Suppose that ψ1 and ψ2 satisfy the time independent Schrödinger equation for some potential. Which of the following necessarily also satisfies the Schrödinger equation?

Question

Cψ1ψ2
aψ1+bψ2
a|ψ1|2+b|ψ2|2
Cψ1ψ∗2

Answer 1

The correct answer is Cψ1ψ∗2.

The time independent Schrödinger equation can be written as:

Hψ = Eψ

where H is the Hamiltonian operator, ψ is the wavefunction, E is the energy, and the complex conjugate of ψ is denoted by ψ∗.

Taking the complex conjugate of the Schrödinger equation, we have:

Hψ∗ = Eψ∗

Multiplying both sides of the equation by ψ1 and ψ2, we get:

ψ1Hψ∗2 = Eψ1ψ∗2

Since both ψ1 and ψ2 satisfy the Schrödinger equation, we can replace ψ1Hψ∗2 with Eψ1ψ∗2:

Eψ1ψ∗2 = Eψ1ψ∗2

This shows that Cψ1ψ∗2 also satisfies the Schrödinger equation.

Therefore, the correct answer is Cψ1ψ∗2.