
The
Hydrogen Atom: After a lot of buildup, we have
finally arrived at the quantum mechanical solution to explain the
hydrogen atom. This solution is found by solving the
Schrodinger equation by finding a wavefunction that fits the
boundary conditions, just like we did for the particleinabox.
First, we are going to put our system into spherical polar
coordinates, because that is the coordinate system which most
succinctly describes a system with the symmetry of a sphere.
Thus, instead of have coordinates x,
y, z, we have coordinates r,
theta, and phi
to describe the position of our electron around the nucleus (which
is fixed at the origin). Our boundary condition is that the
wavefunction psi goes to
0 as r goes to 0, which
makes sense because we need to explain the system in the condition
where the electron is pretty close to the nucleus.
The Schrodinger equation can be solved exactly for the hydrogen
atom, and it's actually not that hard, but it is a lot of calculus
that is out of the scope of this class. Therefore we will
not solve this ourselvs, but simply deal with the outcomes of the
solutions, the wavefunction and energy of the hydrogen atom.
To solve the Schrodinger equation, it is necessary to break
up the solution into a radial part, dependent on r,
and an angular part, dependent on theta
and phi. This
turns out to be really helpful for two reasons: 1) the total
energy of the system, E,
depends only on r, not
on any angle; and 2) this makes our wavefunctions much easier to
visualize.
The solutions to the Schrodinger equation give us 3 quantum
numbers:
n = 1, 2,
3... principal quantum number
l = 0, 1,
2...n1 angular momentum
quantum number
m = l,
l+1, l+2...0...l2,
l1, l
magnetic quantum number
This means that certain quantum numbers are allowed for each value
of n. However, the
energy of the system depends on n
only, so states that have the same n
but different l and m
are said to be "degenerate."
The wavefunctions that solve the Schrodinger equation for the
hydrogen atom are:
psi_{n,l,m}(r,theta,phi)
= R_{n,l}(r)Y_{l,m}(theta,phi)
The function R is the radial part of the wavefunction, and Y is
the angular part of the wavefunction. R and Y for the first
few quantum numbers are tabulated in Table 5.2 in your book.

