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

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

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