Eigenvalues containing the variable x

1. Aug 25, 2014

kini.Amith

1. The problem statement, all variables and given/known data
The wave function ψ(x)=Ae-b2x2/2 where A and b are real constants, is a normalized eigenfunction of the schrodinger eqn for a free particle of mass m and energy E. Then find the value of E

2. Relevant equations

3. The attempt at a solution
Substituting the wave function in the time independent schrodinger equation in 1D for free particle (V(x)=0), I got
$$E=\frac{\hbar^2 b^2 (1-b^2 x^2)}{2m}$$
But i thought eigenvalues should not contain the variable x and i don't know how to get rid of it. Apparently, the correct answer is
$$E=\frac{\hbar^2 b^2 }{2m}$$
which is obtained by taking x=0 in my answer. But how can i simply substitute x=0?

2. Aug 25, 2014

nrqed

That wave function is not an eigenstate for the free Hamiltonian, it is an eigenstate for the 1D harmonic oscillator. If they said that it describes a free particle, they made a mistake. Use the 1D harmonic oscillator and you will see that you get an x independent energy, as it should be (you are correct about that point)