Expected value of Q(x,p)

In summary, to find the expected value of a particular operator, one can use the Fourier transform to expand the wave function and operator in momentum eigenstates. Alternatively, one can diagonalize the operator and compute the average using the eigenstates.
  • #1
nicksauce
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To find the expected value of Q(x,p) we evaluate [itex]<\psi|Q(x,-i\hbar \frac{\partial }{\partial x})|\psi>[/itex]. 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?
 
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  • #2


nicksauce said:
To find the expected value of Q(x,p) we evaluate [itex]<\psi|Q(x,-i\hbar \frac{\partial }{\partial x})|\psi>[/itex]. 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.
 
  • #3


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 [tex]\sqrt{ \langle p^3\rangle}[/tex], but you would have to put absolute values somewhere.

...edit: go with count iblis' suggestion instead
 
  • #4


Fourier transform the wave function to the momentum representation, getting
[tex]\phi(p)[/tex]. Then p is just like a c number.
 
  • #5


Okay, thanks for the replies.
 
  • #6


Next problem:

How would you handle:

<x^(1/2)p^(3/2)>
 
  • #7


Count Iblis said:
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

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

also when [tex]n=\sqrt{pi}/i^5.8[/tex], for example. So the <x^(1/2)p^(3/2)> involves a tripple integral.
 
  • #8


Count Iblis said:
Next problem:

How would you handle:

<x^(1/2)p^(3/2)>
Fourier transform [tex]\psi(x)[/tex] and [tex]x^{1/2}\psi(x)[/tex] separately.
 
  • #9


Ok, that was too easy for you two. :smile: 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.:


[tex]f(p,x)=\sqrt{x^2 + p^2}[/tex]

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. :smile:
 
  • #10


For that you may need a Taylor expansion.
 
  • #11


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
 
  • #12


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.
 

1. What is the expected value of Q(x,p)?

The expected value of Q(x,p) is a statistical measure that calculates the average value of a random variable, taking into account the probability of each possible outcome. It is denoted as E[Q(x,p)].

2. How is the expected value of Q(x,p) calculated?

The expected value of Q(x,p) is calculated by multiplying each possible outcome of the random variable by its corresponding probability and then summing up all the products. This can be represented mathematically as E[Q(x,p)] = Σ x * P(x), where x represents the possible outcomes and P(x) represents their probabilities.

3. What is the significance of the expected value of Q(x,p) in science?

The expected value of Q(x,p) is an important concept in science as it allows researchers to make predictions and decisions based on the average outcome of a random variable. It is commonly used in fields such as economics, physics, and engineering to analyze data and make informed conclusions.

4. How does the expected value of Q(x,p) differ from the actual value?

The expected value of Q(x,p) is a theoretical value that represents the average outcome of a random variable. It may or may not match the actual value, which is the observed result of a specific experiment or event. However, as the number of trials increases, the expected value tends to converge towards the actual value.

5. Can the expected value of Q(x,p) be negative?

Yes, the expected value of Q(x,p) can be negative. This means that there is a higher probability of getting a negative outcome than a positive one. However, if the expected value is negative, it does not necessarily mean that all the outcomes will be negative. It is simply the average result taking into account the probabilities of all possible outcomes.

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