
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.

