. $H\Psi=\hbar\frac{\partial\Psi}{\partial t}$

c. $H\Psi=i\frac{\partial\Psi}{\partial t}$

d. $H\Psi=-i\hbar\frac{\partial\Psi}{\partial t}$

63: In the probabilistic interpretation of wave function \Psi the quantity \left|\Psi\right|^2 is:

a. Probability density

b. Probability amplitude

c. Negative probability

d. 1

64: The probability of finding a particle in differential region dx is :

a. $\Psi\left(x,t\right)$ dx

b. $\Psi\left(x,t\right)^\ast$ dx

c. $\frac{\Psi\left(x,t\right)^\ast}{\Psi(x,t)}$ dx

d. $\mathbf{\Psi}\left(\mathbf{x},\mathbf{t}\right)^\ast\mathbf{\Psi}\left(\mathbf{x},\mathbf{t}\right)\mathbit{dx}$

65: A physical requirement on wave functions is that they should be:

a. Reliable

b. Square Integrable

c. Zero

d. Discrete

66: The charge on an electron is represented by e which of the following charges can exist:

a. 2.0 e

b. 2.5\ e

c. 3.6\ e

d. 5.2\ e

67: The energy of phonons depend upon its

a. Amplitude

b. Speed

c. Temperature

d. Pressure

68: When an electron jumps from an orbit where n=1 to n=4 its energy in terms of the energy of ground level E1 is:

a. $\frac{E1}{9}$

b. 2E1

c. 4E1

d. 16E1

69: How much of the universe is comprised of matter and energy that is explained by current physics theory:

a. 95%

b. 75%

c. 50%

d. 5%

70: The Kinetic energy of photo-electrons depend on the:

a. Speed of light

b. intensity of light

c. number of incident photon

d. photon frequency

71: The dynamical variables are real quantities that are measurable and so they are represented by:

a. Hermitian operator

b. Anti-Hermitian Operator

c. Linear operator

d. Non-linear Operaor

72: The maximum number of linearly independent vectors of a vector space is called……….. of vector space:

a. Dimension

b. basis

c. modulus

d. none of these

73: A set of vectors in a vector space is called:

a. Dimension

b. basis

c. modulus

d. probability

74: Heisenberg uncertainity principle is a direct consequence of ………….. of operators:

a. Commutivity

b. Non-commutivity

c. Probability

d. Conductivity

75: The eigen values of anti-hermitian operator are always:

a. Complex

b. Real

c. Imaginary

d. All of these

76: The eigen values of hermitian operator are always:

a. Complex

b. Real

c. Imaginary

d. All of these

77: The eigen values of operators corresponding to dynamical variables are:

a. Complex

b. Real

c. Imaginary

d. All of these

78: The bra space is ………………….. to ket space:

a. Hilbert space

b. Vector space

c. Dual space

d. All of these

79: Which one of the following is not a hermitian operator:

a. Position

b. Momentum

c. Projection

d. Unitary

80: If two vectors are normalized and orthogonal we called them ……….. vectors:

a. Normalized

b. Orthogonal

c. Orthonormal

d. None of these

81: If two operators commute with each other they possess a…………. eigen state:

a. Degenerate

b. Non-degenerate

c. Simultaneous

d. Orthogonal

82: In equation $\hat{A}\Psi=\lambda$ Psi is called:

a. Expectation value

b. Operator

c. Wave function

d. Eigen value

83: Expectation value of an observable is always:

a. Imaginary

b. Complex

c. Real

d. Commutative

84: In schrodinger picture the wavefunctions are time dependent while ………. Are time independent:

a. Operators

b. Wavefunctions

c. Eigen values

d. None

85: In Heisenberg picture the operators are time dependent while……… are time independent:

a. Wavefunctions

b. Operators

c. Eigen values

d. None

86: In interaction picture both …………….. are time dependent:

a. Eigen values, wavefunctions

b. Operators, wavefunctions

c. Eigen values, operators

d. None

87: The solution of time dependent Schrodinger equation for a time independent solution is …..… state:

a. Bound

b. Unbound

c. Stationary

d. All of these

88: The states corresponding to discrete spectra are called …………….. states:

a. Bound

b. Unbound

c. Stationary

d. All of these

89: The states corresponding to continuous spectra are called …………… states:

a. Bound

b. Unbound

c. Stationary

d. All of these

90: If transformation from one orthonormal set into another is unitary such transformation is:

a. Linear

b. Non-linear

c. Unitary

d. Similarity

91: In one dimensional problem the energy levels of a bound state system are discrete and:

a. Degenerate

b. Non-degenerate

c. Complete

d. None

92: In one dimensional problems the energy levels of bound state system \left(E>V\right) are continuous and:

a. Degenerate

b. Non-degenerate

c. Complete

d. None

93: Unbound states occur in those cases where the motion of system is not confined; example is:

a. Step potential

b. Potential barrier

c. Free particle

d. All of these

94: In one dimensional problem the energy levels of unbound state system \left(V_1<E<V_2\right) are continuous and:

a. Degenerate

b. Non-degenerate

c. Complete

d. None

95: Bound states occur in those cases where the motion of a system is confined; example is:

a. Particle in a box

b. Infinite square well

c. Free particle

d. Both a & b

96: When $V\left(x\right)$ is even the corresponding Hamiltonian is:

a. Odd

b. Even

c. Degenerate

d. Non-degenerate

97: The number operator N is ……………. Operator:

a. Parity

b. Projection

c. Identity

d. Hermitian

98: The sum of reflection and transmission coefficients is equal to:

a. 0

b. 1

c. Both a and b

d. None

99: The action a on |n> generates a new state |n-1> with value:

a. n

b. n-1

c. $\sqrt\mathbit{n}<strong>$

d. $\sqrt{n-1}$

100: The action a on |n> generates a new state |n+1> with value:

a. n

b. $\sqrt n$

c. $\sqrt{\mathbit{n}+\mathbf{1}$

d. n+1

101: The position and momentum operator are always :

a. Commutative

b. Anti-commutative

c. Normalized

d. Orthogonal

102: Expression for $-\mathbit{i}\hbar\frac{\partial}{\partial\phi}$ of angular momentum is:

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

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

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

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

103: Components of angular momentum are:

a. Orthogonal

b. Normal

c. Commute

d. Anti-commute

104: Value of $L\times L$ is:

a. 0

b. 1

c. $i\hbar L$

d. All of these

105: The angular momentum of an isolated system is:

a. Conserved

b. Non-conserved

c. Variable

d. None

106: Spin does not depend on:

a. $J_x$

b. $J_y$

c. Spatial degrees of freedom

d. All of these

107: For hydrogen atom that are in ground state the orbital angular momentum will be:

a. 1

b. 2

c. 0

d. Unknown

108: If azimuthal quantum number is 2 the number of values of magnetic quantum number will be:

a. 2

b. 3

c. 4

d. 5

109: Square of angular momentum is:

a. Hermitian

b. Anti-Hermitian

c. Linear

d. Projection Operator

110: The application of barrier tunneling are:

a. Radioactive decay

b. Semiconductor devices

c. Both a & b

d. None

111: The energy of free particle in 3D is:

a. Triply degenerate

b. Non-degenerate

c. Infinitely degenerate

d. None

112: The centrifugal potential depends on…….. quantum number:

a. Orbital

b. Magnetic

c. Principle

d. All of these

113: Three dimensional problems often exhibit degeneracy which occur whenever …….is symmetric:

a. Eigen values

b. Wavefunctions

c. Potential

d. None

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

a. Null

b. Diagonal

c. Identity

d. None

115: As the particle angular momentum increases the particles becomes less and less:

a. Repulsive

b. Unbound

c. Bound

d. Both a and b

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

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

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

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

d. $2S(S+1)\hbar^2$

117: Radial equation for central potential depends on……… quantum number:

a. Azimuthal

b. Magnetic

c. Spin

d. None

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

119: The quantized energy of rigid rotator is:

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

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

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

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

120: A photocell is illuminated by a source 1m away. When a source is taken 2m away:

a. The number of electrons emitted is quarter of the initial number

Hint: $\left(\mathbf{As}\ \ \mathbit{I}\propto\frac{\mathbf{1}}{\mathbit{r}^\mathbf{2}}\right)$

a. The number of electrons is emitted is a half of initial number

b. Each electron emitted carries one quarter of the initial energy

c. Each electron emitted carries double of the initial energy

121: Photoelectric effect supports quantum nature of light because:

a. There is minimum of light below which no photoelectrons are emitted

b. Electric charge of photoelectrons is quantized

c. Maximum kinetic energy of photoelectrons depends only on frequency of light and not on its intensity

d. a and c

122: The process of photoelectric emission depends upon:

a. Wavelength of incident light

b. Work function of surface

c. Nature of surface

d. All of above

123: Which of the following achieve conversion of electromagnetic wave energy into electrical energy:

a. Photocell

b. Coolidge tube

c. C.R.O

d. Vacuum tube

124: The wavelength of matter waves does not depend on:

a. Mass

b. Charge

c. Momentum

d. Velocity

125: If value of plank’s constant is more than present value the De-Broglie wavelength associated with a material particle will be:

a. Less

b. More

c. Same

d. None of these

126: The De-Broglie wavelength of an electron is 0.2A^o. The de-celerating potential to stop it will be:

a. -3762V

b. -1000V

c. -4000V

d. -2000V

127: Which of the following will have the shortest wavelength while moving with the same speed:

a. Proton

b. Neutron

c. Deuteron

d. Electron

128: The waves associated with moving particles are known as:

a. Sound waves

b. Matter waves

c. Electromagnetic waves

d. Stationary waves

129: The wave nature of electrons was verified by:

a. Electron diffraction experiments

b. Compton effect

c. Photoelectric effect

d. None of these

130: The ratio of velocities of a proton and alpha particle is 4:1. The ratio of their De-Broglie wavelength will be:

a. 1:2

b. 1:4

c. 1:1

d. 4:1

132: The De-Broglie wavelength of a neutron at 27 degc . Its wavelength at 927\degc will be:

a. $\frac{\lambda}{9}$

b. $\frac{\lambda}{4}$

c. $\frac{\lambda}{3}$

d. $\frac{\mathbit{\lambda}}{\mathbf{2}}$

Hint: $\left(\mathbf{\lambda}=\frac{\mathbf{h}}{\sqrt{\mathbf{2m}\ \mathbf{k}_\mathbf{B}\mathbf{T}}}\right)$

133: The hypothesis regarding dual nature of material was proposed by:

a. De-Broglie

b. Davisson

c. Germer

d. Heisenberg

134: The waves associated with electrons revolving in various Bohr’s orbits in an atom are:

a. Stationary

b. Transverse

c. Longitudinal

d. Progressive

135: Which of the following properties of electron is made use of electron microscope:

a. High velocity

b. Wave nature

c. Diffraction

d. Interference

135: The wavelength of a photon, electron and uranium nucleus are equal. The minimum energy is associated with:

a. Electron

b. Photon

c. Uranium

d. All of these

136: An electron in hydrogen atom makes a transition from an excited state to ground state. Which of the following statement is true:

a. Its kinetic energy decreases, potential energy increases and total energy remains same

b. Its kinetic energy increases, potential energy and total energy decreases

c. Its kinetic energy and total energy decreases, potential energy increases

d. Its kinetic, potential and total energies decrease

137: Three photons coming from excited atomic hydrogen sample are picked up their energies are 12.1eV, 10.9eV and 1.9eV. These photons must come from:

a. Single atom

b. Two atoms

c. Three atoms

d. Either two or three atoms

138: The quantity $\left|\Psi(x,t)\right|^2$ is called:

a. Probability density

b. Probability current density

c. Probability

d. All of these

139: The quantity $\left|\Psi(x,t)\right|^2$ dx is called:

a. Probability density

b. Probability current density

c. Probability

d. All of these

140: The condition $\int_{-\infty}^{+\infty}{\Psi^\ast(x,t)\Psi(x,t)dx}=1$ is called:

a. Normalization condition

b. Probability density condition

c. Probability

d. All of these

141: The Schrodinger wave equation is:

a. $\hat{H}\Psi=E\Psi$

b. $-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi=E\Psi$

c. $-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi=i\hbar\frac{\partial\Psi}{\partial t}$

d. all of these

142: The dynamical variables are real quantities that are measurable and so they are represented by:

a. Hermitian operators

b. Anti-hermitian operators

c. Non-linear operators

d. None

143: A measurement of identical states need not give identical results but only identical:

a. Probability distributions

b. Charge distributions

c. Momentum distributions

d. Velocity distributions

144: Position and momentum of an electron cannot be measured simultaneously. This is statement of:

a. Uncertainty principle

b. Ehrenfest theorem

c. Bohr postulate

d. None

145: Heisenberg uncertainty principle is a direct consequence of ………….. of operators corresponding to canonically conjugate variables:

a. Commutivity

b. Non-commutivity

c. Conductivity

d. None of these

146: Heisenberg has proposed certain though experiments known as………… to illustrate the principle of uncertainty:

a. Gedanken experiments

b. Bohr experiments

c. Deuteron experiments

d. Electron experiments

147: A given state of a physical system is represented by a vector in multi-dimensional complex linear vector space with an orthonormal basis called:

a. Vector space

b. Euclidean space

c. Inner product space

d. Hilbert space

148: The result of measurement of an observable quantum mechanical system in a particular state is given by…………. Value of corresponding Hermitian operator in that state:

a. Eigen

b. Expectation

c. Characteristic

d. None of these

149: The time evaluation of state vector \Psi of a system is governed by equation:

a. $-\mathbf{i}\hbar\frac{\partial\Psi\left(\mathbf{t}\right)}{\partial t}=\hat{\mathbf{H}}\mathbf{\Psi}(\mathbf{t})$

b. p=mv

c. $\lambda=\frac{h}{mv}$

d. All of these

150: In equation $\hat{A}\Psi=\lambda\Psi$ , $\lambda$ is called:

a. Expectation value

b. Eigen value

c. a & b

d. None of these

150: The Eigen values of operators corresponding to dynamical variables are:

a. Complex

b. Integers

c. Real

d. All of these

151: Any arbitrary wave function of physical interest can always be written as……….. of eigen vectors of operators corresponding to dynamical variables of a system:

a. Linear combination

b. Superposition

c. a & b

d. None of these

152: The Eigen values of Hermitian operator are always:

a. Complex

b. Integers

c. Real

d. All of these

153: It is necessary for two operators to commute with each other for having a………..

a. Degenerate Eigen value

b. Simultaneous Eigen functions

c. Orthogonal Eigen functions

d. All of these

154: The quantum theory will give the results identical to classical theory if masses and dimensions of system under consideration are made to approach the ………….. size:

a. Microscopic

b. Macroscopic

c. a & b

d. None of these

155: Corresponding to every observable there exists a Hermitian operator. The only measurable values of a physical observable are the various………… values of corresponding operator:

a. Expectation

b. Eigen

c. a & b

d. None of these

156: For a system consisting of a particle moving in a field of conservative force there is an associated complex wave function $\Psi(x,y,z,t)$ where x,y,z are space coordinates and “t” is time. This function enables us to obtain a description of the behavior of the system consistent with:

a. Uncertainty principle

b. Ehrenfest theorem

c. Bohr postulate

d. None of these

157: Let V₁ & V2 be two distinct vector spaces. The vector space obtained by multiplying element of V₁by every element of V₂ is called………… of V1 with V₂:

a. Density product

b. Direct product

c. Probability

d. All of these

158: The number of basis vectors of a vector space is called……….. of vector space:

a. Dimensionality

b. Current density

c. Probability

d. None of these

159: The positive square root of scalar product of a vector with itself is called………… of vector:

a. Normalization condition

b. Probability condition

c. Norm

d. Modulus

160: If two vectors are normalized and orthogonal also we call them…….. vectors:

a. Normalized

b. Orthogonal

c. Orthonormal

d. None

161: The Bra space is ………… to Ket space:

a. Hilbert space

b. Vector space

c. Dual space

d. None

162: If determinant of a matrix is zero it is called:

a. Non-singular matrix

b. Singular matrix

c. Eigen value

d. Velocity distribution

163: The matrix transforming one orthonormal set into another is unitary and the transformation of this type is termed as:

a. Unitary transformation

b. Similarity transformation

c. Bohr postulate

d. None

164: In Schrodinger picture the…………. Are time dependent while ……… are time independent:

a. Wave function, Operator

b. Operator, Wave function

c. a & b

d. None

165: In Heisenberg picture the……….. are time dependent while………. Are time independent:

a. Wave function, Operator

b. Operator, Wave function

c. a & b

d. None

166: The operator $\hat{a}=\frac{1}{\sqrt{2\hbar m\omega}}(m\omega x+ip)$ is called:

a. Raising operator

b. Lowering operator

c. Inner product operator

d. Hilbert operator

167: A free particle cannot exist in a stationary state or to put it another way there is no such thing as a free particle with a definite:

a. Energy

b. Momentum

c. Acceleration

d. All of these

168: Generating function for Hermite polynomials is:

a. $e^{-t^2+2xt}=\sum_{n=0}^{\infty}\frac{t^n}{n!}H_n(x)$

b. $e^{-t^2-2xt}=\sum_{n=0}^{\infty}\frac{t^n}{n!}H_n(x)$

c. $e^{-t^2+2xt}=\sum_{n=0}^{\infty}\frac{t^n}{n!}H_n\left(ax\right)$

d. None of these

169: In one dimensional problem the energy levels of a bound state system are discrete and………..:

a. Degenerate

b. Non-degenerate

c. Orthogonal

d. Complete

170: Unbound states occur in those cases where the motion of a system is not confined, a typical example is:

a. Step potential

b. Potential barrier

c. Free particle

d. All of these

171: The ladder operator $\hat{a}=\frac{1}{\sqrt2}\left(\hat{q}-i\hat{p}\right)$ is called:

a. Raising operator

b. Lowering operator

c. a & b

d. None of these

172: The number operator $\hat{N}={\hat{a}}^\dag\hat{a}$ is……….. operator:

a. Unitary

b. Projection

c. Hermitian

d. All of these

173: A wave function \psi(x) is acceptable if it is:

a. Integrable

b. Single valued and continuous

c. Differentiable

d. All of these

174: For a particle subject to delta potential $V\left(x\right)=-V_o\delta\left(x\right), V_o>0$ , for non-negative energies the particle has:

a. Two unbound states

b. Only one bound state

c. a & b

d. None of these

174: Applications of barrier tunneling are:

a. Radioactive decays

b. Semiconductor devices

c. a & b

d. None of these

175: In case of harmonic oscillator the conditions $\psi\left(x_{min}\right)=0, \psi\left(x_{min}\right)=0$ , are satisfied as follows:

a. $\mathbf{\psi}\left(-\infty\right)=\mathbf{0}=\mathbf{\psi}\left(+\infty\right)$

b. $\psi\left(-a\right)=0=\psi\left(+a\right)$

c. $\psi\left(-\frac{a}{2}\right)=0=\psi\left(+\frac{a}{2}\right)$

d. None of these

176: The operator for angular momentum is:

a. $-i\hbar r\hat{r}\times\nabla$

b. $-i\hbar r\times\nabla$

c. – $\frac{\hbar}{i}r\hat{r}\times\nabla$

d. All of these

177: Components of angular momentum are:

a. Orthogonal

b. Anti-commute

c. Do not commute

d. None of these

178: Value of $\hat{L}\times\hat{L}$ is:

a. Zero

b. One

c. $-\mathbf{i}\hbar\hat{\mathbf{L}}$

d. None of these

179: Square of angular momentum is:

a. Hermitian

b. Projection operator

c. Unitary operator

d. All of these

180: Expression of z-component of angular momentum is:

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

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

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

d. None of these

MCQ's with Answers

1: The principle that all microscopic physical entities have both wave and particle properties is called the wave-particle

3: In the probabilistic interpretation of wave function , the quantity is

4: The probability of finding a particle in differential region dx is

5: A physical requirement on wavefunctions is that they should be

6: The charge on an electron is represented by e. Which of the following charges can exist

7: The energy of a photon depends on its

8: When an electron jumps from an orbit where n = 1 to n = 4, its energy in terms of the energy of the ground level E1 is

9: How much of the universe is comprised of matter and energy that is explained by current Physics theory

10: The kinetic energy of photoelectrons depends on the

11: The dynamical variables are real quantities that are measurable and so they are represented by

12: The maximum number of linearly independent vectors of a vector space is called − − − − − of vector space

13: A set of vectors in a vector space is called

14: Heisenberg uncertainty principle is a direct consequence of − − − − − of operators

15: The eigen values of Apnti Hermitian operator are always

16: The eigen values of Hermitian operator are always

17: The eigen values of operators corresponding to dynamical variables are

18: The bra space is − − − − − − − to kept space

19: Which one of the following is not a Hermitian operator

20: If two vectors are normalized and orthogonal, we called them − − − − − − − vectors

21: If two operators are commute with each other, they possess a −−−−−−− eigenstate

23: Expectation value of an observable is always

24: In Schrodinger Picture, the wavefunctions are time dependent while − − − − − − − are time independent

25: In Heisenberg Picture, the operators are time dependent while −−−−−−− are time independent

26: In Interaction Picture, both − − − − − − − are time dependent

27: The solution of time dependent Schrodinger equation for a time independent potential is − − − − − − − state

28: The states corresponding to discrete spectra are called − − − − − − − states

29: The states corresponding to Continuous spectra are called − − − − − − − states

30: If transformation from one orthonormal set into another is unitary, such transformation is

31: In one dimensional problem the energy levels of a bound state system are discrete and

32: In one dimensional problem the energy levels of unbound state system are continuous and

33: Unbound states occur in those cases where the motion of system is not confined example is

34: In one dimensional problem the energy levels of unbound state system (V1 < E < V2) are continuous and

35: Bound states occur in those cases where the motion of system is confined; example is

36: When V (x) is even, the corresponding Hamiltonian is

37: The Number operator N is − − − − − − − operator

38: The sum of Reflection (R) and Transmission (T) Coefficients is equal to

39: The action of a on |n > generates a new state |n-1 > with value

40: The action of a on |n> generates a new state |n + 1> with value

41: The Position and Momentum operators are always

43: Components of angular momentum are

44: Value of L X L is

45: The angular momentum of an isolated system is

46: Spin does not depend on

47: For Hydrogen atom that are in ground state, the orbital angular momentum will be

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

49: Square of angular momentum is

50: The square of angular momentum (J2) commutes with

51: Application of Barrier Tunneling are

52: The energy of free particle in 3D is

53: The Centrifugal potential depends on − − −− quantum number

54: Three dimensional problems often exhibit degeneracy, which occur whenever − − −− is symmetric

55: Value of square of any Pauli spin matrix is − − −− matrix

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

57: Eigen values of square of spin angular momentum are

58: Radial equation for central potential depends on − − − − − − − quantum number

59: The Hamiltonian of rigid rotator

60: The quantized energy of rigid rotator is

61: The principle that all microscopic physical entities have both wave and particle properties is called the wave-particle:

62: The full Schrodinger’s equation in compact form is:

63: In the probabilistic interpretation of wave function \Psi the quantity \left|\Psi\right|^2 is:

64: The probability of finding a particle in differential region dx is :

65: A physical requirement on wave functions is that they should be:

66: The charge on an electron is represented by e which of the following charges can exist:

67: The energy of phonons depend upon its

68: When an electron jumps from an orbit where n=1 to n=4 its energy in terms of the energy of ground level E1 is:

69: How much of the universe is comprised of matter and energy that is explained by current physics theory:

70: The Kinetic energy of photo-electrons depend on the:

71: The dynamical variables are real quantities that are measurable and so they are represented by:

72: The maximum number of linearly independent vectors of a vector space is called……….. of vector space:

73: A set of vectors in a vector space is called:

74: Heisenberg uncertainity principle is a direct consequence of ………….. of operators:

75: The eigen values of anti-hermitian operator are always:

76: The eigen values of hermitian operator are always:

77: The eigen values of operators corresponding to dynamical variables are:

78: The bra space is ………………….. to ket space:

79: Which one of the following is not a hermitian operator:

80: If two vectors are normalized and orthogonal we called them ……….. vectors:

81: If two operators commute with each other they possess a…………. eigen state:

83: Expectation value of an observable is always:

8: When an electron jumps from an orbit where n = 1 to n = 4, its energy in terms of the energy of the ground level E₁ is

34: In one dimensional problem the energy levels of unbound state system (V₁ < E < V₂) are continuous and

50: The square of angular momentum (J²) commutes with

126: The De-Broglie wavelength of an electron is 0.2A^o. The de-celerating potential to stop it will be: