Number of allowed pathways for electron transitions in hydrogen

[KaiserBrandon]

Homework Statement

for my comp physics course, we have to create a program that outputs all the possible electron transitions in hydrogen in the for n2 -> n1, where n1 < n2, and n2 < 21. We also have to determine the number of allowed pathways for the transition, and the energy and wavelength of the emitted photon.

Homework Equations

n= 1,2,3,...,20
l= 0,1,2,...,19 m= -l,...,0,...,l (angular momentum quantum numbers)

The Attempt at a Solution

I figured out how to do everything but the allowed pathways. I know the calculation of them has something to do with the angular momentum quantum numbers, but I can't even figure out how the number of pathways are determined, let alone create a program to determine them. All I know is that the output should be 3 for 2 -> 1, 3 for 3 -> 1, 15 for 3 -> 2, 15 for 4 -> 2, etc. Any help would be greatly appreciated.

 

[Jhero]

Dear student,

Determining the number of allowed pathways for electron transitions in hydrogen is a complex task, but it can be broken down into a few key steps.

First, it is important to understand the concept of quantum numbers in relation to electron transitions. The quantum numbers n, l, and m represent the energy level, orbital angular momentum, and magnetic quantum number, respectively. In order for an electron transition to be allowed, there are certain rules that must be followed. For example, the change in energy level (n) must be equal to 1, and the change in orbital angular momentum (l) must be either 1 or 0. Additionally, the change in magnetic quantum number (m) must be either 0 or ±1. These rules help to determine which transitions are allowed and which are not.

Next, it is important to understand the concept of selection rules. Selection rules are a set of rules that determine which transitions are allowed based on the quantum numbers. For electron transitions in hydrogen, the selection rules are as follows:

1. Δn = 1 (change in energy level must be 1)
2. Δl = ±1 (change in orbital angular momentum must be 1 or 0)
3. Δm = 0, ±1 (change in magnetic quantum number must be 0 or ±1)

Using these selection rules, we can determine the number of allowed pathways for a given transition. Let's take the example of 3 -> 2. From the selection rules, we can see that the change in energy level (n) is 1, the change in orbital angular momentum (l) can be either 1 or 0, and the change in magnetic quantum number (m) can be either 0 or ±1. This means that there are three possible pathways for this transition: 3p -> 2s, 3d -> 2p, and 3d -> 2s.

For transitions where the change in energy level (n) is larger, such as 4 -> 2, there will be more pathways that satisfy the selection rules. In this case, there are 15 possible pathways: 4f -> 2d, 4p -> 2s, 4d -> 2p, 4d -> 2s, and 4f -> 2s.

To create a program that determines the number of allowed pathways for a given transition, you