[mohaah2]

I have been trying to understand the concept of the probability density of an atom's electron. I understand that an electron is in a state of both wave and a particle but what I don't understand is how we get the image for example hydrogens probability graph is very confusing (http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html) what does that graph mean.

[Edit: external link deleted, asked on forum, answered on forum, that's part of the forum rules]

[Borek]

We calculate the density squaring the wave function, which we find by solving the Schroedinger equation.

[mjc123]

The probability density at a given point, obtained by squaring the wavefunction, gives the probability of finding the electron within a small volume element at that point. What your graphs show is the radial probability density, that is, the probability of finding the electron at a distance from the nucleus (in any direction) between r and r+dr. That is equal to the square of the wavefunction at r, multiplied by the volume of a spherical shell of radius r and thickness dr.

[Enthalpy]

The electron acts simultaneously from all the positions it occupies, and with a intensity and phase (or complex amplitude) given by the wavefunction.

Sometimes you consider interactions that can occur only in a volume smaller than the electron's delocalization (which is the atom's volume, for hydrogen). This is typically the case if the other interacting particle is more localized than the atom's electron, and the interaction is efficient in a small volume only. Then the atom's electron can interact only from a fraction of the volume it occupies, and the probability that you observe such an interaction drops as a consequence of the fraction of the wavefunction able to interact.

In that sense, the wavefunction (squared) tells a probability of interaction, what many people call a "probability of presence" - but please remember that the electron is simultaneously in all its possible positions. Quantum mechanics, as it is written and taught, carries a heavy load of historical misinterpretations that are presently abandoned.

The square of the wavefunction can be computed per volume element (for instance per pm³) directly. This is a function of x, y and z. For instance for the 1s, 2s... wavefunctions, it is biggest at the nucleus. Or you may consider a probability as a function of the radius to the nucleus, whatever the direction, as Hyperphysics does at your link. It's a density per length element, for instance per pm. For a spherical wavefunction, the density per radius unit is a product of the surface of the sphere of this radius, and then all such functions are zero at the nucleus.

"Both a wave and a particle": beware the historical misinterpretations there too. All interactions happen within a finite volume, they are never points. From a classical particle, quantum mechanics keeps very few attributes. For instance if an electron interacts by its charge, the interaction force or energy are computed for the whole charge at every possible position, rather than by spreading a charge density over all the occupied volume. Same for the momentum, kinetic energy... if they are defined well enough, an electron interacts with its full momentum and kinetic energy even if the interaction happens in a volume smaller than the electron's delocalization. Such attributes resemble better a classical particle if you wish.

[mohaah2]

Thank you so much all for the response it helped me get through this concept that was kicking my but. Especially your response enthalpy the detail actually helped a lot