The Stark effect describes that two energy
states that have the same energy can be split
up using an electric field. In other words,
an electric field can help to distinguish
degenerate eigenstates in a physical system.
Let's look at the hydrogen atom as an example.
After solving the Schrödinger equation, we
see that the energy levels in the Hydrogen
atom are given by one over n squared. This
means that for instance the 2s- and the 2p-states
should have the same energy. Looking at the
experimentally found spectral lines, it's
true that we only get one line for all n=2
states. However, if we turn on an electric
field, we can separate these lines! For a
small electric field, the lines will be a
little bit apart, but if we have a stronger
electric field, the lines will be even wider
apart. At first, this goes linearly with the
electric field and this is what we will investigate
now:
In order to do calculations in quantum mechanics,
we use our favorite toolkit - perturbation
theory. To use perturbation theory, we must
first think about the potential that will
perturb our Hamiltonian. An electric field
will push the electron in the hydrogen atom
in one way and the nucleus in the other way.
Mathematically, this is described with an
electric dipole d, which has potential energy
minus d times the electric field. The dipole
d is a vector from the negative charge to
the positive charge, multiplied with the corresponding
charge q. In our case, d points from the electron
to the proton and q is the elementary charge
of the electron.
In perturbation theory, the first order corrections
to the energy are given by the eigenvalues
of the perturbing potential. If we want to
investigate the n=2 states, we need to calculate
matrix elements of V with the n=2 states.
To save us some time, the only non-zero results
arise from the 2s and those 2p states with
m=0. Since the potential is a hermitian operator,
these two values must be the complex conjugates
of each other. If we assume that the electric
field points in the z-direction, the inner
product simplifies and the potential is now
given by charge times magnitude of the electric
field times z.
These wave functions are listed in most textbooks,
so we can easily write down the integral representation
of this matrix element. Don't forget to include
every term of the wave functions as well as
the Jacobian of spherical coordinates and
of course the z of the potential. The result
is minus 3a, where a is Bohr's radius. Now
the matrix, whose eigenvalues should give
us the energy corrections, reads zero, 3 a
e E, 3 a e E, zero. You can now either calculate
the eigenvalues directly, or notice that this
matrix is minus 3 a e E times the first Pauli
matrix. The eigenvalues of the Pauli matrices
are plus and minus one, so the eigenvalues
of this matrix are plus and minus 3 a e E.
But which state gets the higher energy and
which state gets the lower energy? Does the
2s now lie higher than the 2p? Or the other
way? Well, neither. The truth is, that the
2s and the 2p states form two new states as
a linear combination. The positive eigenvalue
belongs to the eigenvector (1,1), whereas
the negative eigenvalue belongs to the eigenvector
(1,-1). So the new state with higher energy
is one over the square root of two |200> plus
|210> and the one with lower energy is one
over square root of two |200> minus |210>.
The Stark effect is named after the German
physicist Johannes Stark, who discovered this
effect in 1913. In the same year, Italian
physicist Antonino Lo Surdo also discovered
the very same effect, which is why it is sometimes
called the Stark–Lo Surdo effect. However,
only Johannes Stark was awarded the Nobel
Prize in Physics in the year 1919.
And that's pretty much it for this video,
thanks for watching!
