M.Sc 1st Semester Physics (Quantum Mechanics-1) Paper 2024 Download PDF

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Roll No……..

B010704T
M.Sc. (First Semester)
(NEP) EXAMINATION, 2023-24
PHYSICS
(Quantum Mechanics -I)

Time : 2 Hours ]                                                                                                                [ Maximum Marks: 75

Note : attempt questions from all sections as directed.

Instruction : The candidates are required to answer only in serial order if there are many parts of equation answer them in continuation.

Section – A
(Short Answer Type Questions)

Note : All questions are compulsory each questions carries 5 marks

1 (A) define a hermitian operator to that the eigen values of a hermitian operator are real.

(B) Explain the matrix representation of wave function and operator.

(C ) The base vectors of a representatio are 10 and 01. Construct a transformation Matrix U for transformation to another representation having base vectors 1/√2 1/√2 and -1/√2 1/√2

(D) Distinguish between co-ordinate and momentum representations what are the operators for coordinate and momentum in the two representations ?

(E) Evaluate the most probable distance of the electron of the hydrogen atom in its 2p state.
Given :

(F) write the density operator and density Matrix for (i) a completely polarized electron beam in the pure state |Sz = +½ > and and polarized electron beam described by a mixture of 50% | Sz = -½ > states. Also so that the expectation value of Sz in case (ii) is zero

(G) Evaluate the following commutators :

(H) What are Pauli’s spin Matrices ? Show that :

(I) Write short note on any one of the following :

Ehrenfest Theorem
Exchange Degeneracy
Pauli’s Exclusion Principle

M.Sc 1st Semester Physics (Quantum Mechanics-1) Paper 2024

Section – B
(Long Answer Type Questions)

Note : Attempt anyone questions. Each questions carries 15 marks.

2. (i) define the uncertainty (∆A) in the measurement of a dynamical variable. State and explain the general uncertainty relation.

(ii) The wave function of a particle in a state is :

3. (i) Outline the different postulates of quantum mechanics.

(ii) consider a system whose state is given in terms of complete and Ortho normal set of three vectors | ¢1>, | ¢2 >, | ¢3> as follows :

4. Distinguish between Heisenberg and Schrodinger pictures. Show that the state vectors and operators are the same in both the
Pictures at t = 0. also obtain the equation of motion for an operator corresponding to a dynamical variable using Heisenberg picture.

5. Solve the problem of linear harmonic oscillator using the operator method.

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Section – C
(Long Answer Type Questions)

Note : Attempt anyone questions. Each questions carries 15 marks.

6. What is the symmetry transformation explain how the momentum operator becomes the generator of infinitesimal translation in space?
Prove that the Conservation of linear momentum is a consequence of the translational invariance of the system.

7. Derive the eigen value-eigen vector relations for the operators J² and Jz. Hence obtain the matrices for J² and Jz.

8. Describe symmetric and anti symmetric wave function. Show that the symmetry characters of a wave function does not change with time. Explain how symmetric and anti symmetric wave function are constructed from unsymmetrized solution of the Schrodinger equation of a system of indistinguishable particles.

9. Obtain the C.G. coefficient for a system having :

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