Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.80 Hydrogen ground state Stark effect
This note derives the Stark effect on the hydrogen ground state.
Since spin is irrelevant for the Stark effect, it will be ignored.
The unperturbed ground state of hydrogen was derived in chapter
4.3. Following the convention in perturbation theory to
append a subscript zero to the unperturbed state, it can be summarized
as:
where is the unperturbed hydrogen atom Hamiltonian,
the unperturbed ground state wave function,
the unperturbed ground state energy, 13.6 eV, and is the Bohr
radius, 0.53 Å.
The Stark perturbation produces a change in this wave
function that satisfies, from (A.243),
The first order energy change is zero and can be dropped.
The solution for will now simply be guessed to be
times some spatial function still to be found:
Differentiating out the Laplacian of the product
into individual terms using Cartesian coordinates, the
equation becomes
The first term in this equation is zero since
. Also, now using spherical coordinates,
the gradients are, e.g. [41, 20.74, 20.82],
Substituting that into the equation, it reduces to
Now in polar coordinates, and for the
derivative of to produce something that is proportional
to , must be proportional to . (The Laplacian
in the second term always produces lower powers of than the
derivative and can for now be ignored.) So, to balance the
right hand side, should contain a highest power of equal to:
but then, using [41, 20.83], the term in the
left hand side produces an term. So
add another term to for its derivative to eliminate it:
The Laplacian of is zero so no further terms
need to be added. The change in wave function is
therefore
(This small perturbation
becomes larger than the
unperturbed wave function far from the atom because of the growing
value of . It is implicitly assumed that the electric
field terminates before a real problem arises. This is related to the
possibility of the electron tunneling out of the atom if the potential
far from the atom is less than its energy in the atom: if the electron
can tunnel out, there is strictly speaking no bound state.)
Now according to (A.243), the second order energy change
can be found as
Doing the inner product integration in spherical coordinates produces