Show that function is not an energy eigenfunction

[beth92]

Homework Statement

We are considering the Superposition state:

ψ(x,t) = 1/2 u₁(x)e^{(-i/hbar)E₁t} + √3/2 u₄(x) e^{(-i/hbar)E₄t}

We had to verify that ψ is a solution of the time dependent Schrödinger Equation, which I have done. Next we are asked to show that ψ is NOT an energy eigenfunction and therefore not a solution to the time independent Schrödinger Equation.

Homework Equations

Time Independent Schrödinger Equation: H(ψ(x,t)) = E_nψ(x,t)
Hamiltonian operator: H= (-hbar/2m)∇²+V(x)

The Attempt at a Solution

At first I tried to see if I could somehow show that the first equation did not hold and tried to apply the Hamiltonian on the left hand side, but I was just left with ∇²u(x) terms and I couldn't think of any way of simplifying them. Then I considered the time dependent Schrödinger which states that the Hamiltonian of ψ is equal to (i*hbar)dψ/dt. So I then set the right hand sides of each Schr. equation equal to each other (as the left hand side of both is the Hamiltonian) and tried to go about showing that this equality did not hold. I can't seem to get anywhere with this method. I'm wondering if there's a simpler way of doing this?

 

[tiny-tim]

hi beth92!

hint: H is additive

 

[beth92]

Hmm okay. So can I just say that ψ is a superposition of two states ψ₁ and ψ₄ and that the time independent Schrödinger Equation then becomes:

H(ψ₁+ψ₄) = E_n*(ψ₁+ψ₄)

which, due to the fact that H is additive means that:

H(ψ₁)+H(ψ₄) = E₁ψ₁+E₄ψ₄

And therefore E_n is not an energy eigenvalue but is rather a sum/superposition of two separate energies E₁ and E₄...or does E_n just not exist at all? I feel like I almost understand it but I am struggling to come up with words to explain the conclusion.

 

[tiny-tim]

hi beth92!

… due to the fact that H is additive means that:

H(ψ₁)+H(ψ₄) = E₁ψ₁+E₄ψ₄

that's right …

now you have to prove that that is not a scalar multiple of ψ₁ + ψ₄