Constructing time-independent wave function with given energies

[droedujay]

Does anyone know how to construct a Time-independent wave function with given energies and probability on obtaining energies.

 

[G01]

Are you asking about constructing an initial wave function, $$\Psi (x,0)$$, for a particle in a potential given that information?

If so, what you are describing can be done by using the expansion theorem. Using that theorem, you can express a general time-independent wave function as an infinite sum of the energy eigenstates:

$$| \Psi > = \sum_n^{\infty} C_n |n>$$

where $$|n>$$ is the wave function for the energy E_n and C_n is the probability for measuring that energy.

I can't offer anything more specific given that information, but if you have a homework problem or something similar involving this, please post it in the homework help forum and I'll help you if I can.

 

[droedujay]

I found out the Coefficient expansion theorem and constructed the following wavefunction:

Ψ(x,0) = 1/sqrt(2)*φ1 + sqrt(2/5)*φ3 + 1/sqrt(10)*φ5
where φn = sqrt(2/a)*sin(n*pi*x/a)

Is this unique why or why not? I'm thinking that it has something to do with all odd Energies.