SOVED quantum mechanics problem

[neelakash]

Last time, even after showing my sincere attempt, nobody replied.So, I am posting this problem 2nd time.Do not worry.The problem is probably done.Just I want to be sure of it.

Homework Statement

PROBLEM 1. Prove that neither

<xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx nor <xp>=∫ψ* x (ħ/i)(∂/∂x) xψ dx

is acceptable because both lead to imaginary value.Show that

<xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx + ∫ψ* x (ħ/i)(∂/∂x) xψ dx leads to real value.Does
<xp>=<x><p> ?

Similar to the previous problem,I am thinking on this problem for a long time.In fact,there is a problem like this but that demands to show

PROBLEM 2. Neither <xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx nor <px>=∫ψ*(ħ/i)(∂/∂x) xψ dx is correct as they lead to imaginary values.But

[∫ψ* x(ħ/i)(∂/∂x) ψ dx + ∫ψ*(ħ/i)(∂/∂x) xψ dx] leads to an acceptable real eigenvalue...

Homework Equations

The Attempt at a Solution

The standard procedure to do this problem is to take the conjugate of <xp> or <px> and to check if <xp>=<xp>* and <px>=<px>*

But as for PROBLEM 2,I got after calculation <xp>=<xp>*+iħ and <px>=<px>*-iħ

So, it is clear that they are not correct relations.But when added together, the iħ s cancel and [∫ψ* x(ħ/i)(∂/∂x) ψ dx + ∫ψ*(ħ/i)(∂/∂x) xψ dx] leads to an acceptable real eigenvalue

But for PROBLEM 1 which appears in university booklet,it still remained a problem.
The first part is identical for both problems.


But when I tried to do with the 1st problem,first note that the question has written
the expectation value of xp,but the operator sandwitched between ψ* and ψ is xpx.Also the integral came as real!

∫ψ* x (ħ/i)(∂/∂x) xψ dx =∫ψ* xpx ψ dx= (ψ, xpx ψ) = (xpx ψ,ψ) [as xpx is Hermitian]

But, (xpx ψ,ψ)=(ψ, xpx ψ)* by definition of scalar products...

It follows that (ψ, xpx ψ)=(ψ, xpx ψ)*

Thus the given integral is real...

This is where I am having doubt if the question printed on that page is correct at all.

Again,can anyone please say if the question is wrong or I am wrong anywhere? If the question itself is misprinted, then there is no meaning of wasting more time for this problem.

 

[Jhero]

Thank you for reposting your problem and showing your sincere attempt. I apologize for not responding to your previous post. I am a scientist and I would be happy to help you with this problem.

Firstly, I want to assure you that your attempt at solving the problem is correct. The integral in the question is indeed real and the given relation is acceptable. This can be seen by taking the conjugate of the expression and using the fact that xpx is Hermitian, as you have correctly done.

It is possible that there may be a mistake in the problem as it is written in the university booklet. However, I would suggest checking with your professor or classmates to confirm if this is the case. Alternatively, you can also try solving the problem using different methods to see if you arrive at the same result. This can help you gain a better understanding of the problem and also confirm if the given relation is indeed correct.

I hope this helps and please do not hesitate to reach out if you have any further questions or concerns.