A particle is in the three-dimensional box of Section 41.2 . (a) Consider the cubical volume def

A particle is in the three-dimensional box of Section 41.2 ....

Key Concepts

Particle in a Box

This concept refers to a quantum mechanical model in which a particle is confined within an impenetrable potential well. In three dimensions, the Schrödinger equation is solved using separation of variables, yielding quantized energy eigenstates and corresponding eigenfunctions. The solutions are sine functions in each dimensional coordinate, which vanish at the boundaries of the box, and serve as a fundamental example in quantum mechanics to illustrate the quantization of energy and wavefunction behavior.

Ground State

The ground state is the lowest energy state of a quantum system. For a particle in a three-dimensional box, the ground state corresponds to the lowest set of quantum numbers (n_X = 1, n_Y = 1, n_Z = 1), leading to a symmetric wavefunction. In this state, the probability density is distributed according to the square of the wavefunction, which reflects how likely the particle is to be found in any given region of the box.

Probability Density

Probability density in quantum mechanics is the square of the absolute value of the wavefunction. It indicates how probability is spatially distributed throughout the region where the particle is confined. Calculating the probability that a particle is found in a subregion involves integrating this probability density over that volume, emphasizing the connection between the normalization of the wavefunction and measurable probabilities.

Volume Fraction

Volume fraction is a measure of the proportion of a larger volume occupied by a smaller specified subvolume. In the context of a particle in a three-dimensional box, it involves calculating the ratio of the subregion's volume compared to the total volume of the box. This concept is important for understanding statistical and probabilistic distributions over different spatial regions.

Excited State

An excited state refers to any quantum state of the system that has a higher energy than the ground state. For a particle in a three-dimensional box, excited states are characterized by higher quantum numbers in one or more dimensions. These states have different spatial distributions of probability density, which can lead to different probabilities for finding the particle in any given subregion when compared with the ground state.

Transcript

00:01 In general, what we're going to use for parts b and part c is the fact that the maxborn normalization condition for this cubicle volume of a box is equal to the integral over dx, d, d, z, of the wave function, psi, squared, is equal to one, where the wave function psi is equal to some constant c, right, times the sign, and i'm using the notation this s here is equal to sign.

00:28 So this is the sign of what's ever in the parentheses.

00:32 So sign of nx, pi x over l, sign of n sub y, pi y over l, and then sign of n sub z, pi z over l for the stationary state wave function of the particle inside the cubicle box, where nx, n y, n z are just some, you know, some integer value.

00:51 And c is just a normalization constant.

00:53 For part a, though, to find the fractional volume, we're given the fact that the cubicle volume is defined by, the parameters of x being between 0 and l over 4, y being between 0 and l over 4, and z being between 0 and l over 4.

01:07 So the volume of the cube is v, and it's equal to l over 4 cubed, right, for x, y, z.

01:14 Well, the total volume of the box at v prime is equal to l cubed.

01:19 So the fractional of the total volume of the box in this cubicle volume is equal to v divided by v prime.

01:29 So we plug in values for vmv prime.

01:34 Of course the l values cancel because we have l over 4 to the third divided by l to the third.

01:51 This is equal to 1 over 64.

01:57 You can write it as a numeric like as a decimal value or as a fractional value.

02:02 I'm going to leave it as a fractional value.

02:06 For part b this is when the probability function and the wave function come in because we're going to find the particle in the ground state, which is when n y, x, and n z are all equal to one.

02:18 To calculate that probability found in the cubicle volume defined, so in part a.

02:24 So this is the x, y, and z defined between 0 and l over 4.

02:28 So those are the numerical bounds of our integration.

02:32 Okay.

02:37 So let's get into it.

02:40 So what we have here is let's go ahead and pull the constant out front.

02:44 So what we have is c squared.

02:47 So we're just plugging that value of si in for our probability equation.

02:53 So c squared, you have integrals from 0 to l over 4, right? for every single one, for x, y, and z, 0, l over 4, and then you have all the sign functions now squared.

03:24 So we have sine squared of n sub x, pi x over l, sine squared n sub y pi y over l, and then the same thing, sine squared n sub z pi z over l.

04:10 Okay, and then you'd write the n sub z pi z over l.

04:13 We won't write that out there.

04:14 That's what we go there.

04:17 And then the volume, of course, is dx, d, y, dz is equal to one.

04:36 Equal to one.

04:37 Okay.

04:38 So what we have here is c, the constant c squared.

04:45 And each one of these comes out to be l over two.

04:48 L over two, integrating sine squared nx pi x over l with respect to dx from zero to l over four gives you l over two, l over two, l over two...