chem3322_hwk2_soln - Chem 3322 homework #2 solutions out of...

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Chem 3322 homework #2 solutions – out of 32 marks Problem 1 – 4 marks Consider the operator ˆ A = x d dx - d dx x (1) What does this operator do to a function f ( x )? Based on your answer, express this operator in a simpler form. Solution: ˆ Af ( x ) = ± x d dx - d dx x ² f ( x ) (2) = xf 0 ( x ) - d dx ( xf ( x )) = xf 0 ( x ) - ( f ( x ) + xf 0 ( x )) = - f ( x ) from which we conclude that ˆ A = - 1. Namely the operator A acts on a function by multiplying the function by - 1. Problem 2 – 9 marks Consider a particle in a one-dimensional box of length L in its lowest energy (ground) stationary state. Calculate the probability that the particle is a – 3 marks) in the left half of the box solution: For the lowest energy state, we have ψ ( x ) = r 2 L sin πx L (3) The probability density of this state is thus 2 L sin 2 πx L (4) You need to integrate this expression. Normally this integral is found in textbooks as Z sin 2 x dx = 1 2 x - 1 4 sin 2 x (5) Changing variables from x to u = πx/L , we have Z b a 2 L sin 2 πx L dx = 2 π Z πb/L πa/L sin 2 u du (6) 1
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For a = 0 and b = L/ 2 we get a probability of 1/2. This makes sense because of symmetry – see Fig 1. b – 3 marks)
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chem3322_hwk2_soln - Chem 3322 homework #2 solutions out of...

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