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 Schrodinger 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 Schrodinger Equation.

Homework Equations

Time Independent Schrodinger 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 Schrodinger 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 Schrodinger 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

 

[paranormal]

I would first like to clarify the definition of an energy eigenfunction. An energy eigenfunction is a solution to the time independent Schrodinger Equation, where the energy of the system is well-defined and does not change with time. In other words, the energy eigenfunction is a stationary state of the system.

In this case, we are given a superposition state which is a linear combination of two different wavefunctions (u1 and u4). This means that the state is not a single, well-defined energy state, but rather a combination of two different energy states. Therefore, ψ is not an energy eigenfunction.

To show this mathematically, we can use the time independent Schrodinger Equation and the Hamiltonian operator. If ψ was an energy eigenfunction, then it would satisfy the equation H(ψ(x,t)) = Enψ(x,t), where En is the energy eigenvalue. However, when we apply the Hamiltonian operator to ψ, we get a combination of the two different energy eigenvalues E1 and E4. This means that ψ is not an eigenfunction of the Hamiltonian and therefore not an energy eigenfunction.

In conclusion, we have shown that ψ is not an energy eigenfunction by using the definition of an energy eigenfunction and the time independent Schrodinger Equation.