Quantum Mechanics - I

Course: M.Sc. Code: EF-514 Branch: Physics Year: 2021

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Total No. of Questions : 11
[Total No. of Printed Pages : 8
EF-514
M.Sc. Ind Semester (New/ATKT)
Examination, 2021-22
Physics
Paper - III
Quantum Mechanics - I
Time : 3 Hours]
[Maximum Marks : 85
Note :- Attempt all questions.
SECTION - 'A'
Objective Type Questions
1.
Choose the correct answer. 10×1.5=15
(i) Photon density is proportional to
(a) A2
(b) A2
(c) A2/3
(d) None of them
(ii) The equation of continuity probability.
(a) dp/dt = -∇⋅J⃗
(b) dp/dt = -∇⋅J
(c) ∇+∇⋅J = 0
(d) J+∇P = 0
(iii) For function of the variable X the parity operator x is defined as
(a) πψ(x) = ψ(x)
(b) πψ(x) = ψ(-x)
(c) πψ(x) = φ(x)
(d) None of them
(iv) According to quantum mechanics the energy levels of a particle excecuting one dimensional simple harmonic motion are:
(a) Continuous
(b) Discrete but equispaced
(c) Discrete but not equispaced
(d) Nothing cab be said
(v) Eigen value of L2 is given by
(a) m (m + 1) ℏ2
(b) m2 (m + 1) ℏ2
(c) l (l + 1) ℏ2
(d) l2 + (l + 1) ℏ2
(vi) The quantity |ψ|2 represents
(a) Probability density
(b) Charge density
(c) Energy density
(d) Intensity of wave
(vii) From definition of unitary matrix A† A = AA† =
(a) I
(b) 2I
(c) 5I
(d) None of them
(viii) Commutation relation of J+ and J- mutually [J+, J-]:
(a) 0
(b) 2I
(c) 5I
(d) None of them
(ix) Angular momentum operator is
(a) L = r × (ℏ/i) ∇
(b) L = - (ℏ/i) × x
(c) L = r . (ℏ/i) ∇
(d) Both a and b
(x) Any mathematical operation, differentiation integration, division, multiplication, addition, subtraction etc. can be represented by :
(a) Wave function
(b) Operator
(c) Both a and b
(d) None of them
SECTION - 'B'
Short Answer Type Questions 5×5=25
2.
Explain normality, orthogonality and closure properties of eigen functions?
OR
Find the energy value and energy function of a particle for infinite deep potential well. (onedimensional)
3.
Explain and discuss concept of Hilbert space?
OR
Show that the equation of motion is x = p̂/m where p̂ is the opertor associated with momentum.
4.
Discuss Hydrogen like atom.
OR
Discuss low energy nuclear state.
5.
Prove the following commutation relation.
[Lx, Ly] = iℏLz, [Ly, Lz] = iℏLx
and [Lz, Lx] = iℏLy
OR
Determine the matrices for ̂Lx and ̂Ly
6.
Write short notes on any one
(i) Equation of continuity
(ii) Deutron
OR
(iii) Bra and ket notation for state vector and its properties.
(iv) Parity operator.
SECTION - 'C'
Long Answer Type Questions 9×5=45
7.
State and prove Ehrenfet's theorem.
OR
Calculate the transmission coefficient of electron through one dimensional rectangular potential barrier.
8.
What do you mean by change of basis, similarity and unitary transformations explain it in brief.
OR
Explain Heisenberg Uncertainty through operators.
9.
Calculate the discrete energy levels of a particle in one dimensional square well potential with.
(i) Perfectly rigid walls
(ii) Finite potential step
OR
Establish schrodinger equation for a linear harmonic oscillator and solve it to obtain its eigen values and eigen functions. Discuss the significance of zero point energy.
10.
Discuss time independent perturbation theory for non-degenerate case and obtain expression for the first order correction to energy and eigen function ?
OR
Derive the eigen value and eigenfunction of Lz and L2 in terms of spherical harmonics.
11.
Write short notes on any two of the following :-
(i) Time dependent perturbation theory in short.
(ii) Bra and ket notation for state vector and its properties.
(iii) Unitary operator
(iv) Parity operator.