For a particle in a 1-dimensional box confined between 0<x<a , the initial state of a particle is given by
phi = phi_1 + 3phi_2 + 2phi_3 (all phi's are functions of x).
a) Normalize this wave function.
c) Is this expectation value of energy an energy eigenvalues of this system? Why or why not?
I'll assume that the psi's are the normalized eigenfunctions:
psi_n(x) = sqrt(2/a) sin(n pi x/a)
The energy eigenvalue of psi_n is:
E_n = pi^2 hbar^2 n^2/(2 m a^2) (1)
We want to normalize the wavefunction
psi = psi_1 + 3 psi_2 + 2 psi_3
This can be done without much computation using linear algebra techniques as follows. You consider the wavefunction to be a vector in a vector space (that vector space is called a Hilbert space). One defines a complex inner product on this vector space as follows:
<psi|phi> = Integral of ...
A detailed solution is given.