Hermitian Properties of Matrices with Differential Operators

[ma18]

Homework Statement

Are the following matrices hermitian, anti-hermitian or neither

a) x^2

b) x p = x (hbar/i) (d/dx)

Homework Equations

The Attempt at a Solution

For a) I assume it is hermitian because it is just x^2 and you can just move it to get from <f|x^2 g> to <f x^2|g> but I am not sure how to do it for b) because there is the x term and the d/dx term I can't just differntiate it like normal for these problems.

Any help would be appreciated.

 

[Lisa_304]

Hello ma18, I agree with a part, if your functions are in configuration space yes x^2 as an operator is real and multiplies your function.
About b part here's my proposed solution, I hope to help you :)

<f | i x d/dx | g>= ∫ f* (i x dg/dx) dx Now we use the commutator

[x,p]g=ihbar g= -ihbar x dg/dx +ihbar d(gx)/dx
And replace in the integral,, the ihbar x dg/dx =ihbar d(gx)/dx - ihbar g
So you have
=i hbar ( ∫ f* ( d(gx)/dx ) dx - ∫ f* g dx)
the integral of f*g is one (1), if the functions are normalized or a constant if they are not. suppose now that they are normalized.
so now you apply integration by parts in the first integral setting f* as u and d(gx)/dx as v. And you have

= i hbar ∫ df*/dx g x dx -i hbar, but this is , = - <i xd f* /dx | g> -i hbar

Please check my signs , but whatever the sign is it can't be hermitian because you have an extra constant so xp is not a hermitian and therefore can't be mesured in the lab.

 

[ma18]

Thanks for your help Lisa!