LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – PHYSICS
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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 [ JX,JY] = iħ JZ and its cyclic, verify that [ J+, J–] = 2ħ JZ, where J+ = JX + iJY and J– = JX– iJY.
- 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 ( um, x un) where un’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 (σx +i σy)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 xop and pop 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, Jz ,J+ and J–, starting with the postulate [ Jx, Jy] = iħ Jz and its cyclic.
- Outline the perturbation theory of degenerate case with specific reference to the two-dimensional harmonic oscillator.