# 851HW13_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

This preview shows pages 1–2. Sign up to view the full content.

PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 13 Topics Covered: Spin Please note that the physics of spin-1/2 particles will ±gure heavily in both the ±nal exam for 851, as well as the QM subject exam. Spin- 1 / 2 : The Hilbert space of a spin-1/2 particle is the tensor product between the in±nite dimen- sional ‘motional’ Hilbert space H ( r ) and a two-dimensional ‘spin’ Hilbert space, H ( s ) . The spin Hilbert space is de±ned by three non-commuting observables, S x , S y , and S z . These operators satisfy angular momentum commutation relations, so that simultaneous eigenstates of S 2 = S 2 x + S 2 y + S 2 z and S z exist. According to the general theory of angular momentum, these states can be designated by two quantum numbers, s , and m s , where s must be either an integer or half integer, and m s ∈ { s, s 1 , . . . , s } . The theory of spin says that for a given particle, the value of s is ±xed. A spin-1/2 particle has s = 1 / 2, so that m s ∈ {− 1 / 2 , 1 / 2 } . Since s never changes, we can label the two eigenstates of S z as | ↑ z a and | ↓ z a , where | ↑ z a = | s = 1 2 , m s = 1 2 a and | ↓ z a = | s = 1 2 , m s = 1 2 a , so that S z | ↑ z a = p 2 | ↑ z a (1) S z | ↓ z a = p 2 | ↓ z a (2) and S 2 | ↑ z a = 3 p 4 | ↑ z a (3) S 2 | ↓ z a = 3 p 4 | ↓ z a . (4) As eigenstates of an observable, these states must satisfy the orthonormality conditions A↑ z | ↑ z a = 1,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

851HW13_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online