Easy way to get the expectation value of momentum squared?

In summary, the expectation value of momentum squared, denoted by ⟨p^2⟩, can be calculated using the formula ⟨p^2⟩ = ∫ p^2 ψ(x)^2 dx, where ψ(x) is the wave function and dx is the differential element. This quantity is significant in quantum mechanics as it helps determine the uncertainty in the momentum of a particle. It cannot be negative and is one of the quantities used in the Heisenberg uncertainty principle. It can be calculated for any wave function that satisfies the Dirichlet boundary conditions and is normalized.
  • #1
chi-young
9
0
Hello, I've been trying to define <p2> in terms of <x2>, much the same way that you can write <p> = m d<x>/dt, because it would be easier in my calculations.

Is this possible, or am I on a fools errand?

Edit: For Gaussian distributions.
 
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  • #2
To get p^2 you apply the position-space operator twice.
What's the problem?

Oh you want the classical relation - so you'd need the relationship between <p>^2 and <p^2>?
 

1. How do I calculate the expectation value of momentum squared?

The expectation value of momentum squared, denoted by ⟨p^2⟩, can be calculated using the formula ⟨p^2⟩ = ∫ p^2 ψ(x)^2 dx, where ψ(x) is the wave function and dx is the differential element.

2. What is the significance of the expectation value of momentum squared?

The expectation value of momentum squared is an important quantity in quantum mechanics as it helps determine the spread or uncertainty in the momentum of a particle.

3. Can the expectation value of momentum squared be negative?

No, the expectation value of momentum squared cannot be negative as it is the average of the square of the momentum, which is always a positive quantity.

4. How does the expectation value of momentum squared relate to the Heisenberg uncertainty principle?

The expectation value of momentum squared is one of the quantities used in the Heisenberg uncertainty principle. It, along with the expectation value of position squared, helps determine the minimum uncertainty in the measurement of momentum and position of a particle.

5. Can the expectation value of momentum squared be calculated for any wave function?

Yes, the expectation value of momentum squared can be calculated for any wave function as long as the wave function satisfies the Dirichlet boundary conditions and is normalized.

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