Expectation Values <E> and <E^2>

In summary: Otherwise you can expand ##\psi## in a series of energy eigenfunctions as suggested above and evaluate the integral. In summary, the conversation discusses the process of finding the expectation value of energy for a particle in a box using the given wavefunction. The steps involve normalizing the wavefunction, using the Hamiltonian operator to find <E>, and then using the same equation to calculate <E^2>. A suggestion is also given to expand the wavefunction in terms of energy eigenfunctions to better understand the question. The issue of d^2psi/dx^2 being 0 is addressed and other methods are suggested to calculate the expectation value of energy.
  • #1
a1234
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6
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)) ?
 
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  • #2
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
$$\hat{H}=\frac{\hat{p}^2}{2m}=\ldots$$
 
  • #3
vanhees71 said:
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.
 
  • #4
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.
 
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  • #5
a1234 said:
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/
 
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  • #6
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!
 
  • #7
a1234 said:
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.
 
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FAQ: Expectation Values <E> and <E^2>

1. What are expectation values and ?

Expectation values and are mathematical concepts used in quantum mechanics to describe the average value of a physical quantity, such as energy or momentum, in a given system. They represent the most probable outcome of a measurement for that particular quantity.

2. How are expectation values and calculated?

Expectation values and are calculated by taking the sum of all possible outcomes of a measurement, multiplied by their corresponding probabilities. In quantum mechanics, this is represented by the integral of the wave function squared over all possible values of the physical quantity.

3. What is the significance of expectation values and in quantum mechanics?

Expectation values and play a crucial role in quantum mechanics as they provide a way to predict the most probable outcome of a measurement for a particular physical quantity. They also help in understanding the behavior of quantum systems and their evolution over time.

4. Can expectation values and be measured experimentally?

Yes, expectation values and can be measured experimentally by performing repeated measurements on a quantum system and calculating the average value of the physical quantity. However, due to the probabilistic nature of quantum mechanics, the measured values may not always match the calculated expectation values.

5. How do expectation values and relate to uncertainty principle?

Expectation values and are related to the uncertainty principle in that they represent the average values of a physical quantity, while the uncertainty principle states that it is impossible to know both the exact value of a physical quantity and its conjugate variable at the same time. This means that the expectation values are subject to uncertainty and may not always match the measured values.

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