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...

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 2nd part,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.

Do you wish to have another -ve feedback besides your name?Talking about this nonsense irrelevant stuffs does not help anyone.Not even you.

Hi kolahal_b,

I do not see any errors in your work. The first question has a typographical error in which the editor forgot to delete the x in the old position after adding it in the new location. This is a common editing error, especially easy to make if the typist is not a mathematician or physicist. You are correct that xpx is Hermitian and therefore has real eigenvalues; however, it cannot represent <xp> since it has the wrong units and would not scale correctly if one were to double the value of x.

Perhaps you trust your text too much, which allows mistakes to waste your valuable time. I used to trust my books, until I found enough small errors that I became wary. Textbooks are of course usually trustworthy, but even the best authors make occasional errors, and it can often take some time to determine that the error is in the book and not in your own understanding. What works for me is to assume that the text is correct until I come to something that doesn't seem right. I then try to convince myself that my way is right, and see if I can force myself into a contradiction. If I cannot make a contradiction out of my understanding of the situation, then I assume (temporarily) that I am right and the text is wrong. Then I try to think of how the mistake could have been made, perhaps a typographical error or a simple miscalculation, or ambiguity in the phrasing. If I can see a clear possibility of how a mistake could be made here and can find nothing wrong with my understanding, then I assume (confidently) that the text is wrong and proceed to the next question.

Of course, on rare occasions, I will still be wrong, because something subtle was wrong with my understanding. However, if that happens, it is always easier to correct your understanding LATER when it becomes more obvious where the error lies, than to try to correct it now when the situation is unclear. Even though it is not perfect, this method allows you to make progress without wasting undue time on problems which turn out to be in error.

Hope that helps!

--Stuart Anderson

I agree with you mr homm.

You also got the correct picture about me that I trust my books very much. Therefore,while starting a course, I always try to get a very good book on that subject.

As for Electrodynamics by Griffiths, I found many places in that book that require more explanation.Griffiths simply stated them.And for many first time reader,those would simply look wrong or at least unplausible.But after carefully reading many times,I got that almost in all cases what Griffiths meant to say.Two examples currently in my hand can be found in

http://www.physicsforums.com/showthread.php?t=222614

and

http://www.physicsforums.com/showthread.php?t=222427

Here I ["neel akash" (means Blue Sky in Bengali)] posted whatever I understood.

So,unconsciously I have developed an idea that "printed materials" are generally correct,though my conscious mind knows that is not always the case.(And infact, the authors may also do the mistakes.I can remember that in an occasion,I am sure that Griffiths made a mistake in his book 5th chapter.And I emailed him to tell that!)

The same happened with these ones.Before posting them here,I discussed with my friends.One of them told he had done the present problem!!! For the previous problem,our Professor told that it was a so important problem that we must do it!!!

I should keep all these influences away and rely only on my logic...as you suggested for yourself.And that is the correct way to learn things.As I do while reading Griffiths and other books.This would give confidence which is a reflection of sheer knowledge.

Even then,I rely on you the most in my recent academic years.