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851HW13_09

# 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 figure heavily in both the final 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 infinite dimen- sional ‘motional’ Hilbert space H ( r ) and a two-dimensional ‘spin’ Hilbert space, H ( s ) . The spin Hilbert space is defined 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 fixed. 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 ) and | ↓ z ) , where | ↑ z ) = | s = 1 2 ,m s = 1 2 ) and | ↓ z ) = | s = 1 2 ,m s = 1 2 ) , so that S z | ↑ z ) = planckover2pi1 2 | ↑ z ) (1) S z | ↓ z ) = planckover2pi1 2 | ↓ z ) (2) and S 2 | ↑ z ) = 3 planckover2pi1 2 4 | ↑ z ) (3) S 2 | ↓ z ) = 3 planckover2pi1 2 4 | ↓ z ) . (4) As eigenstates of an observable, these states must satisfy the orthonormality conditions (↑ z | ↑ z ) = 1, (↓ z | ↓ z ) = 1, and (↑ z | ↓ z ) = (↓ z | ↑ z ) = 0. The two vectors {| ↑) , | ↓)} must therefore form a complete basis that spans H ( s ) , so that I = | ↑ z )(↑ z | + | ↓ z )(↓ z | . (5) 1

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1. In this problem you will derive the 2 × 2 matrix representations of the three spin observables from first principles: (a) In the basis {| ↑ z ) , | ↓ z )}
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851HW13_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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