QM. eigenfunction help!

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  • #26
malawi_glenn
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click on that hyperlink: expected value

Or wait until it is introduced in class, I thought you have done that.
 
  • #27
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what hyperlink. is it hard? i might understand if u give me the instruction on here.
 
  • #28
malawi_glenn
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In your post #25, there was a hyperlink created for the library item

https://www.physicsforums.com/library.php?do=view_item&itemid=206

It is very easy, we learned operators, eigenfunction, expectation values etc, in one lecture I recall..

This should be introduced to you next in your class I think.

The example I gave you will learn you to master the operators and taking integrals with Gaussian functions.
 
  • #29
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I can give you this nice excerice, if you want.

Given the wavefuction [tex]\psi(x) = (1/(\pi ^{1/4}\sqrt{d}))\exp (ikx - x^2/(2d^2))[/tex]

Evaluate the expectation values of x, x^2, p and p^2

This wavefunction minimizes the Heisenberg Uncertainty principle:
[tex]\Delta x \Delta p = \hbar / 2[/tex]

hey glenn, do u mind evaluating for x and p^2 for me, showing the steps. so i can learn how to do it. i went through the library item and didnt quite get the end.
to find x do i integrate x|ѱ(x)2| ?? what does ѱ(x)2 mean??
 
  • #30
malawi_glenn
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you mean a physical interpretation or what it is mathematically?

Since ѱ is a complex valued function, in order to get something with is postive and real, one has to take the MODULUS square. Now for a complex number a, a* means that you take the complex conjugate of a.

a = Re(a) + Im(a)
a* = Re(a) - Im(a)

So, |a|^2 means a* a

I thought complex number algebra was a pre requirement for Quantum Mechanics.. what kind of Quantum Class do you attend? :P I mean, no book :P

That was the mathematical explanation what |ѱ(x)|^2 is. The physical is that |ѱ(x)| ^2 is the probability DENSITY. Just as you have in mathematical statistics, which is also an requirement for quantum mechanics.

<x> = int psi* x psi dx

and

<p^2> = (-i hbar)^" int psi* (d/dx)^2 psi dx
 
  • #31
malawi_glenn
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actually, this was written in the post

[tex]
< x > = \int _{\text{All space}} \psi^*(x)x\psi(x) dx = \int x|\psi (x)|^2 dx
[/tex]

So I have no clue why you asked. Maybe this is too difficult for you? I suggest you wait. You still don't seem to follow my advise that taking one small step a head =(
 
  • #32
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i just didnt know the notation. i know the complex conjugate. ok, so heres the question.
for yesterdays O(with hat) i got the eigenfunction(ѱ) after normalization to be,
ѱ=exp((x2 -x)/2ih') how would i go on about finding the expectation value of x, px and px2?
would you mind telling me the process step by step??
 
  • #33
malawi_glenn
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I don't think you understand this thing with operators and eigenfunction...

It is a difference of FINDING the eigenfunctions to a given operator and finding the eigenvalues of a given operator on a given wavefunction.

The example was:

Given the wavefuction [tex]\psi(x) = (1/(\pi ^{1/4}\sqrt{d}))\exp (ikx - x^2/(2d^2))[/tex]

Evaluate the expectation values of x, x^2, p and p^2

------

So for the first part, <x> is int ѱ* x ѱ dx

What you need to know here are two parts, first what the complex conjugate of ѱ is and then what integral of a Gaussian integral is.

http://en.wikipedia.org/wiki/Gaussian_integral

See " Generalizations "

This is too hard for you, I give up.
 

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