Energy Probability of Electron in 1d box

In summary, the homework statement asks for the probability of finding an electron in a lowest energy state. The student tried different methods for calculating this, but none of them worked. They eventually used a Fourier expansion to find the coefficient of the ground state for the electron in the box.
  • #1
Lamebert
39
1

Homework Statement


We're given an unnormalized state function ψ(x) of an electron in a 1 dimensional box of length pi. The state function is a polynomial. We're asked to find the probability that a measurement of its energy would find it in the lowest possible energy state.

Homework Equations


H = -h2 / 2m d2/dx2
E = h2n2π2/ 2mL2

The Attempt at a Solution


I took several approaches to finding the lowest possible energy state probability (n=1). I attempted to do it in the same way that you would do a position probability, except I also included the Hamiltonian. So I tried something along the lines of <E> = ∫(0 to pi) ψ(x)*⋅ H⋅ ψ(x) dx which I thought might give me a number that I could then divide into the second equation above (using n=1 to find energy). This returned a number that is clearly not a probability. I'm not going to explain every approach I took because I don't want to take away from my main questions. But another approach I took is finding the expectation value and dividing the first energy level value by this expectation value, which returned a very small value for n=1 (I assume n=1 should be somewhat probable, with decreasing values for n=2 and so forth).

I do not understand the energy dependence on position. How does energy only depend on position? Also, is it necessary to normalize my energy equations (since it has to do with probability). I'm not sure I'm even on the right track with expectation values or my integration of ψ(x)*⋅ H⋅ ψ(x), there's some kind of connection between the quantum mechanical ideas that I'm just not grasping. Any help is appreciated.
 
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  • #2
What you are trying to do is to find the expectation value of the energy. You cannot find the probability of being in a particular energy eigenstate by doing this. In fact, a system can have an expected energy equal to one of the eigenstate energies with probability zero of actually being found in that state.

Instead, you need to find the coefficient of the lowest eigenstate wavefunction when you expand the wavefunction in the eigenstates.

And yes, you should normalise your wave function or you will have to take into account at every alternate step that you are working with a non-normalised wave function. Better just do it once and for all.
 
  • #3
Look up probability amplitude in your textbook.
 
  • #4
yeah. expectation values are not the right track to be on.
"I do not understand the energy dependence on position. How does energy only depend on position?" In this case, the Hamiltonian is just the kinetic energy of the particle. And the problem is implicitly assuming that the particle has zero chance of appearing outside the box. Hence the Hamiltonian does not include any potential energy terms.
In this problem, the main thing is that you are given the wavefunction ψ(x), and you also need to know the wavefunction of the ground state. From there, how do you calculate the probability of collapsing from ψ(x) into the ground state? (this is what vela was hinting at).
 
  • #5
Orodruin said:
What you are trying to do is to find the expectation value of the energy. You cannot find the probability of being in a particular energy eigenstate by doing this. In fact, a system can have an expected energy equal to one of the eigenstate energies with probability zero of actually being found in that state.

Instead, you need to find the coefficient of the lowest eigenstate wavefunction when you expand the wavefunction in the eigenstates.

And yes, you should normalise your wave function or you will have to take into account at every alternate step that you are working with a non-normalised wave function. Better just do it once and for all.

I ended up using a Fourier expansion to solve for each wavefunction coefficient. Is this what you were hinting at?
 
  • #6
Yes. The coefficient of the ground state is the amplitude of being in that state.
 
  • #7
Orodruin said:
Yes. The coefficient of the ground state is the amplitude of being in that state.
Looking back on this two weeks later, what you were saying makes sense. I guess I just hadn't made connections between the math and concepts at that point. Thanks for your help.
 

What is the "Energy Probability of Electron in 1d box"?

The "Energy Probability of Electron in 1d box" refers to a theoretical model that describes the energy levels and probability distribution of an electron confined in a one-dimensional box. This model is often used in quantum mechanics to understand the behavior of particles in a confined space.

How is the energy probability calculated in this model?

In this model, the energy probability is calculated using the Schrödinger equation, which takes into account the size and shape of the box, as well as the mass and charge of the electron. The solution to this equation gives the energy levels and probability distribution of the electron in the box.

What is the significance of this model in understanding the behavior of electrons?

The "Energy Probability of Electron in 1d box" model is important because it helps us understand how the energy and position of an electron are related. It also provides insights into the quantum nature of particles and how they behave in confined spaces, which has applications in various fields such as nanotechnology and materials science.

Can this model be applied to other particles besides electrons?

Yes, this model can be applied to other particles with some modifications. For example, the Schrödinger equation can be adapted to describe the energy probability of a proton in a one-dimensional box. However, the results may vary depending on the mass and charge of the particle.

How does temperature affect the energy probability of an electron in a 1d box?

Temperature can affect the energy probability of an electron in a 1d box by altering the distribution of the electron's energy levels. At higher temperatures, the electron is more likely to occupy higher energy levels, resulting in a broader probability distribution. Conversely, at lower temperatures, the electron is more likely to occupy lower energy levels, resulting in a narrower probability distribution.

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