12b-prob2-10

12b-prob2-10 - 1 Ph 12b Homework Assignment No. 2 Due: 5pm,...

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1 Ph 12b Homework Assignment No. 2 Due: 5pm, Thursday, 21 January 2010 1. Quantized rotor . A wheel spinning in a plane can be described as a Hamiltonian dynamical system with one degree of freedom: the coordinate is the angular orientation θ taking values in the interval [0 , 2 π ), and the conjugate momentum is the angular momentum L . The Hamiltonian H is H = L 2 / 2 I, where I is the moment of inertia. a ) What are the Hamilton equations of motion for this system? Is there a conserved constant of the motion? What is the associated symmetry? In quantum mechanics, the Hilbert space for this system is the space of square-integrable periodic functions of θ , i.e. functions with the properties ψ ( θ + 2 π ) = ψ ( θ ) , i 2 π 0 | ψ ( θ ) | 2 < . The angular momentum operator becomes ˆ L = - i ¯ h d . b ) Find the eigenvalues and normalized eigenfunctions of the operator ˆ L . That is, ±nd all values of λ and functions ψ λ ( θ ) such that ˆ λ ( θ ) = λψ ( θ ) , ψ λ ( θ + 2 π ) = ψ λ ( θ ) , i 2 π 0 | ψ λ ( θ ) | 2 = 1 .
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This note was uploaded on 01/28/2011 for the course PH 2 taught by Professor Dudko during the Spring '09 term at UCSD.

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12b-prob2-10 - 1 Ph 12b Homework Assignment No. 2 Due: 5pm,...

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