Enclosed charge of an electron cloud with a given charge density function.

[geophysics10]

I have a conceptual problem.

Homework Statement

I was given a charge distribution for an electron cloud of a hydrogen atom in the ground state - ignoring the nucleus.

Homework Equations

charge density: ρ=charge of electron/(pi*Bohr radius^3)*exp(-2r/Bohr radius)

The Attempt at a Solution

Whenever I integrate over the volume of the sphere to find the enclosed charge, I got a value other than the charge of an electron. Why isn't the enclosed charge equal to the charge of an electron? What concept am I missing here? The charge density function comes from quantum mechanics. Is there some weird quantum effect there that is taken into account?

 

[Simon Bridge]

The charge distribution does not terminate inside a finite volume - plot ##\rho(r)## vs ##r## and compare with the radius of the volume you are integrating over.

The "weird quantum effect" is the statistical nature of that charge distribution ... for a sphere radius R (finite volume) about the nucleus, there is always a non-zero probability of finding the electron outside that volume. Integrate to infinity and you should get ##q_e##.

 

[geophysics10]

Thank you. I took a drive and realized that some of the distribution must lie outside the Bohr radius because it is an average value for the distance of the electron, and that when r is at infinity the electron and proton must both be point charges at the origin. I'm glad I came back to someone confirming that!

 

[Simon Bridge]

Yah - this treatment of the electron as having its charge smeared out over the wave0-function is only an average treatment. An actual interaction may measure the position of a particular electron more accurately than that... but, for a large number of hydrogen atoms, it all averages out.

 

[paranormal]

It is important to remember that the charge density function for an electron cloud in an atom is a probability distribution, not a physical distribution of charge. This means that while it can give us information about the likelihood of finding an electron at a specific point in space, it does not represent the actual physical distribution of charge. Additionally, the charge of an electron is a fundamental constant and is not dependent on the position or state of the electron. Therefore, the enclosed charge may not necessarily equal the charge of an electron, as it is determined by the specific distribution and probability of the electron's location within the atom. It is also worth noting that the Bohr radius used in the charge density function is a classical approximation and may not accurately represent the quantum effects of the electron cloud. I would suggest consulting with your instructor or a more advanced textbook for further clarification on this concept.