1(a) Assume that a particle is
represented by a wavepacket and show that the uncertainty principle follows
naturally from the de Broglie hypothesis. (6 Marks)

(b) What is the significance of commutability
of operators? Discuss Heisenberg’s commutation relations. (6 Marks)

(c) What is momentum
wavefunction? Explain its significance. (4 Marks)

(d) Use the uncertainty principle
to calculate the ground state energy of an electron in hydrogen atom. (4 Marks)

2(a) Generalize the expression
for transmission coefficient to a potential hill of any shape. (6 Marks)

(b) Discuss the solution of the
SchrÃ¶dinger equation for a three-dimensional potential box and show that the
degeneracy depends on the symmetry of the box. (10 Marks)

(c) What are creation and
annihilation operators? (4 Marks)

3(a) Setup and solve the
SchrÃ¶dinger equation for a rigid rotator with free axis for its eigenvalues and
eigenfunctions. (10 Marks)

(b) Find the average value of the
distance of the electron from the nucleus in the ground state of a hydrogen
atom. (6 Marks)

(c) Write Laguerre differential
equation. Comment on its solutions. (4 Marks)

4(a) What is Hilbert space? (5
Marks)

(b) What is Hermitian matrix?
Show that the eigenvalues of a Hermitian matrix are real and that the
diagonalising matrix of a Hermitian matrix is unitary. (10 Marks)

(c) Write a note on secular
equation. (5 Marks)

5(a) Use the time independent
perturbation method to obtain the ground state energy of a helium atom. Compare
the result with the experimental value. (8 Marks)

(b) Explain the Heitler-London
theory of hydrogen molecule and discuss the results obtained. (6 Marks)

(c) Use the WKB approximation to
calculate the decay constant of nuclei emitting alpha particles. (6 Marks)

6(a) What is sudden
approximation? What is the nature of eigenfunctions in this process? (6 Marks)

(b) Explain the Hamiltonian of a
charged particle in an electromagnetic field and use it to find the transition
rate in dipole approximation. (6 Marks)

(c) What is Born approximation?
(4 Marks)

(d) Write a note on Eikonal
approximation. (4 Marks)

7(a) Show that three Pauli
matrices along with a 2 x 2 unit matrix form a complete basis of algebra. (6
Marks)

(b) What is the central field
approximation? Discuss the Thomas-Fermi statistical model used for the purpose.
(6 Marks)

(c) What are pure quantum state
and mixed quantum state? (4 Marks)

(d) Write a note on
symmetrisation of wavefunctions. (4 Marks)

8(a) Derive Klein-Gordon
equation. (6 Marks)

(b) Show that the spin-orbit
interaction is a consequence of the Dirac relativistic equation. (6 Marks)

(c) How does the quantization of
an electromagnetic field in vacuum lead to the concept of photon? (4 Marks)

(d) Find the commutation
relations for the electric and magnetic vectors of the electromagnetic field.
(4 Marks)

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