Eigenvalues containing the variable x

[kini.Amith]

Homework Statement

The wave function ψ(x)=Ae^-b2x2/2 where A and b are real constants, is a normalized eigenfunction of the Schrödinger eqn for a free particle of mass m and energy E. Then find the value of E

Homework Equations

The Attempt at a Solution

Substituting the wave function in the time independent Schrödinger 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?

 

[nrqed]

Homework Statement

The wave function ψ(x)=Ae^-b2x2/2 where A and b are real constants, is a normalized eigenfunction of the Schrödinger eqn for a free particle of mass m and energy E. Then find the value of E

Homework Equations

The Attempt at a Solution

Substituting the wave function in the time independent Schrödinger 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?

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)