PC4201 QUANTUM MECHANICS



PC4201 QUANTUM MECHANICS

Tutorial 3 (Angular Momentum)

1. Show that if the (2k + 1) operators [pic] satisfy the commutation relations

[pic] , [pic] [pic].

the (2k + 1) operators [pic] have the same property.

2. We denote by [pic] the following function of the operator A and the components of angular momentum: [pic][pic].

Show that if [pic] is a k th-order irreducible tensor operator, its components verify the relation

[pic]

3. Let [pic] be the annihilation and creation operators of a two-dimensional, isotropic, harmonic oscillator: [pic]

We write

[pic]

[pic]

[pic]

J1, J2, J3 may be considered as the Cartesian coordinates of a certain vector operator [pic].

Show that the components of [pic]verify the commutation relations [pic]characteristic of an angular momentum, and that one has

[pic]

(therefore [pic]) .

OCH/RW/ 99/00

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