Hermitian operators in quantum mechanics

[sams]

Hello everyone,

There's something I am not understanding in Hermitian operators.

Could anyone explain why the momentum operator:
p_x = -iħ∂/∂x
is a Hermitian operator? Knowing that Hermitian operators is equal to their adjoints (A = A^†), how come the complex conjugate of p_x (iħ∂/∂x) = p_x (-iħ∂/∂x) ?

Thank you so much for your support...

 

[Demystifier]

http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf

https://www.eng.fsu.edu/~dommelen/quantum/style_a/nt_px.html

https://www.quora.com/How-do-you-prove-the-momentum-operator-is-Hermitian

 

[vanhees71]

A hermitean operator $\hat{p}$ fulfills
$$\langle \psi_1|\hat{p} \psi_2 \rangle=\langle \hat{p} \psi_1|\psi_2 \rangle$$
for all vecotrs $|\psi_1 \rangle$ and $|\psi_2 \rangle$ in the domain of the operator. In the position representation for the momentum operator you have $\hat{p}=-\mathrm{i} \partial_x$ and the scalar product is the usual $\mathrm{L}^2$ product. Now indeed $\hat{p}$ is hermitean since via integration by parts you get
$$\langle \psi_1|\hat{p} \psi_2 \rangle=\int_{\mathbb{R}} \mathrm{d} x \psi_2^*(x)(-\mathrm{i} \partial_x) \psi_1(x) = \int_{\mathbb{R}} \mathrm{d} x +\mathrm{i} \partial_x \psi_1^* \psi_2(x) = \int_{\mathbb{R}} \mathrm{d} x [-\mathrm{i} \partial_r \psi_1(x)]^{*} \psi_2(x)=\langle{\hat{p} \psi_1}|\psi_2 \rangle.$$
So $\hat{p}$ is hermitean.

 

[sams]

OK, so according to http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf

We proved that: (-iħ∂/∂x)^* = iħ∂/∂x

Thus, (p_x)^* = -p_x

and not (p_x)^* = p_x

 

[vanhees71]

OK, so according to http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf

We proved that: (-iħ∂/∂x)^* = iħ∂/∂x

Thus, (p_x)^* = -p_x

and not (p_x)^* = p_x

No, of course I didn't prove an obviously wrong result. Please read more carefully what I posted! The linked document doesn't make sense to me.

 

[DrDu]

What makes you think that the adjoint of an operator is the same as the complex conjugate of an operator?

 

[vanhees71]

Well, I guess that's due to the misleading notation of the linked paper :-(.

 

[sams]

No, of course I didn't prove an obviously wrong result. Please read more carefully what I posted! The linked document doesn't make sense to me.

I am sorry Dr. vanhees71 for this confusion. My reply was for response #2.
Thank you for your reply and for valuable information.

 

[sams]

What makes you think that the adjoint of an operator is the same as the complex conjugate of an operator?

I think any operator should be inserted in an eigenvalue equation. If this operator is Hermitian, it should leads to a physical observable and gives the same eigenvalue as its complex conjugate.

However, the definition of a Hermitian operator is:
A = A^*

If the latter equation is true, can we state that the following:
p_x = -iħ∂/∂x = iħ∂/∂x = (p_x)^* ?

We can clearly realize that p_x ≠ (p_x)^*

 

[vanhees71]

This is very misleading, as this discussion shows. The hermitean conjugation of an operator is not simply its complex conjugation. It's not even clear, how to define it, but the adjoint operator $\hat{A}^{\dagger}$ to an operator $\hat{A}$ is defined such that
$$\langle \psi_1 |\hat{A} \psi_2 \rangle=\langle \hat{A}^{\dagger} \psi_1|\psi_2 \rangle.$$
Note that I leave out the somewhat subtle questions on domains and codomains of the operators here. This we can discuss as soon as the usual physicists' meaning of hermitean conjugation is understood.

 

[DrDu]

I think any operator should be inserted in an eigenvalue equation. If this operator is Hermitian, it should leads to a physical observable and gives the same eigenvalue as its complex conjugate.

Congratulations! You just rediscovered the fact that complex conjugation of an operator depends on the chosen basis.

 

[sams]

Dear Sirs,

Thank you so much for this valuable discussion.

Best wishes