Solution to Schrodinger Equation

[kdndk]

Homework Statement

I need Part B of this question

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

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 don't think it works out

Anyone have any idea how to solve Part B?

Thanks!

 

[Simon Bridge]

Welcome to PF;
How did you go for part (a)?
You are provided with a hint for part (b), how did you apply it?
Why did you think your proposed solution may be correct, and what leads you to think that it doesn't work out after all?

How to type math equations on PF:
https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

Schrödinger - you can write the umlaut in with the compose key.