Description

From basics of quantum mechanics to relativistic quantum mechanics
1. States, measurements and uncertainties: Consider a system that is initially in the state

,
where Ylm(θ,φ) are the spherical harmonics.
(a) Evaluate Lˆ+ in the above state. 3 marks
(b) If Lˆz is measured, what are the values one would obtain and what are the corresponding probabilities? 2 marks
(c) If, after measuring Lˆz, one obtains Lz = −, calculate the corresponding uncertainties ΔLx
and ΔLy as well as their product ΔLx ΔLy. 5 marks
2. An operator relation involving angular momentum: Show that

Lˆ2 = xˆ2pˆ2− (xˆ · pˆ)2 + ixˆ · pˆ. 10 marks
3. Interacting spins: A system of two particles with spins and is described by the
Hamiltonian
Hˆ = αSˆ1 · Sˆ2,
with α being a given constant. The system is initially (say, at t = 0) in the following eigenstate of
Sˆ12, Sˆ22, Sˆ1z and Sˆ2z:
|s1 s2;m .
(a) Find the state of the system at times7 marks
(b) What is the probability of finding the system in the state ? 3 marks
4. States and energies of non-interacting bosons and fermions: Consider a system of N non-interacting,

identical particles that are confined to a one-dimensional box with its walls at x = 0 and x = L. As usual, the potential V (x) is assumed to be zero inside the box and infinity outside. Determine the energy and the wavefunction of the ground state, when the particles are:
(a) identical bosons, 3 marks
(b) identical spin- fermions. 7 marks
5. Non-relativistic and relativistic spin- particles: Consider non-relativistic and relativistic spin-
particles.
(a) What is the spin operator Sˆ describing a non-relativistic spin- particle? What are the eigen values of the corresponding Sˆ2 operator? Show that all spinors are eigen functions of the Sˆ2
operator? 1+1+1 marks
(b) What is the spin operator Sˆ describing a relativistic spin- particle? What are the eigen values of the corresponding Sˆ2 operator? Show that all bispinors are eigen functions of the Sˆ2
operator? 1+3+3 marks

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