Help with this Hamiltonian

  • Thread starter eljose79
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  • #1
eljose79
214
1
given a classic hamiltonian of the form H=f(x)P**n what would be it quantum version of it?..how do you quantizy this?... (n is an integer)
Should you take all the possible permutation of it?..i have this problem...thanks.
 

Answers and Replies

  • #2
Sonty
108
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I think first of all you have to expand f(x) into a Taylor series and then make every xm*pn symmetric.My first impulse would be to write it (xm*pn + pn*xm)/2, but I have strong doubts about it as I seem to have heared my teacher say "stick to those simple ones".
 
  • #3
nbo10
418
5
hint. The Energy has to equal n*hbar
 
  • #4
Originally posted by Sonty
I think first of all you have to expand f(x) into a Taylor series and then make every xm*pn symmetric.My first impulse would be to write it (xm*pn + pn*xm)/2, but I have strong doubts about it as I seem to have heared my teacher say "stick to those simple ones".

I don't see the point of expanding f(x). Here's what I'd do

H = [f(x)*P^n + P^n*f(x)]/2

Pete
 
  • #5
Sonty
108
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Originally posted by pmb
I don't see the point of expanding f(x).

Well, as a first thing I have to make sure H is a linear operator.
 
  • #6
Originally posted by Sonty
Well, as a first thing I have to make sure H is a linear operator.
Of course it's linear. Regardless of what particular form it takes so long as it's a Hamiltonian I.e.

H(a|Psi1> + b|Psi2>) = aH|Psi1> + bH|Psi2>

I.e. you take H and multiply it through. And example of a non-linear operator is O where O(A) = A^2

In the present case

H(Psi) = [f(x)P^n][a*Psi1 + b*Psi2]
= [f(x)P^n](a*Psi1) + [f(x)P^n](b*Psi2)
= a*[f(x)P^n]Psi1 + b*[f(x)P^n]Psi2
= a*H(Psi1) + b*H(Psi2)

However I don't see what possible physical system this Hamiltonian could belong to and thus the reason to call it a Hamiltonian. But that's par for the course sometimes. Goldstein's text has an example of a Hamiltonian in the problem section for which I can't see what physical system it describes either.

Pmb0
 

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