An electron moves in a three-dimensional box of edge length...

An electron moves in a three-dimensional box of edge length $L$ and volume $L^{3}$. The wave function of the particle is $\psi=A \sin \left(k_{x} x\right) \sin (k y) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation 42.18 ,$$E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{2}^{2}\right)$$ where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1$ Suggestion: The Schrödinger equation in three dimensions may be written $$\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}\right)=(U-E) \psi$$

An electron moves in a three-dimensional box of edge length $L$ and volume $L^{3}$. The wave function of the particle is $\psi=A \sin \left(k_{x} x\right) \sin (k y) \sin \left(k_{z} z\right) .$ Show that its energy
is given by Equation 42.18 ,$$E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{2}^{2}\right)$$
where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1$ Suggestion: The Schrödinger equation in three dimensions may be written $$\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}\right)=(U-E) \psi$$

Key Concepts

Schrödinger Equation in Three Dimensions

This is the fundamental equation of quantum mechanics that describes how the quantum state of a physical system changes with time. In the context of a particle in a three-dimensional potential (such as a box), the time-independent Schrödinger equation is used to determine the allowed energy levels and corresponding wave functions by solving a partial differential equation with respect to three spatial variables.

Particle in a Box

This is a model used in quantum mechanics where a particle is confined within an impenetrable boundary (a box) with a defined geometry. The model helps illustrate key quantum concepts such as quantization of energy and the importance of boundary conditions, allowing the simplest view of how confinement leads to discrete energy levels.

Quantization of Energy Levels

Due to the boundary conditions imposed on the confined particle, only specific (quantized) energy levels are allowed. These quantized energies emerge from the solution of the Schrödinger equation, and they depend on the dimensions of the box and the quantum numbers that describe the wave function solutions.

Boundary Conditions and Wave Functions

The requirement that the wave function must vanish at the boundaries of the box plays a crucial role in determining its form. These boundary conditions ensure that the particle is confined to the box and lead to the use of sine functions in the solution, which in turn result in quantized wave numbers and energy levels.

Quantum Numbers

Quantum numbers are integers that arise from solving the Schrödinger equation under specific boundary conditions. They characterize the permissible states of the particle and directly relate to the energy levels in the system. In a three-dimensional box, each spatial dimension contributes its own quantum number, reflecting the quantization along that direction.

Transcript

00:01 So in this problem, we're looking at the schroeder equation for a three -dimensional cube.

00:06 It's a three -dimensional analog of the infinite square well potential.

00:12 So the problem doesn't explicitly say it, but the potential inside the cube is zero, and outside its length l, the potential energy is infinite, so the weight function is zero at l or greater distance.

00:33 From the origin.

00:40 So it wants us to find the energy in this particular form where it's h bar instead of h.

00:49 So hold on a sec.

01:06 So what we want to do is note that we're given this wave function a times sine kx, sine ky, sine kz, where the ks are different.

01:18 What this is really saying is that these side functions are one is dependent on x one is dependent on y one is dependent on z there's no cross dependence so they're separate from each other so the key of this problem is that you can decompose that wave function is a product of one -dimensional wave functions and for notational simplicity i'm just going to call those big x y and z so those aren't variables x, y, and c, those are functions x, y, and z.

02:04 So if you write it that way, you can rewrite the schrodinger equation.

02:12 It's a h bar squared 2m.

02:19 So the point of this is that the schrodinger equation was partial derivatives here.

02:27 Because we're doing a separation of variables where things depend on only one dimension.

02:39 Means we're turning the three -dimensional partial derivative partial differential equation into three ordinary differential equations, which is essentially just doing the infinite square well, the three infinite square well problems simultaneously.

03:20 So these functions are effectively constant because they don't, y and z here don't have any x dependent, so they're effectively constant.

03:41 So this is supposed to be times size, so that is xyz.

03:47 So if you were to then divide x, y, z through on both sides, you get three ordinary differential equations of this form.

04:31 Then if we notice that the potential u is zero inside the cube and infinite outside, where the whole thing goes the whole way of function goes to zero if you is infinite then we can just since this is zero we can write that as equal to minus e um so first thing i want to point out is that since these x y and z are sign functions uh we do two derivatives on them what you're really doing so you do the first, you do the river the first time, you pull out a kx, so the constant factor inside the sign function.

05:54 But then, so then it turns into a cosine kx.

05:57 Then the second derivative turns the cosine into a minus side kx and brings out another kx.

06:05 So that becomes this...

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