An electron is confined to a one-dimensional box of length...

An electron is confined to a one-dimensional box of length $L$. (a) Sketch the wave function for the third excited state. (b) What is the energy of the third excited state? (c) The potential energy can't really be infinite outside of the box. Suppose that $U(x)=+U_{0}$ outside the box, where $U_{0}$ is large but finite. Sketch the wave function for the third excited state of the electron in the finite box. (d) Is the energy of the third excited state for the finite box less than, greater than, or equal to the value calculated in part (b)? Explain your reasoning. [Hint: Compare the wavelengths inside the box.] (e) Give a rough estimate of the number of bound states for the electron in the finite box in terms of $L$ and $U_{0}$.

Key Concepts

Estimation of Bound States

In a finite potential well, the number of bound states is not infinite and can be roughly estimated by comparing the well's spatial dimensions and the height of the potential barrier. This estimation involves comparing the kinetic energy associated with the confinement to the potential energy outside the well, and counting the number of quantized levels that can be accommodated within the energy range where the particle remains bound.

Particle in a Box Model

This concept deals with the quantum mechanical model where a particle is confined within a perfectly rigid and impenetrable region and the potential energy is taken as zero inside the box and infinite outside. It serves as a fundamental example that illustrates quantization of energy levels, as the Schrödinger equation yields solutions only for discrete energy values with specific boundary conditions.

Wave Function and Nodal Structure

The wave function describes the quantum state of the particle and its spatial probability distribution. In problems like the particle in a box, the wave function exhibits a characteristic sinusoidal form with nodes—points where the probability density is zero—which increase in number with higher energy states. The excited states are characterized by additional nodes that correspond to higher quantum numbers.

Energy Quantization

Energy quantization in confined systems arises since only specific wavelengths satisfy the boundary conditions imposed by the confinement. In the context of the box model, the allowed energy levels are determined by the integer multiples of half wavelengths that fit perfectly within the box, leading to energy values that scale as the square of the quantum number.

Finite Potential Well

When the potential barrier is finite rather than infinite, the walls of the well allow for some tunneling of the wave function beyond the nominal 'box' boundaries. This adjustment in boundary conditions affects the exact form of the wave function, especially near the edges, and can result in modified energy levels compared to those calculated in the idealized infinite case.

Transcript

00:01 In this exercise, we have an electron that's confined to a box of length l.

00:04 And in question a, we have to sketch the wave function of the electron for the third excited state.

00:15 Well, remember that for the third excited state, we have three nodes.

00:23 So basically, if this here is the wave function in the y -axis and the distance in the x -axis, and here we have l, then the wave function vanishes at the boundaries, that is at the walls of the box, and there are three nodes in between the walls.

00:50 So stars, then we have one, two, and three.

00:54 Actually, it's better to do a little one, two, and three nodes.

01:03 And notice that there are exactly three notes inside the box and the function vanishes at the boundaries and for that reason this is a good sketch for the third excited state in question b we have to to calculate what is the energy of the electron in the third excited state and we know that the elect the energy of the nth state is equal to n squared h squared over 8 m l squared for the third excited state n equals 4 so this is equal to 16 h squared divided by 8 m l squared which is equal to 2 h squared divided by m l squared so this here is the energy of the fourth state, that is the third excited state.

02:07 Let me just write this as equal to e4.

02:18 Okay, in question c, we have to assume that we have a finite box, that is that the potential at the walls is not infinite and it is actually finite with a value of eu0.

02:35 And then we have to sketch the wave function for the third excited state again.

02:41 In this case, the wave function does not vanish at the boundaries, but actually what happens is that the wave function tends to zero, so very far away from the block.

02:54 So here we have the boundaries that's the zero and l.

02:59 Inside the boundaries, there still will be three nodes, but far away from the wall, the wave function tends to zero, then it grows, and we have have one, two, three nodes, and then it tends to zero as, uh, i'm sorry, and it tends to zero as the, the distance from the box grows.

03:32 Let me just make the wave function get closer to zero...

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