Quantum mechanics, wavefunction problem

[fluidistic]

Homework Statement

Consider the wavefunction \Psi (x,t)=c_1 \psi _1 (x)e^{-\frac{iE_1t}{\hbar}}+c_2 \psi _2 (x)e^{-\frac{iE_2t}{\hbar}} where \psi _1 (x) and \psi _2 (x) are normalized and orthogonal. Knowing \Psi (x,0), find the values of c_1 and c_2.

Homework Equations

C^2 \int _{-\infty}^{\infty} |\Psi (x,t)|^2 dx=1. I also know that the product of psi 1 by psi 2 is worth 0 (they are orthogonal) so this simplifies the expression to integrate.
But I'm still left with the integration of both lower case psi functions that I don't know how to handle.

The Attempt at a Solution

I'm thinking on how to use the fact that I know Psi at t=0 but so far I'm out of ideas.

 

[ehild]

\psi _i (x)-s are orthonormal, what does it mean on the integrals of \psi _i ^* (x) \psi _j (x)?

ehild

 

[fluidistic]

\psi _i (x)-s are orthonormal, what does it mean on the integrals of \psi _i ^* (x) \psi _j (x)?

ehild

Sorry for being almost 1 month late. That they are worth 1?

Edit: Is what I've done right?:
\int _{-\infty}^{\infty} \Psi _1 (x) \Psi (x,0)dx=c_1 \int _{-\infty}^{\infty} \Psi _1 ^2 (x)dx=c_1.
In the same fashion I get c_2=\int _{-\infty}^{\infty} \Psi _2 (x) \Psi (x,0)dx.
But I didn't use any complex conjugate... I must be missing something.

 

[vela]

That's almost correct. It should be
c_i = \int \psi_i^*(x)\Psi(x,0)\,dxWhen you calculate the inner product, you use the complex conjugate of the first function in the integrand.

 

[fluidistic]

That's almost correct. It should be
c_i = \int \psi_i^*(x)\Psi(x,0)\,dxWhen you calculate the inner product, you use the complex conjugate of the first function in the integrand.

Let's see if I understand:
\int _{- \infty}^{\infty} \Psi _1 ^* (x) \Psi (x,0)dx=c_1 \int _{- \infty}^{\infty} \Psi _1 ^* (x) \Psi _1 (x)dx=c_1 \int _{-\infty} ^{\infty} |\Psi _1 (x)|^2 dx=c_1.
Am I right if I say that since \Psi _1(x) and \Psi _2 (x) are orthogonal, so is \Psi _1^* (x) with \Psi _2 (x)? Because I'm using this fact!
Thanks for your help guys.

 

[vela]

Yup. When you say the two are orthogonal, you mean that
\int \psi_1^*(x)\psi_2(x)\,dx = 0.It does not mean that
\int \psi_1(x)\psi_2(x)\,dx = 0.The complex conjugation is a vital part of taking the inner product.

 

[fluidistic]

Yup. When you say the two are orthogonal, you mean that
\int \psi_1^*(x)\psi_2(x)\,dx = 0.It does not mean that
\int \psi_1(x)\psi_2(x)\,dx = 0.The complex conjugation is a vital part of taking the inner product.

Oh, I understand. Wow, thank so you much. My understanding was so poor before I asked help for this problem... now things are getting clearer.