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

A particle is in the three-dimensional cubical box of Section $41.2 .$ (a) Consider the cubical volume defined by $0 \leq x \leq L / 4$ $0 \leq y \leq L / 4,$ and $0 \leq z \leq L / 4 .$ What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state $\left(n_{X}=1, n_{Y}=1, n_{Z}=1\right),$ calculate the probability that the particle will be found in the cubical volume defined in part (a). (c) Repeat the calculation of part (b) when the particle is in the state ny $$ =1 $$

A particle is in the three-dimensional cubical box of Section $41.2 .$ (a) Consider the cubical volume defined by $0 \leq x \leq L / 4$ $0 \leq y \leq L / 4,$ and $0 \leq z \leq L / 4 .$ What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state $\left(n_{X}=1, n_{Y}=1, n_{Z}=1\right),$ calculate the probability that the particle will be found in the cubical volume defined in part (a).
(c) Repeat the calculation of part (b) when the particle is in the state ny $$ =1 $$

Key Concepts

Particle in a Three-Dimensional Box

In quantum mechanics, a particle confined in a three-dimensional box (also known as the infinite potential well) has quantized energy levels and corresponding eigenfunctions. The system is solved by applying the Schrödinger equation with boundary conditions that restrict the particle to a finite region in each spatial dimension. This leads to discrete allowed quantum numbers and wavefunctions that are typically products of sinusoidal functions.

Volume Fraction

The concept of volume fraction involves determining the ratio of the volume of a subregion to the total volume of the container. In problems involving spatial probabilities, it is important to calculate this fraction as it often provides a straightforward estimation of the probability of finding a particle in that region, especially when the probability distribution is uniform or constant.

Probability Density and Normalization

The probability density in quantum mechanics is obtained by taking the square modulus of the wavefunction. To calculate the probability that a particle is found in a specific spatial region, one must integrate the probability density over that region. Normalization is critical, ensuring that the total probability over all space sums to one, which is a fundamental requirement of quantum theory.

Separation of Variables and Eigenfunctions

Separation of variables is a method used to solve the Schrödinger equation in multiple dimensions, where the solution is written as a product of functions, each depending on a single coordinate. This technique simplifies the problem by reducing it to one-dimensional equations, each of which can be solved to yield eigenfunctions associated with their respective quantum numbers. This approach underpins the determination of both energy levels and spatial probability distributions.

Transcript

00:02 Okay, so in this problem we have a particle inside a three -dimensional box and we must calculate two things, actually three things in this problem.

00:15 So let's make one thing at a time.

00:19 The first thing we must calculate is what fraction of the volume the specific volume we must consider represents.

00:28 So, first of all, this is item a, okay? this is the easiest one.

00:35 So the total volume, the total volume of the box is v equals l cubics, because it's a cubical box.

00:52 And the specific volume we have to consider, so the specific volume, let's call this ve.

01:07 Is basically l to the 4 cubics.

01:14 Okay? therefore the answer, or the first question, which is what fraction of the total volume is this cubic box represents? well, we just need to divide ve by v, which gives us 1 divided by 64.

01:38 So this is the fraction, or if we, if we want in percentage this is just 1 .56 % of the entire volume of the box okay so but the fraction is 1 divided by 64 this is the fraction this is the answer now let's move on to the item b so item b of this problem we must discover what is the probability of finding this particle inside this specific volume.

02:18 So first of all, how we discover probability in quantum mechanics.

02:26 So the probability is described by the integral from minus infinity to infinity of the modular square, absolute value square, of the wave function of the particle.

02:47 Okay, the solution for the wave function of a particle inside a box is already no.

02:54 We can look in the book or we can look in other questions, but since we want the wave function square, well the probability for our particular volume is going to be 2 divided by l this is the normalization cubics that multiplies the integral from 0 to l4 this is the this is the x -axis so the solution is is sine square of pi x l d x and we repeat this for the entire for all the three axes because i forgot to mention that as we can see in the problem we have to consider that the particle is in the ground state so therefore the ground state is n x n y n z equals 1 okay that's why the solution for the wave particle the wave function sorry is this one in here.

04:16 So the solution for the y is sine square pi i l dy and the solution for the z 0 to l to the 4 sine square pi z l d z okay so this is the definition of the probability of finding a particle inside the specific volume.

04:53 Well as we can see all the three integrals are the same with the same interval.

05:01 Therefore let's solve only one of them because the solution will be the multiplication of all the three solutions.

05:10 So let's begin with this one in here.

05:14 Well the solution for the first integral we can use a trigonometric relation and rewrite the sine square as 1 minus cosine of 2 pi x l, all this divided by 2...

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