How Do You Quantize a Classical Hamiltonian of the Form H=f(x)P^n?

[eljose79]

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.

 

[Sonty]

I think first of all you have to expand f(x) into a Taylor series and then make every x^m*pⁿ symmetric.My first impulse would be to write it (x^m*pⁿ + pⁿ*x^m)/2, but I have strong doubts about it as I seem to have heared my teacher say "stick to those simple ones".

 

[nbo10]

hint. The Energy has to equal n*hbar

 

[pmb]

Originally posted by Sonty
I think first of all you have to expand f(x) into a Taylor series and then make every x^m*pⁿ symmetric.My first impulse would be to write it (x^m*pⁿ + pⁿ*x^m)/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

 

[Sonty]

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.

 

[pmb]

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