How Do You Compute Non-Integer Powers of Quantum Operators?

[nicksauce]

To find the expected value of Q(x,p) we evaluate $<\psi|Q(x,-i\hbar \frac{\partial }{\partial x})|\psi>$. But what do you do if you want to find, say, <p^(3/2)>. How do you raise the derivative operator to the three-halves?

 

[Count Iblis]

To find the expected value of Q(x,p) we evaluate $<\psi|Q(x,-i\hbar \frac{\partial }{\partial x})|\psi>$. But what do you do if you want to find, say, <p^(3/2)>. How do you raise the derivative operator to the three-halves?

Expand the state in momentum eigenstates.

 

[Euclid]

You would make like you do for angular momentum. Since you can't take square roots of operator, you have to study the square of the quantity you are interested in. So in your case, you would have to settle for $$\sqrt{ \langle p^3\rangle}$$, but you would have to put absolute values somewhere.

...edit: go with count iblis' suggestion instead

 

[Meir Achuz]

Fourier transform the wave function to the momentum representation, getting
$$\phi(p)$$. Then p is just like a c number.

 

[nicksauce]

Okay, thanks for the replies.

 

[Count Iblis]

Next problem:

How would you handle:

<x^(1/2)p^(3/2)>

 

[per.sundqvist]

Next problem:

How would you handle:

<x^(1/2)p^(3/2)>

Formally if f(x) is a nice function you could always calculate the n'th derivative using Fourier transforms, like

$$\frac{d^nf[x]}{dx^n}=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk(ik)^ne^{-ikx}<br /> \int_{-\infty}^{\infty}dxf[x]e^{ikx}$$

also when $$n=\sqrt{pi}/i^5.8$$, for example. So the <x^(1/2)p^(3/2)> involves a tripple integral.

 

[Meir Achuz]

Next problem:

How would you handle:

<x^(1/2)p^(3/2)>

Fourier transform $$\psi(x)$$ and $$x^{1/2}\psi(x)$$ separately.

 

[Count Iblis]

Ok, that was too easy for you two. Let me think of a more difficult problem. Well, why not just consider <f(x,p)> where f is some arbitrary function, like e.g.:


$$f(p,x)=\sqrt{x^2 + p^2}$$

Now, you can just modify the Fourier transform method and write this too as a triple integral. However, that may not be the simplest way, particularly not in the case when f is given as above.

 

[Meir Achuz]

For that you may need a Taylor expansion.

 

[Count Iblis]

Diagonalizing the operator x^2 + p^2 is easier. The eigenstates are the harmonic oscillator eigenstates, let's denote them by |n>. You can thus compute the average as:

<psi|sqrt(x^2 + P^2)|psi> =

sum over n of <psi|sqrt(x^2 + p^2)|n><n|psi>

sqrt(x^2 + p^2)|n> = sqrt[(n+1/2)C]

were C follows from the usual H.O. algebra (i'm too lazy to compute it right now) So, the average is:

sum over n of sqrt[(n+1/2)C] |<n|psi>|^2

 

[Meir Achuz]

Expanding in SHO states may or may not be easier than Taylor expansion.
This would depend on the original wave function and operator.
SHO works for the particular operator x^2+p^2, but TE works for most operators.