Suppose that ψ1 and ψ2 satisfy the time independent Schrödinger equation for some potential. Which of the following necessarily also satisfies the Schrödinger equation?
Cψ1ψ2
aψ1+bψ2
a|ψ1|2+b|ψ2|2
Cψ1ψ∗2
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.