7The hydrogen atom
IB Quantum Mechanics
7.3 Comments
Is this it? Not really. When we solved the hydrogen atom, we made a lot of
simplifying assumptions. It is worth revisiting these assumptions and see if they
are actually significant.
Assumptions in the treatment of the hydrogen atom
One thing we assumed was that the proton is stationary at the origin and the
electron moves around it. We also took the mass to be
µ
=
m
e
. More accurately,
we can consider the motion relative to the center of mass of the system, and we
should take the mass as the reduced mass
µ =
m
e
m
p
m
e
+ m
p
,
just as in classical mechanics. However, the proton mass is so much larger and
heavier, and the reduced mass is very close to the electron mass. Hence, what
we’ve got is actually a good approximation. In principle, we can take this into
account and this will change the energy levels very slightly.
What else? The entire treatment of quantum mechanics is non-relativistic.
We can work a bit harder and solve the hydrogen atom relativistically, but the
corrections are also small. These are rather moot problems. There are larger
problems.
Spin
We have always assumed that particles are structureless, namely that we can
completely specify the properties of a particle by its position and momentum.
However, it turns out electrons (and protons and neutrons) have an additional
internal degree of freedom called spin. This is a form of angular momentum, but
with
`
=
1
2
and
m
=
±
1
2
. This cannot be due to orbital motion, since orbital
motion has integer values of
`
for well-behaved wavefunctions. However, we still
call it angular momentum, since angular momentum is conserved only if we take
these into account as well.
The result is that for each each quantum number
n, `, m
, there are two
possible spin states, and the total degeneracy of level
E
n
is then 2
n
2
. This agrees
with what we know from chemistry.
Many electron atoms
So far, we have been looking at a hydrogen atom, with just one proton and one
electron. What if we had more electrons? Consider a nucleus at the origin with
charge +
Ze
, where
Z
is the atomic number. This has
Z
independent electrons
orbiting it with positions x
a
for a = 1, ··· , Z.
We can write down the Schr¨odinger equation for these particles, and it looks
rather complicated, since electrons not only interact with the nucleus, but with
other electrons as well.
So, to begin, we first ignore electron-electron interactions. Then the solutions
can be written down immediately:
ψ(x
1
, x
2
, ··· , x
Z
) = ψ
1
(x
1
)ψ
2
(x
2
) ···ψ
Z
(x
Z
),
where
ψ
i
is any solution for the hydrogen atom, scaled appropriately by
e
2
7→ Ze
2
to accommodate for the larger charge of the nucleus. The energy is then
E = E
1
+ E
2
+ ··· + E
Z
.
We can next add in the electron-electron interactions terms, and find a more
accurate equation for
ψ
using perturbation theory, which you will come across
in IID Principles of Quantum Mechanics.
However, there is an additional constraint on this. The Fermi-Dirac statistics
or Pauli exclusion principle states that no two particles can have the same state.
In other words, if we attempt to construct a multi-electron atom, we cannot
put everything into the ground state. We are forced to put some electrons in
higher energy states. This is how chemical reactivity arises, which depends on
occupancy of energy levels.
–
For
n
= 1, we have 2
n
2
= 2 electron states. This is full for
Z
= 2, and this
is helium.
–
For
n
= 2, we have 2
n
2
= 8 electron states. Hence the first two energy
levels are full for Z = 10, and this is neon.
These are rather stable elements, since to give them an additional electron, we
must put it in a higher energy level, which costs a lot of energy.
We also expect reactive atoms when the number of electrons is one more
or less than the full energy levels. These include hydrogen (
Z
= 1), lithium
(Z = 3), fluorine (Z = 9) and sodium (Z = 11).
This is a recognizable sketch of the periodic table. However, for
n
= 3 and
above, this model does not hold well. At these energy levels, electron-electron
interactions become important, and the world is not so simple.