HW03 & Solutions

# HW03 & Solutions - ECE 306 Homework Set#3 Spring...

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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.

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• Spring '05
• TANG
• Ly, eigen values

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