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

CALC A particle in the three-dimensional box of Section 41.2 is in the ground state, where $n_{X}=n_{Y}=n_{Z}=1 .$ (a) Calculate the probability that the particle will be found somewhere between $x=0$ and $x=L / 2 .$ (b) Calculate the probability that the particle will be found somewhere between $x=L / 4$ and $x=L / 2$ . Compare your results to the result of Example 41.1 for the probability of finding the particle in the region $x=0$ to $x=L / 4$ .

CALC A particle in the three-dimensional box of Section 41.2 is in the ground state, where $n_{X}=n_{Y}=n_{Z}=1 .$ (a) Calculate the probability that the particle will be found somewhere
between $x=0$ and $x=L / 2 .$ (b) Calculate the probability that the particle will be found somewhere between $x=L / 4$ and $x=L / 2$ . Compare your results to the result of Example 41.1 for the probability of finding the particle in the region $x=0$ to $x=L / 4$ .

Key Concepts

Separation of Variables in Multi-dimensional Systems

For systems like the three-dimensional box, the Schrödinger equation is solved by writing the overall wavefunction as a product of functions, each depending on a single coordinate (x, y, and z). This method simplifies the problem by reducing it to one-dimensional cases, which are easier to solve and analyze, especially when determining probabilities along a single axis.

Integration of Probability Density

Calculating the probability of finding a particle in a specified region involves integrating the normalized probability density over that region. This process relies on the properties of the wavefunction and its squared amplitude, and it is a standard technique in quantum mechanics to evaluate spatial probabilities and expectation values.

Normalization of the Wavefunction

Normalization is a key principle that ensures the total probability of finding the particle within the entire box is unity. This condition is used to determine the constant factors in the wavefunction and is fundamental when integrating the probability density over any subregion to compute the probability of particle localization.

Wavefunction and Probability Density

The wavefunction is a mathematical function that encapsulates the quantum state of a particle. Its square (or the modulus squared in the case of complex functions) represents the probability density, which tells us the likelihood of finding the particle at a specific point in space. This concept is crucial when calculating the probability of the particle’s position over a spatial region through integration.

Quantum Particle in a Box

This concept involves a quantum particle that is confined within an impenetrable boundary (the box) where the potential energy is zero inside and infinite outside. The boundary conditions force the particle’s wavefunction to vanish at the edges, leading to quantized energy levels. This idealized problem serves as an essential model for understanding confinement and quantization in quantum mechanics.

Transcript

00:01 To find the probability here, we're going to have the max born normalization condition, which gives the integral over dx, d, y, d, z, which is the volume times si squared, right? so, si squared over the volume integral is equal to one, where si is equal to some constant c times the sign.

00:19 So i'm using the abbreviation just s for sign times the sign of nx, pi, x over l, times a sign n y, pi, y, over l, times the sign mz, pi, c over l.

00:29 So for a and b, we're giving the area over which we're integrating here.

00:34 So let's go ahead and do part a.

00:37 So the probability in part a, p, is the integral from x equals 0 to l over 2.

00:52 For y, we're integrating from 0 to l, and from z we're also going from 0 to l.

01:05 Okay, and c is just a constant, so we can pull that out front.

01:11 So let's move this stuff over so we can pull the constant outside of the integration.

01:18 So we have the constant c getting squared.

01:21 So we have c squared times sine squared in x pi x over l times sine squared in y, pi y over l, and then same thing for z.

02:18 This is supposed to be sine squared times that value.

02:22 So sine squared times the z value, just as above.

02:27 So that goes there.

02:28 Okay, and again, this is integrated with dx, d, d, x, d, y, dz.

02:36 So treat the integration with respect to x, y, and z separately.

02:40 And again, the integration is from 0 to l over 2 for x, 0 to l for y, and 0 to l for z.

02:46 So we find that the probability here, so according to the max born rule, of course, just this equation, if it's equal to 1, gives us a value of c equal to 2 over l.

03:03 And this is done in the book.

03:05 You could do the proof again if you wanted to, but not necessary.

03:09 So when this is equal to 1, c is equal to 2 over l to the 3 halves.

03:16 So we'll replace c with 2 over l to the 3 halves.

03:18 Of course, c is getting squared there.

03:20 So what we have here is the probability is equal to...

Please give Ace some feedback