Schrödinger's Cormorant

6.1 ① We start with our standing wave expression:

Since ψ(x) does not depend on time, it just behaves as a constant when we differentiate with respect to t, so we can write:

So we can just focus on differentiating cos(2πvt/λ). Use the chain rule: set u = 2πvt/λ

First differentiation:

Second differentiation (use the chain rule again):

Hence, substituting back into the full expression (VI-x), we get:

② Since Ψ = ψ (x). cos(2πvt/λ), this simplifies to:

③ The wave equation states that

Substituting in our expression from ②, we get:

So, helpfully, one of those variables we might have struggled to deal with, the wave velocity, v, has cancelled out of the expression. Progress!

④ Since Ψ(x,t) = ψ(x). cos(2πvt/λ) and the cos(2πvt/λ) term does not depend on x,

we can rewrite our result from ③ as:

Cancelling the cos(2πvt/λ) term, this simplifies to:

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