LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FG 30

SECOND SEMESTER – APRIL 2008

PH 2808 – QUANTUM MECHANICS

 

 

 

Date : 22/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART A                                               (10 x2 m = 20 m)

 

• Prove that the momentum operator is self adjoint.
• In what way is the orthonormal property of eigenfunctions belonging to discrete set of eigenvalues different from those of continuous set?
• Define parity operator. What is its action on a wave function ψ ( r, θ, φ)?
• What is a rigid rotator and what are its energy eigenvalues?
• Expand an arbitrary state vector in terms of certain basis vectors. Define projection operator.
• Explain the term the ‘wave function’ of a state vector ׀ψ>.
• Given that [ J_X,J_Y] = iħ J_Z and its cyclic, verify that [ J₊, J_–] = 2ħ J_Z, where J₊ = J_X + iJ_Y and J_– = J_X– iJ_Y.
• Prove that the Pauli spin matrices anticommute.
• Write down the Hamiltonian of a hydrogen molecule.
• Explain the terms classical turning points and the asymptotic solution in the context of WKB approximation method.

PART B                                            ( 4×7 ½ m = 30 m)

ANSWER ANY FOUR QUESTIONS

 

• (a) Verify the identities [ AB, C) = A [ B, C] + [ A, C] B and [ A, BC] = B[A, C] + [ A, B] C . (b) Determine [ x², p²], given that [ x, p] = iħ.
• Evaluate ( u_m, x u_n) where u_n’s are the eigenfunctions of a linear harmonic oscillator.
• Prove that “the momentum operator in quantum mechanics is the generator of infinitesimal translations”.
• (a) Prove that ( σ.A) (σ.B) = B + i σ. ( A xB) where σ’s are the Pauli spin matrices , if the components of A and B commute with those of σ. (b) Determine the value of (σ_x +i σ_y)².
• Estimate the ground state energy of a two-electron system by the variation method.

PART C                                          ( 4x 12 ½ m=50 m)

ANSWER ANY FOUR QUESTIONS

 

• (a) State and prove closure property for a complete set of orthonormal functions. (b) Normalize the wave function ψ(x) = e ^{– ׀x׀}
• Discuss the simple harmonic oscillator problem by the method of abstract operators and obtain its eigenvalues and eigenfunctions.
• (a) The position and momentum operators x_op and p_op have the Schrödinger representations as x and –iħ ∂/∂x”-Verify this statement.(b) Explain the transformation of Schrödinger picture to Heisenberg picture in time evolution of quantum mechanical system.
• Determine the eigenvalue spectrum of the angular momentum operators J², J_z ,J₊ and J_–, starting with the postulate [ J_x, J_y] = iħ J_z and its cyclic.
• Outline the perturbation theory of degenerate case with specific reference to the two-dimensional harmonic oscillator.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2012

PH 2811 / 2808 – QUANTUM MECHANICS

 

 

Date : 19-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

 

 

PART – A

Answer ALL questions:                                                                                   (10×2=20)

 

1. Prove [ [A,B], C]+[ [B,C], A]+[ [C, A], B] = 0
2. State Heisenberg’s uncertainty principle
3. What are spherical harmonics? Are they mutually orthogonal?
4. Prove that the square of the angular momentum commutes with its z-component.
5. If A and B are two operators, then show that [A^-1[A,B]] = 2B
6. What are unitary transformations?
7. Show that commuting operators have simultaneous eigenfunctions.
8. What are indistinguishable particles?
9. What is Rayleigh ratio?
10. Find the separation between any two consecutive energy levels of a rigid rotator.

 

 

PART – B

Answer any FOUR questions:                                                                     (4×7.5 = 30)

 

11. Obtain the equation of continuity in Quantum mechanics.
12. Obtain the normalized wave function for a particle trapped in the potential
    V(x) = 0 for 0 < x < a and V(x) =
13. (a) With an example explain linear operator (b) A and B are two operators defined by AY(x) = Y(x) + x and BY(x) = + 2Y(x) check for their linearity                                     (2.5 +5)
14. If + μω²x then show that
15. and          ii. x                                                      (4+3.5)
16. Obtain the second order correction for a non-degenerae energy level.

 

 

 

 

 

 

 

PART – C

Answer any FOUR questions:                                                                     (4×12.5 = 50)

 

16. State and prove Ehernfest’s theorem
17. Solve the Schrodinger equation for a linear harmonic oscillator. Sketch the first two eigenfunctions of the system.
18. Determine the eigenvalue spectrum of angular momentum operators J^z and J_z
19. What are symmetric and antisymmetric wave functions? Show that the symmetry character of a wave function does not change with time. Explain how symmetric and antisymmetric wave functions are constructed from unsymmetrized solution of the schrodinger equation of a system of indistinguishable particles. (3+3+6.5)
20. Using perturbation theory, explain the effect of an electric field on the energy levels of an atom (Stark effect).

 

 

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