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

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Chem 5314 homework #1 – out of 35 marks Problem 1 – 6 marks For a particle in free space ( V = 0), the angular frequency ω and the wave number k of its associated wave function are related by ~ ω = ~ 2 k 2 2 m (1) a – 4 marks) Verify that, if a monochromatic wave of the form ψ = e i ( kx - ωt ) is substituted into the Schr¨odinger time dependent equation, the above relation is reproduced. Solution: taking partial derivatives gives ∂ψ ∂t = - iωψ (2) and 2 ψ ∂x 2 = - k 2 ψ (3) Substituting into Schoedinger’s Equation and canceling ψ from both sides and rearranging gives the desired relationship between k and ω . b – 2 marks) Show that ψ = cos( kx - ωt ) fails to satisfy the Schr¨ odinger time dependent equation. Solution: In this case, ∂ψ ∂t = ω sin( kx - ωt ) (4) and 2 ψ ∂x 2 = - k 2 ψ = - k 2 cos( kx - ωt ) (5) If we substitute into the Schr¨ odinger equation, we get tan( kx - ωt ) = - i ~ k 2 2 = constant (6) 1
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which doesn’t 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 http://vergil.chemistry.gatech.edu/notes/quantrev/node14.html 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: ˆ A ( f + g ) = ˆ Af + ˆ Ag (8) and ˆ A ( cf ) = c ˆ 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 doesn’t matter if its a partial derivative or not) operator is linear. We must begin with the definition of a derivative: f 0 ( x 0 ) lim h 0 f ( x 0 + h ) - f ( x 0 ) h (10) To show linearity, we must establish that ˆ A ( af + bg ) = a ˆ Af + b ˆ Ag (11) where ˆ A = d/dx , and where a,b are constants, and f,g are functions f(x), g(x). Consider the
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