chem5314_hwk1_soln - Chem 5314 homework #1 out of 35 marks...

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Unformatted text preview: Chem 5314 homework #1 out of 35 marks Problem 1 6 marks For a particle in free space ( V = 0), the angular frequency and the wave number k of its associated wave function are related by ~ = ~ 2 k 2 2 m (1) a 4 marks) Verify that, if a monochromatic wave of the form = e i ( kx- t ) is substituted into the Schrodinger time dependent equation, the above relation is reproduced. Solution: taking partial derivatives gives t =- i (2) and 2 x 2 =- k 2 (3) Substituting into Schoedingers Equation and canceling from both sides and rearranging gives the desired relationship between k and . b 2 marks) Show that = cos( kx- t ) fails to satisfy the Schr odinger time dependent equation. Solution: In this case, t = sin( kx- t ) (4) and 2 x 2 =- k 2 =- k 2 cos( kx- t ) (5) If we substitute into the Schr odinger equation, we get tan( kx- t ) =- i ~ k 2 2 m = constant (6) 1 which doesnt work because x and t are supposed to be independent variables, yet we have found an equation relating them. Problem 2 4 marks Show that 2 x 2 (7) is a linear operator. Hint: see for a detailed explanation of why d/dx is a linear operator. Solution: From the definition of linearity (see the hint), we need to verify that: A ( f + g ) = Af + Ag (8) and A ( cf ) = c Af (9) If you want to be completely safe, you should work from the definition of the derivative, which I was not expecting in your solution, but I thought I would prove it so you see exactly what is involved: here is the complete solution for the first derivative (it doesnt matter if its a partial derivative or not) operator is linear. We must begin with the definition of a derivative: f ( x ) lim h f ( x + h )- f ( x ) h (10) To show linearity, we must establish that A ( af + bg ) = a Af + b Ag (11) where A = d/dx...
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chem5314_hwk1_soln - Chem 5314 homework #1 out of 35 marks...

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