Expectation Values <E> and <E^2>

[a1234]

Homework Statement

Calculate <E> and <E^2> for the particle in a box problem, given the wavefunction psi(x) = n*(|x - a/2| - a/2) at time = 0.

Relevant Equations

psi(x) = n*(|x - a/2| - a/2)
<E> = pi^2* h_bar^2* n^2/(2*m)

I first normalized the given wavefunction and found the value of n that satisfies the normalization condition. I then used E = <E> = pi^2* h_bar^2* n^2/(2*m) to get the expectation value of energy. Assuming that this was the right process, I'm now trying to find <E^2> using the same equation. Would <E^2> then be (pi^2* h_bar^2* n^2/(2*m))^2, or simply n^4 * (pi^2* h_bar^2/(2*m)) ?

 

[vanhees71]

How do you come to the expression for $E$? What you wrote down are the eigenvalues (but check your formula again!). Since you don't have an energy eigenfunction here, you need to employ the Hamilton operator
$$\hat{H}=\frac{\hat{p}^2}{2m}=\ldots$$

 

[a1234]

How do you come to the expression for $E$? What you wrote down are the eigenvalues (but check your formula again!). Since you don't have an energy eigenfunction here, you need to employ the Hamilton operator
$$\hat{H}=\frac{\hat{p}^2}{2m}=\ldots$$

According to the page below, the average energy of a particle in the nth state is given by n^2*pi^2*h_bar^2 / 2m.

https://cnx.org/contents/pZH6GMP0@1.185:Ek7FZdR3@3/The-Quantum-Particle-in-a-Box

I initially tried finding <E> using the Hamiltonian operator, but the integral required to calculate <p^2> contains a factor of d^2psi/dx^2, which turns out to be 0 for the given wavefunction. That led me to believe that using the Hamiltonian operator to find <E> might not work for time-independent wavefunctions.

 

[vanhees71]

Indeed, the question is not trivial, given the singular character of the wave function. As an alternative you can first expand the function in terms of the energy eigenfunctions and then try to calculate the expectation values with that.

 

[PeroK]

Homework Statement:: Calculate <E> and <E^2> for the particle in a box problem, given the wavefunction psi(x) = n*(|x - a/2| - a/2) at time = 0.
Relevant Equations:: psi(x) = n*(|x - a/2| - a/2)
<E> = pi^2* h_bar^2* n^2/(2*m)

I first normalized the given wavefunction and found the value of n that satisfies the normalization condition. I then used E = <E> = pi^2* h_bar^2* n^2/(2*m) to get the expectation value of energy. Assuming that this was the right process, I'm now trying to find <E^2> using the same equation. Would <E^2> then be (pi^2* h_bar^2* n^2/(2*m))^2, or simply n^4 * (pi^2* h_bar^2/(2*m)) ?

A little Latex would go a long way here:

https://www.physicsforums.com/help/latexhelp/

 

[vanhees71]

It's a pretty tricky question, but it's well worth working it out and think about it. It's related to the somewhat subtle mathematics of the domain and co-domain of self-adjoint operators! The idea to first expand the given wave function in energy eigenfunctions works well to see the issues!

 

[vela]

I initially tried finding <E> using the Hamiltonian operator, but the integral required to calculate <p^2> contains a factor of d^2psi/dx^2, which turns out to be 0 for the given wave function.

$\psi''$ isn't 0 at $x=a/2$. If you're familiar with the Dirac delta function, you can use it to express $\psi''$ and calculate $\langle E \rangle$ using the usual integral.