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Momentum operator hermiticity

So, I'm trying to prove that the hermitian operator is hermitian in one spatial dimension.

So, I took my vector space as

\mathbb{C}

.

This is what I have: (

Let

a,b \in \mathbb{C}

Unparseable latex formula:

<a,\hat{p} b> = \int_{-\infty}^{\infty} (\bar{a} . - i \hbar \dfrac{db}{dx}) dx \\ [br]\\[br]= -i \hbar \left( \left[ \bar{a}b \right]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} (w . {\dfrac{d\bar{b}}{dx}})dx \right)

Then I just 'presumed' that the left part of the above is 0 and then I don't seem to find what I want. Obviously it's hermitian, what's going wrong? I'm probably doing something really stupid and assuming something that can't be true.

(edited 12 years ago)

Reply 1

around

The momentum operator actually acts on the space of complex-valued functions which decay sufficiently quickly at infinity so that the integral over all space is defined. This is why the left part of your integral is 0.