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## Homework Statement

I need Part B of this question

http://physics.wustl.edu/classes/FL2013/217/homework/ps03.pdf [Broken]

Recall that the free particle Schr¨odinger equation,

i~

∂

∂tψ(x, t) = −

~

2

2m

∂

2

∂x2

ψ(x, t) (1)

has solutions of the “plane wave” form

ψk(x, t) = exp[ikx − iω(k)t] , (2)

where ω(k) = ~k

2/2m.

(a) (10 points)

Consider φ1(x, t) = cos(k0x − ω(k0)t). Show that φ1(x, t) is not a solution of the

Schr¨odinger equation, i.e. when plugged into both sides of the equation, identity

does not hold for all x and t as long as k0 6= 0.

(b) (10 points)

Find a solution φ2(x, t) to the Schr¨odingier equation that also satisﬁes φ1(x, 0) =

φ2(x, 0). (Hint: Write cos(k0x) as the sum of a right moving and a left moving

plane wave the way we did in class, and use the superposition principle. That

is, if you know the correct time dependence for each term in the sum, see class

notes, then you also know the correct time dependence of their sum.)

## Homework Equations

## The Attempt at a Solution

I thought the solution may be φ(x, t)=Ae

^{(-k1x-ω1t)}+Ae

^{(-k2x-ω2t)}

but I dont think it works out

Anyone have any idea how to solve Part B?

Thanks!

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