CP A photon is emitted when an electron in a...

CP A photon is emitted when an electron in a three-dimensional box of side length $8.00 \times 10^{-11} \mathrm{m}$ makes a transition from the $n_{X}=2, n_{Y}=2, n_{Z}=1$ state to the $n_{X}=1, n_{Y}=1$
$n_{Z}=1$ state. What is the wavelength of this photon?

Key Concepts

Quantum Numbers

Quantum numbers are sets of numerical values that describe the quantized energy states and other properties of particles in quantum systems. In the context of a three-dimensional box, each spatial dimension has an associated quantum number that determines the contribution of that dimension to the total energy of the state.

Particle in a Box

This concept describes a quantum system where a particle is confined within an infinitely deep potential well with well-defined boundaries. In such a situation, the particle can only occupy specific energy states, which are determined by the dimensions of the box and the quantum numbers associated with motion in each spatial direction.

Energy Quantization

Energy quantization refers to the idea that in quantum mechanics, a system such as a confined particle can only have discrete values of energy. These specific energy levels arise due to the boundary conditions imposed on the particle’s wavefunction, and transitions between these levels involve well-defined changes in energy.

Photon Emission and Energy Transition

When an electron transitions from a higher energy state to a lower energy state, the difference in energy is released in the form of a photon. This process is at the heart of various spectroscopic phenomena and relies on the conservation of energy, connecting the quantized energy levels of the electron to the energy (and hence wavelength) of the emitted photon.

Planck's Relation

Planck's relation, expressed as E = hc/?, provides a direct connection between the energy of a photon and its wavelength. This fundamental relation underpins the quantum nature of light and allows one to determine the wavelength of the emitted photon by comparing it to the energy difference between quantum states.

Transcript

00:01 We'll be using the concept of the particle in the box to find the energy of the photon that is released in a transition between two energy levels that are given here.

00:09 So the energy level of a cubicle box, which has x, y, and z components is given as follows.

00:16 Nx, the energy level in the x squared plus ny, energy level in the y plus nz, energy level in the z, all squared values, multiplied by plank's constant divided by eight times the mass of the particle times the length of the box.

00:30 L squared.

00:33 So we are told that this is going from the 221 state to the 1 -1 -1 state for as far as nx, and y, and then z are concerned.

00:43 So we also know that the change in energy levels gives the energy of the photon delta e which is equal to planks constant times the speed of light divided by the wavelength.

00:53 So if we find delta e we can find the wavelength.

00:56 So delta the e here is going to be equal to the energy level in the 221 state.

01:04 So we can call this e221 minus the energy level in the 111 state.

01:16 So we just simply plug in 221 for nx, and y, and z accordingly, and then 111 for nx, and y, and z accordingly, and carry out the operation.

01:26 So 221 gives you 2 squared plus 2 squared plus 1 squared, which is 9.

01:30 So this is 9 minus 1 squared plus 1 squared plus 1 squared, which is 3.

01:36 So this is 9 minus 3.

01:38 And then both values are multiplied by planx constant divided by aml.

01:42 So that's why i pulled that out...