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 $\psi=A \sin \left(k_{x} x\right) \sin \left(k_{y} y\right) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation 43.20 , $$E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)$$ where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1 .$ Suggestion: The Schrodinger 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 $\psi=A \sin \left(k_{x} x\right) \sin \left(k_{y} y\right) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation 43.20 ,
$$E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)$$
where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1 .$ Suggestion: The Schrodinger 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

Particle in a Box

This concept describes a quantum mechanical model in which a particle is confined within a well-defined region of space with infinite potential barriers at the boundaries. The model demonstrates how spatial confinement leads to discretized energy levels due to the imposition of boundary conditions on the wave function, highlighting one of the fundamental departures of quantum mechanics from classical physics.

Schrödinger Equation in Three Dimensions

The Schrödinger equation is the fundamental equation of quantum mechanics that governs the behavior of quantum systems. In three dimensions, it incorporates the Laplacian operator to account for spatial variations, and when applied to confined systems, it yields eigenfunctions and eigenvalues that encode the quantized energy states of the particle.

Separation of Variables

This mathematical technique is used to reduce a complex partial differential equation into simpler, one-dimensional ordinary differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. In quantum mechanics, this approach is particularly useful for solving the Schrödinger equation in systems with separable potentials or geometries, like a three-dimensional box.

Quantum Numbers and Energy Quantization

Quantum numbers arise naturally when solving the Schrödinger equation under specific boundary conditions. They label the discrete energy eigenstates and determine the quantized values of physical quantities such as energy. This quantization results from the requirement that the wave function must satisfy certain conditions at the boundaries, leading to a set of allowed wave vectors and corresponding energies.

Wave Function Normalization

Normalization is the process of adjusting the wave function so that its total probability density integrates to one over the entire volume. This is a central requirement in quantum mechanics, ensuring that the interpretation of the wave function as a probability amplitude is mathematically consistent and physically meaningful.

Transcript

00:01 In this question, we are given the wave function of a particle or an electron defined by this wave function here.

00:07 We're given the schrodinger equation in terms of the x, y, z components of the particle's position, and we are also given, well, we're told to express the energy in this form here.

00:19 So what we need to do is work out these differentials.

00:24 So we'll start with the partial differential in x.

00:29 So working out the first derivative, we see that, deep sigh over the x will simply be equal to well the only term the only factor we need to really consider is this time with x in so we end up with a k x because the k x comes out we get a cos k x x sine k y y sine k z z z when we take the second derivative of this well the difference of cos will be minus sign.

01:18 So we can hopefully spot a trend here whereby we don't need to write out all of this expression all over again because the second derivative will simply be minus kx squared because the kx comes out so we get a kx squared.

01:38 However we're left with a sign kxx again which means that we have the original expression as before.

01:45 So simply it will just be equal to minus kx squared and then the original expression for bsi.

01:52 Now, because we have a wave function that's described in x, y, and z in the same way, we can simply say that the partial differential expressions for y and z will be analogous to that of x, as shown by these two here.

02:10 So now we need to consider the boundary conditions of the particle.

02:13 The first of which is that inside the box, no matter where you are inside the box, the potential energy of the particle will always equal zero.

02:23 So all that simply means is that we can just cross off the u in this expression here.

02:29 Secondly, in order to get in this expression in terms of number integers here, as well as the actual dimensions of the box itself, we need to look at what the value of psi will be at the 1 .5.

02:43 Walls of the box.

02:46 So looking at, let's just say in the x direction, we know it's described by a sinusoidal relationship here...

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