# Hydrogen probability distribution

• unscientific
In summary, the conversation discusses the significance of the summations for l=1 and l=2 in relation to the probability distribution in a hydrogen atom. The results show that there are five-thirds as many states for l=2 compared to l=1, which aligns with the number of orbitals for each value of m. The terms in the summations represent the degeneracy for a certain l, and each spin/orbit state contributes a factor of 1/4pi to the overall calculation. The question is asked about the meaning of the terms on the right-hand side and their significance for a particular |n,l,m> state.
unscientific
I have found that:

For l = 1:

$$\sum_{m=-l}^l |Y_l^m|^2 = \frac{3}{4\pi}$$

For l = 2:

$$\sum_{m=-l}^l |Y_l^m|^2 = \frac{5}{4\pi}$$

What significance does this have for the probability distribution in an hydrogen atom?

the significance is that there are five-thirds as many states of orbital angular momentum l=2, than for l=1. This makes sense because there are five orbitals for l=2, and three orbitals for l=1, for all the values of m.

Last edited:
Do you know what those terms mean?
Do you know why you did the sum in each case?

Simon Bridge said:
Do you know what those terms mean?
Do you know why you did the sum in each case?

I think the numerator gives the degeneracy for a certain l?

since -l ≤ m ≤ l, m can take on 2l+1 possible values.

Like possible states for |n,l,m>:

For l = 1:

Possible states are |n,1,-1> and |n,1,0> and |n,1,1>

I think the numerator gives the degeneracy for a certain l?
Each spin/orbit state contributes a factor of ##1/4\pi## to the overall thingy being calculated.

But I meant the terms on the RHS.

For a particular |n,l,m> state, what does |Ylm|2 mean?

Now - what was your question?

## 1. What is a hydrogen probability distribution?

A hydrogen probability distribution is a mathematical function that describes the likelihood of finding a hydrogen atom at a certain distance from its nucleus. It is based on the quantum mechanical model of the atom and takes into account the wave-like behavior of electrons.

## 2. How is a hydrogen probability distribution calculated?

A hydrogen probability distribution is calculated using the Schrödinger equation, which takes into account the energy of the electron, the position of the nucleus, and the shape of the electron's wave function. This calculation results in a three-dimensional graph that shows the probability of finding the electron at different distances from the nucleus.

## 3. What does a hydrogen probability distribution tell us about the atom?

A hydrogen probability distribution gives us information about the electron's location and energy within the atom. It shows us the most likely locations of the electron and the regions where it is less likely to be found. This information is important for understanding the behavior and properties of atoms.

## 4. How does a hydrogen probability distribution differ from other elements?

The hydrogen atom is the simplest atom and has only one electron, so its probability distribution is relatively straightforward. However, for atoms with multiple electrons, the probability distribution becomes more complex as the electrons interact with each other. Additionally, the shape of the probability distribution can vary depending on the energy level of the electron.

## 5. What are some practical applications of hydrogen probability distributions?

Hydrogen probability distributions are used in a variety of fields, including chemistry, physics, and materials science. They can help predict the behavior of atoms and molecules in chemical reactions, understand the properties of materials, and aid in the development of new technologies such as solar cells and fuel cells.

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