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Unformatted text preview: ECE 306 Homework Set #3 Spring 2005 (Due 12 noon, Wednesday, February 16, 2005.) 1. The definition of angular momentum in classical mechanics is L = r x p (a) Give the quantum mechanical operators L x , L y , and L z corresponding to the Cartesian components of the angular momentum in terms of the canonical variables x, y, z, p x , p y and p z . (b) Using these operators, prove the following commutation relationship for angular momentum: [L x , L y ] = i Ñ L z It can be shown that similar commutations relationships hold: [L y , L z ] = i Ñ L x [L z , L x ] = i Ñ L y but you need not repeat the proof for these. (c) Defining the square of the magnitude of the angular momentum as L 2 = L x 2 + L y 2 + L z 2 , show that [L 2 , L x ] = [L 2 , L y ] = [L 2 , L z ] = 0 (Hint: Make use of the commutation relations [x, p x ] = [y, p y ] = [z, p z ] = i Ñ , [x, y] = [y, z] = [x, z] = 0, [x, p y ] = [x, p z ] = 0, etc.) Comment: The commutation relationships given above are of fundamental importance in the quantum theory of atom and molecules, as we shall see later. importance in the quantum theory of atom and molecules, as we shall see later....
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