Quantum mechanics wave function

[gotmilk04]

Homework Statement

One of the quantum mechanics wave functions of a particle of unit mass trapped in an infinite potential square well of width 1 unit is given by

Ψ(x,t)= sin([tex]\pi[/tex]x)e[tex]^{-i(\pi^2\overline{h}/2)t}[/tex] + sin(2[tex]\pi[/tex]x)e[tex]^{-i(4\pi^2\overline{h}/2)t}[/tex]\

where [tex]\overline{h}[/tex] is a certain constant. Calculate |[tex]\Psi[/tex](x,t)|[tex]^{2}[/tex]

Homework Equations

The Attempt at a Solution

I know to calculate |[tex]\Psi[/tex](x,t)|[tex]^{2}[/tex] I need to separate the real and imaginary parts, but I'm not sure how to get started.

 

[Redbelly98]

Use e^iz = cos(z) + i·sin(z)

Alternatively, you could multiply ψ by its complex conjugate.

 

[gotmilk04]

If I multiply by the complex conjugate, I'll get |[tex]\Psi(x,t)|^{2}[/tex]?

 

[gotmilk04]

I multiplied it by the complex conjugate, but there are still i's in the equation. Aren't there supposed to be no i's?

 

[vela]

That's right. If you multiply by the conjugate and simplify, there should be no i's left.

Show us your calculations.

 

[Redbelly98]

Sorry about not responding sooner. Somehow I missed seeing this thread's new activity in my subscribed threads lists, until just now.

If I multiply by the complex conjugate, I'll get |[tex]\Psi(x,t)|^{2}[/tex]?

Yes. This is true of any complex number.

Let z = x + iy, where x and y are real.

Then

z·z* = (x + iy)·(x - iy) = [algebra left as an exercise to the reader] = |z|²

If you post your calculation, we can help either spotting an error or with how to simplify your expression further.