# Atomic Probability Densities Always Even?

• physicsphreak2
In summary, the conversation discusses the parity of hydrogenic wavefunctions and the probability densities for such wavefunctions. The concept of the Stark effect is also brought up, specifically in relation to the absence of a first-order correction for the ground state of atoms with a p-orbital valence electron. The conversation then explores the idea that the probability density must always be even if the underlying state has odd parity. The question is posed whether an energy eigenstate of any potential can have odd parity and if there exists a function that is odd after squaring.
physicsphreak2
I know that for hydrogenic wavefunctions, the parity of a given state is $(-1)^l$. But does this mean that the probability densities for any such wavefunction is ALWAYS even?

I'm trying to understand the Stark effect, and specifically why there is no first-order correction for he ground state of atoms whose valence electron is in a p-orbital. The only reason I can think of is that we must have: $\Delta E^{(1)} = \langle \psi_0 | z | \psi_0 \rangle = 0$ (for a field along z)... so we end up with the integral of $|\psi_0 |^2 z dz$... and if we want to argue that this is the product of an even and odd function (which is odd, therefore integrating to 0), it seems like the probability density (as opposed to the underlying state) must always be even. Is this true?

Can you find a probability density for an energy eigenstate of any potential that has odd parity?
Can you find any function that is odd after squaring?

## What is an atomic probability density?

An atomic probability density is a measure of the likelihood of finding an electron within a certain region of an atom. It is represented by a three-dimensional graph, with the x, y, and z axes representing the position of the electron and the height of the graph representing the probability of finding the electron at that position.

## Why are atomic probability densities always even?

Atomic probability densities are always even because they represent the probability of finding an electron in a given region, and the electron is equally likely to be found on either side of the nucleus. This even distribution is a result of the electron's wave-like nature and the principle of quantum mechanics.

## How are atomic probability densities calculated?

Atomic probability densities are calculated using mathematical equations, such as the Schrödinger equation, which takes into account the energy levels and positions of the electrons in an atom. These equations can be solved using computers to generate the three-dimensional graphs that represent the atomic probability densities.

## Why are atomic probability densities important in chemistry?

Atomic probability densities are important in chemistry because they provide information about the behavior and properties of atoms. They can help predict the chemical reactivity of elements and the formation of chemical bonds. They also play a crucial role in understanding the electronic structure of molecules and their spectroscopic properties.

## Do atomic probability densities change over time?

Yes, atomic probability densities can change over time as the electrons in an atom can move to different energy levels or positions. This is known as electron density fluctuation and is an important factor in chemical reactions and the behavior of atoms in different environments.

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