B Is the Momentum Operator Hermitian? A Proof

hokhani
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How momentum is Hermitian
Momentum operator is ##p=-i\frac{d}{dx}## and its adjoint is ##p^\dagger=i\frac{d}{dx}##. So, ##p^\dagger=-p##. How is the momentum Hermitian?
 
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Remember, the exact definition of an autoadjoint operator is that ##\left<\psi\right|\hat{p}\left|\phi\right>=\left<\phi\right|\hat{p}\left|\psi\right>^*##
You can check that with such a definition the momentum is really autoadjoint.
You can prove indeed that ##\left(\frac{d}{dx}\right)^\dagger = -\frac{d}{dx}##
 
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write this\begin{align*}

(f, p g) = -i \hbar \int_D f^*(x) \frac{\partial g}{\partial x}(x)dx &= i\hbar \int_D \frac{\partial f^*}{\partial x}(x) g(x) dx - {\underbrace{\left[ i \hbar f^*(x) g(x) \right]}_{\overset{!}{=} \, 0}}^{\partial D} \\

&= \int_D \left( - i \hbar \frac{\partial f}{\partial x}(x) \right)^* g(x) dx \\

&= (p f, g)

\end{align*}
 
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hokhani said:
Momentum operator is ##p=-i\frac{d}{dx}## and its adjoint is ##p^\dagger=i\frac{d}{dx}##.
No, this is not correct. The correct equation is ##p^\dagger = (- i)^* \hbar \left( \frac{d}{dx} \right)^\dagger##. Then:

Gaussian97 said:
You can prove indeed that ##\left(\frac{d}{dx}\right)^\dagger = -\frac{d}{dx}##
Which means that ##p^\dagger = (- i)^* \hbar \left( \frac{d}{dx} \right)^\dagger = i \hbar \left(- \frac{d}{dx} \right) = - i \hbar \frac{d}{dx} = p##.
 
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Gaussian97 said:
You can prove indeed that ##\left(\frac{d}{dx}\right)^\dagger = -\frac{d}{dx}##
I can't prove it. Could you please help me with that?
 
please let ##L = \dfrac{d}{dx}## and re-write ##(f, Lg)## in the form ##(Mf, g)## and then tell what is ##M##
 
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hokhani said:
I can't prove it. Could you please help me with that?
Post #3 is a proof of it. If the presence of the factor of ##i\hbar## in post #3 confuses you, just eliminate it; then you have a straightforward proof. The key to the proof is the sign flip that comes with the integration by parts.
 
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PeterDonis said:
Post #3 is a proof of it.
Note, btw, that the proof in post #3 is only valid for functions that vanish at the boundary, i.e., at infinity, so the boundary term in the integration by parts goes away. The usual argument is that any function that can actually be physically realized will have this property; this is what mathematicians often call (somewhat disdainfully) a physicist's level of rigor. :wink:
 
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