p137bp1sol - Physics 137B, Fall 2007, Moore Problem Set 1...

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Physics 137B, Fall 2007, Moore Problem Set 1 Solutions 1. Recall the general commutator rule [ AB, C ] = A [ B, C ] + [ A, C ] B . Applying this, and using [ J y , J z ] = i ¯ hJ x and [ J z , J x ] = i ¯ hJ y , we have [ J 2 , J z ] = [ J 2 x + J 2 y + J 2 z , J z ] = [ J 2 x , J z ] + [ J 2 y , J z ] + [ J 2 z , J z ] = J x [ J x , J z ] + [ J x , J z ] J x + J y [ J y , J z ] + [ J y , J z ] J y + 0 = J x ( - i ¯ hJ y ) + ( - i ¯ hJ y ) J x + J y ( i ¯ hJ x ) + ( i ¯ hJ x ) J y = 0 . 2. (a) Consider a classical particle confined to a ring of radius a that wraps around the z -axis. Its velocity and angular frequency are related by v = ωa . Since the moment arm is fixed at a , its angular momentum (about the z - axis) is L z = ap = μva = μa 2 ω . In the absence of any potential energy terms, the total energy of the particle is therefore simply E = 1 2 μv 2 = 1 2 μa 2 ω 2 = 1 2 2 = L 2 z 2 I . To obtain the Hamiltonian for the corresponding quantum system, we just interpret the dynamical variables (in this case L z ) as quantum operators. Hence the Hamiltonian should be H = L 2 z / 2 I , as claimed. 1
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(b) To find the energy eigenstates and energy eigenvalues, first consider a free particle confined to a line (not a circle), i.e. a free particle in one di- mension. We know that the Hamiltonian for this system is
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This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at University of California, Berkeley.

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p137bp1sol - Physics 137B, Fall 2007, Moore Problem Set 1...

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