Description

Angular momentum
1. Commutation relations involving angular momentum: Evaluate or establish the following commu-

tation relations involving angular momentum:
(a) [Lˆi,xˆj], 2 marks
(b) Lˆ × rˆ+ rˆ× Lˆ = 2irˆ, 3 marks
(c) [Lˆ2,rˆ] = −2iΘˆ , where Θˆ = Lˆ × rˆ− irˆ 3 marks
(d) [Lˆ2x,Lˆ2y] = [Lˆ2y,Lˆ2z] = [Lˆ2z,Lˆ2x]. 2 marks
Note: ijk ljk = 2δil.
2. Rotation matrix for j = 1: Consider a system with j = 1.

(a) Construct the matrix describing Jˆy. 4 marks
(b) Using the matrix representation, show that Jˆy3 = Jˆy. 2 marks (c) As a result, when j = 1, show that we can write 4 marks
.
3. (a) Energy of a spin- particle: A spin- particle is described by the Hamiltonian
,
where α and β are real constants. Determine the energy eigen values of the particle. 5 marks
(b) Particle in a central potential: A particle in a spherically symmetric potential is known to

be in the eigenstate |l,m of the operators Lˆ2 and Lˆz with eigenvalues l(l + 1)2 and m, respectively.
i. Evaluate Lˆx, Lˆy and Lˆz in the state |l,m. 2 marks ii. Similarly, evaluate Lˆ2x, Lˆ2y and Lˆ2z in |l,m. 3 marks
4. Expectation value in the singlet state: Suppose two spin- particles are known to be in the singlet
configuration. Let S1a be the component of the spin angular momentum of the first particle in the direction defined by the unit vector aˆ. Similarly, let S2b be the component of second particle’s spin angular momentum in the direction bˆ. Evaluate the expectation value of the operator Sˆ1a Sˆ2b in the singlet state. 10 marks
5. Addition of spin angular momentum: Consider a system composed of two particles, one with spin-
and another with spin-1.
(a) List all the allowed spin states of the composite system. 3 marks
(b) Express all the spin states of the composite system in terms of the spin states of the individualsystems. 7 marks

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