Electron in a One-Dimensional Box

In summary, the conversation discusses the problem of an electron in a one-dimensional box with walls at x=(0,a) and the quantum state it is in. The solution involves obtaining an expression for the normalization constant, A, and finding the lowest energy of the electron in this state. It is mentioned that the wavefunction is independent of x except around x = a/2, and that the energy eigenfunctions are sin or cosine functions. The conversation also mentions the strangeness and interest of this particular state.
  • #1
Domnu
178
0
Problem
An electron in a one-dimensional box with walls at [tex]x=(0,a)[/tex] is in the quantum state

[tex]\psi(x) = A, 0<x<a/2[/tex]
[tex]\psi(x) = -A, a/2 < x < a[/tex]

Obtain an expression for the normalization constant, [tex]A[/tex]. What is the lowest energy of the electron that will be measured in this state?

Solution

Well, we know that

[tex]\int_{0}^A |\psi(x)|^2 dx = 1 \iff A^2 \cdot a = 1 \iff A = \sqrt{\frac{1}{a}}[/tex]

so this is our normalization constant, [tex]A[/tex]. Now, to find the lowest energy of the electrong that will be measured in this state, we have:

[tex]-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} = E \psi[/tex]

but the left hand side evaluates to 0 since [tex]\psi[/tex] is independent of [tex]x[/tex]. So this is the lowest (and only observed) energy in this state... is this right?
 
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  • #2
Well, [tex]\psi[/tex] is independent of [tex]x[/tex] everywhere except around [tex]x = a / 2[/tex]. Around there things could get a little weird. (In fact, as this problem is stated, [tex]\psi[/tex] changes infinitely quickly from one side of [tex]a / 2[/tex] to the other, so around there things could get very weird.)

Even without doing any calculations, though, you should know that since the electron is stuck in the box, it must have nonzero energy. If nothing else, the box gives it [tex]\Delta x < \infty[/tex], so it must have [tex]\Delta p > 0[/tex] by Heisenberg.

If you've already studied the infinite potential well (one-dimensional box), you might also know that the energy eigenfunctions (or stationary states) are all [tex]sin[/tex] functions (since one of the walls is at [tex]x = 0[/tex]). Now, the given wavefunction [tex]\psi[/tex] is not such a [tex]sin[/tex] function, so to get the measurable energies we have to rewrite it as a sum of the energy eigenfunctions -- that is, as a Fourier series of [tex]sin[/tex] functions.

Hopefully that's enough to get you started.
 
  • #3
Well, its a step function so if you compute the derivative you get a delta function...and then a derivative of a delta function and so on.

By the way, if you solve for the wavefunction of a particle in a 1D box, you get sin or cosine functions. Any state the particle is in can be written as a linear combination of these eigenfunctions. This is a rather strange state...and interesting too :smile:
 

1. What is an "Electron in a One-Dimensional Box?"

An "Electron in a One-Dimensional Box" refers to a simplified model used to study the behavior of electrons in a confined space. It assumes that the electron is confined to a one-dimensional box with infinite potential walls on either side.

2. What are the key assumptions of the "Electron in a One-Dimensional Box" model?

The key assumptions of this model are that the electron has a fixed mass, is confined to a one-dimensional box with infinite potential walls, and experiences no external forces or interactions.

3. How does the "Electron in a One-Dimensional Box" model help us understand quantum mechanics?

This model helps us understand quantum mechanics by showing how the energy levels and wave functions of a confined particle are quantized, meaning they can only take on certain discrete values. It also illustrates the concept of wave-particle duality, where the electron can behave as both a particle and a wave.

4. What is the significance of the energy levels in the "Electron in a One-Dimensional Box" model?

The energy levels in this model represent the possible energies that the electron can have while confined to the one-dimensional box. These discrete energy levels demonstrate the quantization of energy in quantum mechanics and help us understand the stability of atoms and molecules.

5. How does the "Electron in a One-Dimensional Box" model relate to real-world applications?

While the model itself may be simplified, it serves as a basis for understanding more complex quantum mechanical systems, such as atoms and molecules. It also has practical applications in fields such as nanotechnology and semiconductor physics, where particles are confined to very small spaces.

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