p827aps8 - ) can be represented as a Taylor series. (c)....

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Physics 827: Problem Set 8 Due Wednesday, November 21, 2007 1. Shankar, problem 10.3.2. 2. Shankar, problem 10.3.3. 3. Shankar, problem 10.3.6. 4. Prove the following properties of the angular momentum operators L x , L y , L z , and L 2 L 2 x + L 2 y + L 2 z deFned in class: (a). [ L z , L 2 ] = 0. (b). [ L z , L ± ] = ± ¯ hL ± (where L ± = L x ± iL y ). (c). Consider L z = XP y - Y P x . In the coordinate representation, we would have L z = x p - i ¯ h ∂y P - y p - i ¯ h ∂x P . (1) Show that, in a two-dimensional system, if we make the coordinate transformation x = ρ cos φ ; y = ρ sin φ , L z takes the form L z = - i ¯ h ∂φ . (2) 5. (20 pts.) (a). Show that [ L z , P 2 x + P 2 y + P 2 z ] = 0. (b). Show that [ L z , R ] = 0, where R = X 2 + Y 2 + Z 2 . Hint: use the identity [ P x , f ( X )] = - i ¯ hf ( X ), where f ( X ) is a function of the operator X , and the prime denotes a derivative. Optional: prove this identity, assuming that f ( X
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Unformatted text preview: ) can be represented as a Taylor series. (c). Hence, show that [ L z , V ( R )] = 0, where V ( R ) is some function of R . (d). Show that [ L 2 , V ( R )] = 0. 1 6. Shankar, exercise (12.3.8), parts (1) and (3) only. Note that this is a two-dimensional problem with a magnetic feld perpendicular to the XY plane. This problem uses the Fact that the Hamiltonian For a particle in a static magnetic feld is H = ( p-q A /c ) 2 / (2 m ), where p is the canonical momentum and A is the vector potential. You do not have to prove this Fact, but, iF you are interested, you can read the prooF in Shankar, pp. 90-91; I will also discuss it later in the year. 2...
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p827aps8 - ) can be represented as a Taylor series. (c)....

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