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Last modified: January 26, 2018

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.

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