852HW1 - and S y for spin-1/2. e.) Show explicitly that S 2...

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PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 1 1. [25 points] This problem shows you how to derive the matrix representations of spin operators from Frst principles. a.) ±or spin 1/2, use the eigenvalue equation S z | m s a = p m s | m s a to Fnd the components of the two-by-two matrix representation of S z in the basis of its own eigenstates. b.) ±rom the deFnition S ± = S x + iS y , write S x and S y in terms of S + and S - . c.) The relation J ± | j, m j a = p r j ( j + 1) m j ( m j ± 1) | j, m j ± 1 a is valid for any angular momentum operators. Apply this to the spin-1/2 case to Fnd the matrix elements of S + and S - in the basis of eigenstates of S z . d.) ±rom your previous answers, derive the matrix representations of S x
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Unformatted text preview: and S y for spin-1/2. e.) Show explicitly that S 2 = v S v S = p 2 s ( s + 1). 2. [15 points] Consider an electron whose position is held Fxed, so that it can be described by a simple two-component spinor. The initial state of the particle is spin-up ( m s = 1 / 2) with respect to the z-axis. At time t = 0 a uniform magnetic Feld is applied along the y-axis. What is the state-vector of this system at any arbitrary time t > 0. 3. [15 points] Cohen-Tannoudji problem 9.1, page 990 4. [15 points] Cohen-Tonnoudji problem 9.2, page 990 5. [20 points] Cohen-Tannoudji problem 9.3, page 991 1...
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