Hydrogen (atoms, molecules) in external fields
Static electric and magnetic fields
Oscyllating electromagnetic fields
Everything said up to now has to be modified more or less strongly if we
consider atoms (and ions) which are not isolated, but influenced by an external
electromagnetic field.
For low-lying bound states of an atom the influence of external fields can often be
satisfactorily accounted for with perturbative methods, but this is no longer
possible for highly excited states and/or very strong fields, in which case intricate
and physically interesting effects occur,even in the “simple” hydrogen atom.
The study of atoms (and molecules) in strong external fields has been a topic of
considerable interest.
General introduction
In this section we consider a classical electromagnetic field described by
the scalar potential Φ(r, t) and the vector potential A(r, t).
⎧ ^
⎫
⎪⎪[ p + (e / c ) A( r , t )]2
⎪⎪ ^
^
− eϕ ( r , t )⎬ + V
H =⎨
2μ
⎪
⎪
⎪⎩
⎪⎭
An important consequence of external fields is, that the Hamiltonian is in general no longer
rotationally invariant, so that its eigenstates aren’t simultaneously eigenstates of angular
momentum. For spatially homogeneous fields and the appropriate representation of Hamiltonian
it remains invariant under rotations around an axis parallel to the direction of the field, so that
the component of total angular momentum in the direction of the field remains a constant of
motion.
For an electron in a potential V (r) which is not radially symmetric, but invariant under rotations
around the z-axis, say, we can at least reduce the three-dimensional problem to a twodimensional problem by transforming to cylindrical coordinates ρ , z, φ
With the ansatz:
Static homogeneous electric field
We describe a static homogeneous electric field E, which is taken to point in
the direction of the z-axis, by a time-independent scalar potential φ (vector
potential vanishes)
The Hamiltonian then has the following special form:
^2
^
^
p
H=
+ V + ezE z
2μ
The shifts in the energy eigenvalues caused by the contribution of the field
are given in time-independent perturbation theory.
ΔE 1n = eE z ψ n z ψ n
where
ψ n z ψ n = ∫ψ n * zψ ndτ
where ψn are the eigenstates of the unperturbed (Ez = 0) Hamiltonian. These
eigenstates are usually eigenstates of the electron parity operator so that the
expectation value of the operator z, which changes the parity, vanish.
In the unusual case that an eigenvalue of the unperturbed Hamiltonian is degenerate and
has eigenstates of different parity, we already obtain nonvanishing energy shifts in first
order, and this is called the linear Stark effect. The first-order energy shifts are calculated
by diagonalizing the perturbing operator eEzz in the subspace of the eigenstates with the
degenerate (unperturbed) energy.
From the analysis of the second order perturbation one obtains:
ΔE
( 2)
n
= (eE z )
2
ΔE
2
ψ n zψ m
En − Em
( 2)
n
=−
αd
2
where
E 2z
n≠m
αd is dipole polarizability
En and Em are the eigenvalues of the unperturbed Hamiltonian. The energy
shifts depends quadratically on the strength Ez of the electric field and are
known under the name quadratic Stark effect.
For n = 2 there is a non-vanishing matrix element between the l = 0 and l = 1
states with m = 0. The two further l = 1 states with azimuthal quantum
numbers m = +1 and m = −1 are unaffected by the linear Stark effect. Figure
shows the splitting of the n = 2 term in the hydrogen atom due to the linear
Stark effect. For comparison Figure shows the energy shift of the n = 1 level
due to the quadratic Stark effect.
Important extras
1.
The energy shifts in quadratic Stark effect are closely connected with the
dipole polarizability of the atom in an electric field.
The wave functions are no longer eigenfunctions of the electron
parity, and they have a dipole moment induced by the external field and
pointing in the direction of the field (the z-direction).
2.
Field ionization:
-important in spectroscopy of Rydberg states
-large field generated by the lasers can reduce effective energy
for the atom (molecule) ionization.
Atoms in a static, homogeneous magnetic field
A static homogeneous magnetic field pointing in z direction can be described
in the symmetric gauge by a vector potential
In this gauge the Hamiltonian (3.225) keeps its axial symmetry around the
z-axis and has the following special form:
^
2
^
^
p
e
e2 2 2
H=
+V +
Bz L z +
Bz ( x + y 2 )
2μ
2μ
8μ
^
μ is reduced mass
We neglect the term in the Hamiltonian (3.249) which is quadratic in the field
strength Bz, and consider only linear component.
The hamiltonian contains component (Zeeman interaction Hamiltonian)
describing the interaction of the magnetic field with magnetic moment generated
by orbital angular momentum with external magnetic field.
Assuming that the interaction is small enough, first order perturbation theory
gives the Zeeman energy:
normal Zeeman effect
Eigenstates of the unperturbed (field-free) Hamiltonian, in which effects of spin-orbit
coupling are negligible and in which the total spin vanishes, i.e. in which the orbital angular
momentum equals the total angular momentum, remain eigenstates of the Hamiltonian in
the presence of the magnetic field, but the degeneracy in the quantum number ML is lifted.
We generally cannot neglect the contributions of the spin to the energy shifts in a
magnetic field. The most important contribution comes from the magnetic moments
due to the spins of the electrons.
The interaction of these spin moments with a magnetic field is obtained most
directly if we introduce the field into the Dirac equation and perform the transition
to the non-relativistic Schr¨odinger equation. To first order we obtain the following
Hamiltonian for a free electron in an external magnetic field:
^
^
2
^
^
^
p
e
H=
+V +
Bz ( L z + 2 S )
2μ
2μ
The interaction of an atom with a magnetic field is thus given to first order
in the field strength by a contribution.
The magnetic moment now is no longer simply proportional to the total angular
momentum ˆJ = ˆL + ˆS, which means that there is no constant gyromagnetic
ratio. The splitting of the energy levels in the magnetic field now depends not
only on the field strength and the azimuthal quantum number as in the normal
Zeeman effect; for this reason the more general case, in which the spin
of the atomic electrons plays a role, is called anomalous Zeeman effect.
( μJ=gJγJ )
As the strength of the magnetic field increases, the interaction with the field becomes stronger
than the effects of spin-orbit coupling. It is then sensible to first calculate the atomic states
without spin-orbit coupling and to classify them according to the quantum numbers of the zcomponents of the total orbital angular momentum and the total spin: ΨL,S,ML,MS . The
energy shifts due to the interaction with the magnetic field (3.254) are then – without any
further perturbative assumptions – simply
This is the Paschen-Back-Effekt.
Vector model
L
S
S
J
L
Bz
Bz
Under spin-orbit dominance
Paschen-Beck effect
The spin and orbital angular
momentum are decoupled
Schematic illustration of level splitting in a magnetic field for the example
of a 2P1/2 and a 2P3/2 multiplet, which are separated by a spin-orbit splitting
ΔE0 in the field-free case. If the product of field strength B and magneton is
smaller than ΔE0 we obtain the level splitting of the anomalous Zeeman effect
for μBB >ΔE0 we enter the region of the Paschen-Back effect
Atoms in an Oscillating Electric Field
The theory of the interaction between an atom and the electromagnetic field
describes the resonant absorption and emission of photons between stationary
eigenstates of the field-free atom.
But an atom is also influenced by a (monochromatic) electromagnetic field if its
frequency doesn’t happen to match the energy of an allowed transition. For small
intensities one obtain splitting and frequency-dependent shifts of energy levels
(ac Stark shift, frequency dependant polarizability); for sufficiently high intensities
as are easily realized by modern laser technology,multiphoton processes
(excitation, ionization) play an important role.
Extras
Molecular Alignment and Orientation:
From Laser-Induced Mechanisms to Optimal Control
Laser-induced molecular alignment and orientation are challenging control issues
with a wide range of applications, extending from chemical reactivity to nanoscale
design.
They address molecular manipulation, involving external angular degrees of
freedom, aiming at a parallel positioning of the molecular axis with respect to the
laser polarization vector (alignment ) or, even more demanding, with a given
direction (orientation).
E.g., HCN
H=H0 + Hrad
where
The dynamics of the molecular system are then obtained by solving numerically
the time-dependent Schrödinger equation: