The hydrogen atom consists of an electron interacting with a proton through a coloumb force. The hydrogen atom is one of the most accurately described systems in all of physics.
The first postulate of Quantum Mechanics states that the state of a particle is described by its wave function. The probability of finding the particle can be found by the magnitude of the wave function squared. This must be normalized because the probability of finding that particle throughout the entire universe is exactly 1.
The second postulate of Quantum Mechanics states that this wave is described by Schrodinger's Wave Equation:
The normalized solution of the wave equation for an electron in the hydrogen atom is:
n is the quantum number arising from quantization of energy
m is the quantum number arising from quantization of angular momentum
l is the quantum number arising from quantization of angular momentum squared
Lnl is the generalized Laguerre polynomial
Plm is the associate Legendre polynomial
r, theta, phi are the vectors in spherical coordinates as shown to the right
From the solutions to Schrodinger's Wave Equation, the boundary conditions for n, l, m are given as follows:
0 < n < Infinity
0 <= l < n
-l <= m <= l
The probability is a function of n, l and |m|. Consequently, the probability density of m=-1 is the same as m=1. The probability is not a function of the fourth quantum number which represents spin.
Below are visualizations of the probability of finding an electron in the hydrogen atom for a given n, l, m. The colour gradeint from low density (blue) to high density (red) is as follows:
It is important to note that the following visualizations are approximations. The mass ratio of a proton to an electron is approximately 1836. Consequently, the centre of mass of the system is very close to the proton, but not exactly at the centre. In these calculations, it was assumed that the proton was directly at the centre of the system.
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