852_2010hw3 - PHYS852 Quantum Mechanics II, Spring 2010...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 3 Topics covered: Unitary transformations, translation, rotation, vector operators 1. [25] Symmetry : A quantum system is said to posses a symmetry if the Hamiltonian operator, H , is invariant under the associated transformation. In other words, if H = H , where H := U HU . (a) [5] Show that H = H is equivalent to [ H,U ] = 0 (b) [5] Any hermitian operator can be used to generate a unitary operator via U = e iG , where G = G is the generator of the symmetry transformation, and is a free parameter. Show that [ H,G ] = 0 is necessary and sufficient for H to be symmetric under U . (c) [5] Show that when [ H,G ] = 0, the probability distribution over the eigenvalues of G does not change in time. In QM this means that G is a constant of motion. Must a QM constant of motion have a well-defined value? (d) [5] What operator is the generator of translation? If a system possesses translational symme- try what operator is a constant of motion? (e) [5] Consider a particle described by the Hamiltonian H = P 2 2 M + V ( X ) . (1) What operator is the generator of translation? Show thatWhat operator is the generator of translation?...
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852_2010hw3 - PHYS852 Quantum Mechanics II, Spring 2010...

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