- #1

Runei

- 193

- 17

Hi there,

Once again I find myself twiddling around with some quantum mechanics, and I bumped into something I find strange. I can't see what the error of my thinking is, so I hope someone could be able to point it out.

I'm looking at solutions to the infinite square well, and arrive at the simple differential equation

[tex] \frac{d^2\Psi}{dx^2} = -k^2 \Psi [/tex]

The solution to this can be written in terms of complex exponentials or sines and cosines. I bumped into the weird stuff when I use complex exponentials.

So the general solution in that case would be

##\Psi(x) = Ae^{ikx}+Be^{-ikx}##

Now, what I then started thinking was: "Hmmm... This could be viewed mathematically as a sum of two vectors, and solution is simply another vector."

So I drew this picture to illustrate the idea:

So from that perspective it seems that the solution could also be written as

##\Psi(x) = Ce^{ik'x}##

However, using the simple constraints of the infinite square well quickly leads to problems - namely:

##|\Psi(0)|^2 = 0##

##C = 0##

So... Now really what I had hoped for. Where am I going taking a wrong turn? Has it to do with the x? That x should be x' also?

Thanks in advance :)

Once again I find myself twiddling around with some quantum mechanics, and I bumped into something I find strange. I can't see what the error of my thinking is, so I hope someone could be able to point it out.

I'm looking at solutions to the infinite square well, and arrive at the simple differential equation

[tex] \frac{d^2\Psi}{dx^2} = -k^2 \Psi [/tex]

The solution to this can be written in terms of complex exponentials or sines and cosines. I bumped into the weird stuff when I use complex exponentials.

So the general solution in that case would be

##\Psi(x) = Ae^{ikx}+Be^{-ikx}##

Now, what I then started thinking was: "Hmmm... This could be viewed mathematically as a sum of two vectors, and solution is simply another vector."

So I drew this picture to illustrate the idea:

##\Psi(x) = Ce^{ik'x}##

However, using the simple constraints of the infinite square well quickly leads to problems - namely:

##|\Psi(0)|^2 = 0##

##C = 0##

So... Now really what I had hoped for. Where am I going taking a wrong turn? Has it to do with the x? That x should be x' also?

Thanks in advance :)