Quantum Physics - Electron in a 1d Potential Well Question

In summary, this conversation discusses a Quantum Physics problem involving an electron moving in a one-dimensional potential well. The system has energy eigenfunctions and the lowest energy level is given. The conversation then addresses two questions, one about writing the wavefunction as a linear combination of energy eigenfunctions and normalizing it, and the other about the possible results of a measurement of electron energy. The conversation concludes with a clarification about the difference between eigenvalues and wavefunctions in the context of energy measurements.
  • #1
daleklama
33
0

Homework Statement



This is a Quantum Physics problem.

An electron moves in a one-dimensional potential well such that the potential V = 0 for |x| ≤ a, and V = ∞ otherwise.

The system has energy eigenfunctions:

Un = a^(-1/2) cos (n∏x/2a), for n odd, and
Un = a^(-1/2) sin (n∏x/2a) for n even.

(Those are both for |x| ≤ a)

and Un = 0 for |x| > a.

The lowest energy level of the system is 37.6 eV.

At t=0, the wavefunction for the system is

ψ(x, t = 0) = a^(-1/2) (2 cos (∏x/2a) + sin (∏x/a)).

(1) Write the wavefunction as a linear combination of energy eigenfunctions, and hence normalise the wavefunction.

(2) What are the possible results of a measurement of electron energy?


Homework Equations



Not really sure!

The Attempt at a Solution



Okay, so I think I have 1) complete. The wavefunction as a linear combination of energy eigenfunctions is ψ = 2u1 + u2.

Also, I normalised it and got ψ (norm) = ψ/√5. I'm pretty sure I understand all this.

My problem is with (2). I know that the possible results of an energy measurement are either u1 or u2.

And u1 = E1 = 37.6 eV (given in question)
But I don't understand how to get E2. Apparently E2 = 4 (E1)?? Why would that be true.. I don't understand where that 4 came from? :/

Sorry, I'm an absolute beginner to Quantum Physics so I'm going very very slowly!

Any help would be appreciated :)
 
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  • #2
But I don't understand how to get E2. Apparently E2 = 4 (E1)?? Why would that be true.. I don't understand where that 4 came from? :/
... you should have this already in your notes when you covered the infinite square well in general. If not, then you'll have to put U2 through the Schrodinger equation to find the eigenvalue. (H-E2)U2=0.

I know that the possible results of an energy measurement are either u1 or u2.
And u1 = E1 = 37.6 eV
The possible outcomes of a measurement are the eigenvalues, not the wavefunctions. Important not to mix them up. It is never correct to write "u1=E1" for example.
 

Related to Quantum Physics - Electron in a 1d Potential Well Question

1. What is a 1d potential well in quantum physics?

A 1d potential well is a theoretical model used to describe the behavior of an electron confined to a one-dimensional space. It is a potential energy function that forces the electron to remain within a certain region, creating a "well" where the electron can exist.

2. How does an electron behave in a 1d potential well?

In a 1d potential well, the electron's behavior is described by the Schrödinger equation, which predicts that the electron will have quantized energy levels and can only exist within certain energy states. The electron's wave function will oscillate between the boundaries of the well, with a higher probability of being found in the lower energy states.

3. What factors affect the energy levels in a 1d potential well?

The depth and width of the potential well will affect the energy levels of the electron. A deeper well will have lower energy levels, while a wider well will have more energy levels. Additionally, the strength of the potential barrier at the boundaries of the well will also affect the energy levels.

4. What is the significance of the lowest energy level in a 1d potential well?

The lowest energy level, also known as the ground state, is the most stable state for an electron in a 1d potential well. This is because the electron has the lowest possible energy and is confined within the well, making it less likely to escape. The ground state is also important for understanding the behavior of more complex quantum systems.

5. Can an electron escape from a 1d potential well?

Yes, an electron can escape from a 1d potential well if it is given enough energy to overcome the potential barrier at the boundaries. This can happen through various processes such as thermal excitation, tunneling, or absorption of a photon. However, the higher the energy level, the less likely it is for the electron to escape.

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