# Understanding Probability Densities for Hydrogen Wave Functions

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In summary, the conversation discusses a problem involving quantum numbers and radial functions. The probability density function is the sum of the probabilities of being in different states, and the coefficients are assumed to be equally probable. However, in the given solution, the weights for each state are 1/4 and 3/4, which corresponds to 1 l=0 state and 3 l=1 states. This weight distribution is explained and understood by the individual in the conversation.

## Homework Statement

The problem, along with a solution, is attached as an image file.

## The Attempt at a Solution

I have done the problem which was very straight forward. One simply had to look up the Rn,l and then plug in the appropriate quantum numbers. Since for a given n, there are n-1 values of l, there are two corresponding radial functions R2,0 and R2,1 for the n=2 state. So the probability density which is $$\left|R\right|^{2}$$ is the sum of the probability of being in the l=0 state and probability of being in the l=1 state. Because the problem does not indicate the state of the initial wave function, we don't know the coefficients of R2,0 and R2,1 so my TA writes that we should assume they are equally probable. But what I don't understand is why in his probability density function he writes a 1/4 in front of the l=0 function and a 3/4 in front of the l=1 function. Where did these values come from? I'm posting here instead of asking my TA because I have a midterm tomorrow morning and I won't get a response from my TA in time if I were to email him now.

Thanks

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There is 1 $$\ell=1$$ and 3 $$\ell=1$$ states. It appears that he gave each state weight 1/4.

You mean 1 l=0 state, but yes I see now. This makes perfect sense. Thank you very much!

## 1. What is a hydrogen wave function?

A hydrogen wave function is a mathematical expression that describes the behavior of an electron in a hydrogen atom. It represents the probability of finding an electron in a particular location around the nucleus of the atom.

## 2. How is a hydrogen wave function calculated?

A hydrogen wave function is calculated using a mathematical equation called the Schrödinger equation. This equation takes into account the energy of the electron, the mass of the electron, and the potential energy of the nucleus to determine the probability of finding the electron in a specific location.

## 3. What information can be obtained from a hydrogen wave function?

A hydrogen wave function can provide information about the energy levels and orbital shapes of the electron in a hydrogen atom. It can also be used to calculate other properties such as the probability of finding the electron in a certain region of space and the average distance of the electron from the nucleus.

## 4. How does a hydrogen wave function relate to the behavior of electrons?

A hydrogen wave function is a fundamental part of quantum mechanics and is used to describe the behavior of electrons. It provides insight into the probabilistic nature of electrons and how they move and interact with the nucleus in an atom.

## 5. Can a hydrogen wave function be visualized?

Yes, a hydrogen wave function can be visualized using three-dimensional graphs called wave function plots. These plots show the probability distribution of finding an electron in different regions of space around the nucleus of a hydrogen atom.