A particle is in a three-dimensional cubical box that has...

A particle is in a three-dimensional cubical box that has side length $L$. For the state $n_{X}=3, n_{Y}=2,$ and $n_{Z}=1,$ for what planes (in addition to the walls of the box) is the probability distribution function zero?

Key Concepts

Standing Wave Solutions and Separation of Variables

The method of separation of variables is used to solve the Schrödinger equation in systems with separable coordinates, like a cubical box. This technique allows the three-dimensional wavefunction to be expressed as a product of three independent one-dimensional sine functions, each representing a standing wave along one spatial dimension and satisfying the appropriate boundary conditions.

Nodes and Nodal Planes in Wavefunctions

In quantum mechanics, nodes are regions where the wavefunction is zero, meaning the probability of finding the particle there is zero. Nodal planes are surfaces in three dimensions where the wavefunction vanishes, apart from the boundaries of the box. The positions of these nodes are determined by the sine functions’ zeros, which are directly related to the quantum numbers for each spatial dimension.

Particle in a Three-Dimensional Box

This concept involves a quantum system where a particle is confined within a finite region of space, typically with infinitely high potential barriers at the boundaries. In a three-dimensional cubical box, the particle’s wavefunction must vanish at the walls, leading to discrete energy levels and a set of quantized eigenstates.

Quantum Numbers in Bound States

Quantum numbers are integers that label the discrete energy eigenstates of a confined particle. In a three-dimensional box, separate quantum numbers (nX, nY, nZ) correspond to each spatial dimension and determine the number of half-wavelengths that fit along each axis. They play a crucial role in identifying the energy levels and the spatial configuration of the wavefunction.

Transcript

00:02 Okay, so in this problem, we have a particle in a 3d box and the energy state.

00:10 So let's see, the energy state is described by the quantum numbers, nx equals 3, and y equals 2, and n z equals 1.

00:29 So this is our state and we must find what planes the probability function is zero.

00:39 Okay, so how we manage to do this? first of all, we need to remember how we describe the probability function.

00:48 So the probability function is basically the wave function square equals zero because we want to find the probability function when she's zero.

01:03 So the probability function for this particular problem has the solution l divided by 2 to the power of 3.

01:15 So this is the standard solution.

01:16 The only thing we're going to do is to substitute by the states.

01:22 So the first solution for x is sine, square, of 3 because we are in the state 3 pi x l the solution for y is the sign square of 2 pi y l and finally the solution for z of the sine square of pi z so this is the probability function for this problem, for this particle in a 3d box.

02:06 And since we want to describe where this probability, what plane this probability is 0, we just need to analyze these three functions here, all the three signs.

02:21 Okay, so let's begin with the final one.

02:25 Let's begin with the solution for the z.

02:29 So looking to the last solution here, we know that this function will be zero when pi zl is going to be equal to zero to pi and so on.

02:51 Okay, therefore we can took from this solution two planes.

03:01 The planes were z equals zero and z equals well, l.

03:10 And these two planes correspond to the walls of the box, okay? so this is the walls of the box and these two are planes that are equal zero...

Please give Ace some feedback