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

A particle is in the three-dimensional cubical box of...

Key Concepts

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.

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.

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.

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.

Transcript

00:01 In this problem, we're going to talk about a particle in a three -dimensional box.

00:05 So let's consider that we have this box here, and let's assume that this box is a cube, such that the length of the side of the box is up.

00:18 Let's say we have a particle trapped inside this box, so it's confined to move inside the walls of the box.

00:26 And at the walls, the potential at the walls is equal to infinity.

00:36 In that case, the wave function, a psi of x, y, and z of the particle, is equal to a constant a times the sine of n x, pi, x, over l, times the sign of n y pi y over l times the sign of n z pi z over l okay and a in this case is just a normalization constant and it's equal to two over l to the three halves okay you can find this by imposing that the integral of the absolute value of the wave function squared over all space, here i'm going to write d3r, is equal to 1.

01:47 Okay, also remember that the probability of finding the particle in a certain range, let's say between x, 0 and x prime, y0 and y prime, is equal to the integral, the triple integral, over x, y, and z of the absolute value of the wave function squared over all space, oh sorry, over this particular region of the space here.

02:36 And what we have in our problem is a particle inside a box that has a side l and we have to consider the region of the box between 0 and l over 4.

02:53 So x is between 0 and l over 4.

02:59 There's an equal sign here.

03:01 Y is between 0 and l over 4 and z is between 0 and l over 4.

03:08 And in question a we have to find what is the portion of the volume that is occupied by this region of the box.

03:18 So consider a box that instead of having a length l has a length l over 4.

03:25 In a case, the volume of the box is equal to l over 4 cubed.

03:33 So this is l cubed over 64.

03:36 And the original volume, so this is the volume if the box had a line l over 4, the original volume is l cubed.

03:51 So the new volume divided by the old one is equal to 1 over 64.

03:58 And this is 0 .0156.

04:05 This is the answer to our first problem.

04:13 Then in question b, we have to calculate what is the probability that the article will be in this region for the ground state.

04:22 Is when nx is equal to 1, and y is equal to 1, and then z is equal to 1.

04:30 Well, remember that the wave function is 2 over l, cube, actually, 2 the 3 halves, times the sine of n x, which is 1, times pi x over l, sine of pi y overel, times sine of pi y graph.

04:57 Then we need to integrate this over all space and notice that the integral of the actually of the square of this over this virtual region of the space and so this is between let me write it like this.

05:33 Zero and it over four, zero and an all over four, 0 and l over 4 of the x, doid z.

05:43 So this is the integral between 0 and l over 4 times the integral of 0 and l over 4, times interval between 0 and l over 4, of 2 over l to the third times sine squared of pi x over l, sine squared of pi y over l, sine squared of pi z over l.

06:16 Actually, i should multiply this all by d, x, d, y, d z.

06:25 So this is 2 over l to the third times integral from 0 to l over 4 of sine square of pi x, over l, the x, and notice that then i'm going to multiply by a very similar term, substituting y by x, and then by a very similar term, substituting z by x.

06:52 All of these terms will end up being the same...

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