An electron is in a three-dimensional box with side lengths LX=0.600 nm and LY=LZ=2 LX . What ar

step 1 Okay, so in this problem we have an electron inside the 3d box. We know that the length side x,

An electron is in a three-dimensional box with side lengths...

An electron is in a three-dimensional box with side lengths $L_{X}=0.600 \mathrm{nm}$ and $L_{Y}=L_{Z}=2 L_{X} .$ What are the quantum numbers $n_{X}, n_{Y},$ and $n_{Z}$ and the energies, in $\mathrm{eV},$ for the four lowest energy levels? What is the degeneracy of each (includine the dereneracy due in snin)?

Key Concepts

Particle in a Three?Dimensional Box

This is a quantum mechanics model where a particle, such as an electron, is confined in a finite region of space with infinite potential barriers at the boundaries. The model illustrates how the allowed energy states become quantized due to the spatial confinement, and it forms the basis for understanding quantum confinement effects in nanostructures and other quantum systems.

Quantum Numbers

In the context of a three?dimensional box, quantum numbers serve as indices that label the discrete energy eigenstates of the system. Each spatial direction (x, y, and z) is associated with a quantum number, and these integers determine the energy contributions along each axis. The set of quantum numbers (nX, nY, nZ) characterizes a unique state and plays a crucial role in calculating the energy of the state.

Energy Quantization

The quantization of energy in a particle in a box comes from solving the Schrödinger equation with appropriate boundary conditions. The solution reveals that the energy of each state depends on the square of its quantum numbers and the dimensions of the box. Thus, changes in either the quantum numbers or the dimensions of the well directly affect the energy spectrum of the electron.

Degeneracy

Degeneracy refers to the phenomenon where multiple quantum states share the same energy level. In a three-dimensional box, degeneracy can occur when different combinations of quantum numbers yield the same total energy. Understanding degeneracy is important as it leads to multiple states contributing to the physical properties of the system, especially when considering thermal or external field effects.

Spin Degeneracy

In addition to the spatial degeneracy given by combinations of the quantum numbers, electrons, being spin-1/2 particles, contribute an additional factor of degeneracy due to the two possible spin states (spin up and spin down). This spin degeneracy is vital when calculating the total number of accessible states, as it effectively doubles the count of each spatial eigenstate.

Transcript

00:01 Okay, so in this problem we have an electron inside the 3d box.

00:06 We know that the length side x, lx of this box is 0 .6 times 10 to the minus 9 meters, okay? and we also know that this side length y is equal to the side length c that is equal 2 times lx.

00:40 Okay, so this is the setup of the problem, and the problem wants to know what are the quantum numbers nx, n, n, y, and n z.

01:01 We want to know the energy of this state, so energy x, y, z.

01:10 For the four lowest energy levels.

01:18 So what is the energy and the quantum numbers for the four lowest energy levels of the electron inside the box? okay, let me see.

01:39 We know that the definition of energy, which is the first thing we must remember, so the definition of the energy for a particle inside of box, let's call this energy xyz, is described by the quantum numbers, and x, and y, and z, the sum of the square of these numbers, each one divided by the square of the box.

02:16 All this multiply by pi square, the planck constant square divided by two times the mass of the particle, which in this particular case is the electron.

02:38 Okay, we can simplify this equation a little because we know the size of the box.

02:45 We know that lz and ly are two times the length of the side x.

02:54 Therefore, rearranging this, we can simplify and say that the energy for this problem, xyz, is going to be 4nx plus ny square plus nz square.

03:17 All this multiply by pi square times the plank constant square divided by 8 times the mass of the particle times lx square.

03:31 Okay, so this is the simplification of the problem, and we want to find out what are the four lowest energy levels for this, for this hydrogen, for this electron.

03:49 Inside a box.

03:52 Let's begin.

03:54 So this is the equation that we must begin.

03:56 So let's begin...

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