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The Differential Operator, D
----------------------------
Consider:
D|ψ> = ∂/∂x|ψ>
The definition of an Hermitian operator is that
<φ|D|ψ> = <ψ|D|φ>*
∫φ*(∂ψ/∂x)dx = [∫ψ*(∂φ/∂x)dx]*
= ∫ψ(∂φ*/∂x)dx
= -∫φ*(∂ψ/∂x)dx (after integrating by parts)
This doesn't work. We need Dψ = -i∂ψ/∂x. Check:
-i∫φ*(∂ψ/∂x)dx = i[∫ψ*(∂φ/∂x)dx]*
= i∫ψ(∂φ*/∂x)dx
= -i∫φ*(∂ψ/∂x)dx (after integrating by parts)
Find eigenvalues and eigenvectors:
D|ψ(x)> = λ|ψ>
-i∂/∂x|ψ(x)> = λ|ψ>
which is equivalent to,
-i∂ψ(x)/∂x = λψ(x)
Solutions (eigenvectors) are:
ψ(x) = (1/√2π)exp(iλx)
by comparison λ is equivalent to k. Thus,
-i∂ψ(x)/∂x = kψ(x)
∴ -ih∂ψ(x)/∂x = pψ(x)
Thus,
-ih∂/∂x is interpreted as the momentum operator, p.