Zeeman Effect and Approximation Methods

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Zeeman Effect and Approximation Methods

• Kvantfysik
• Eleanor Campbell
• prof. i atom och molekylfysik
• Eleanor.Campbell_at_fy.chalmers.se
• 772 3272, O5155 (Origo)

http//fy.chalmers.se/f3a/
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Zeeman Effect
The explanation of the spectrum of the hydrogen
atom was a leap forward, made possible by quantum
mechanics. However, the spectra of other elements
were not yet explained. Moreover some subtle
effects were observed already at the beginning of
the 20th century, for which no explanation
existed. Zeeman investigated spectral lines in a
magnetic field and observed that a single
emission line split into three or more lines.
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In recognition of the extraordinary service they
rendered by their researches into the influence
of magnetism upon radiation phenomena
Nobel Prize, 1902
Zeeman
Lorentz
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Consider a particle with electric charge q in a
circular orbit with radius r and speed v. This
gives a current
q
v
r
A current loop has an associated magnetic moment
If an external magnetic field is applied, the
current loop experiences a force with the
corresponding potential energy
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Now change from classical to quantum mechanical
treatment - Replace the classical values of
length, angular momentum etc. with their
corresponding quantum mechanical operators
You have already solved the problem
for a central field potential and
obtained the eigenfunctions with
eigenvalues
We now choose the coordinate system with the
z-axis (the quantisation axis) parallel to the
magnetic field
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We therefore obtain
For an electron q - e
There is a splitting of the energy levels in an
external magnetic field
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Normal Zeeman Effect
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Zeemans observation of the splitting of the
sodium D-lines in the presence of an external
magnetic field More than 3 lines!! Anomalous
Zeeman Effect
P. Zeeman, Nature 55 (1897) p. 347
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We need to find ways to solve the Schrödinger
equation for more complicated situations
external influences, multiple electrons etc.
etc. We must use approximations.
2 very important techniques Perturbation
Theory Variational Method
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Variational Method
This is a method that is very suitable for
determining the bound state energies and wave
functions of a time-independent Hamiltonian. The
eigenvalues of the Hamiltonian are denoted by En,
the corresponding eigenfunctions by yn and we
assume that the Hamiltonian has at least one
discrete eigenvalue. Let f be an arbitrary
normalisable function and let Ef be the
functional
If f is equal to an exact eigenfunction, yn, then
Ef will be identical to the corresponding exact
eigenvalue En.
Ef gives an UPPER BOUND to the exact ground
state energy E0
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To prove this, expand the function f in the
complete set of orthonormal eigenfunctions yn of H
then
now subtract E0
Since En? E0 , the right hand side is positive so
that E0 ? Ef The functional gives an upper
bound for the ground state energy.
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Evaluate Ef by using trial functions f which
depend on a number of variational parameters and
then minimise Ef with respect to these
parameters in order to obtain the best
approximation of E0 allowed by the form chosen
for f.
Example find the optimum form of a trial
function e-kr and the upper bound to the
ground-state energy of the H atom.
Need to evaluate following integrals
k variational parameter
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therefore
and
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To minimise Ef we differentiate with respect to
k
The best value of E is therefore
This is actually the exact ground state energy
for the H atom and the corresponding optimised
trial wavefunction is the true wavefunction for
the atom. This is because the trial wavefunction
included the exact wavefunction as a special
case. If you repeat the calculation with
exp(-kr2) instead you will find that the
calculated best energy lies above the true ground
state.
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Rayleigh-Ritz variational method
This replaces the trial wavefunction by a linear
combination of fixed basis functions with
variable coefficients. These coefficients are
then the variational parameters.
assuming all coefficients and basis functions are
real
Minimise by differentiating with respect to each
coefficient
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This expression is satisfied if the numerators
vanish. This means that we need to solve the
secular equations
This is a set of simultaneous equations for the
coefficients ci. The condition for the existence
of solutions is that the secular determinant
should be zero
This leads to a set of values of E as the roots
of the corresponding polynomial and the lowest
value is the best value of the ground state of
the system with a basis set of the selected form.
The coefficients in the linear combination are
then found by solving the set of secular
equations with this value of E.
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Perturbation Theory
This is very useful if the Hamiltonian of the
true system differs by only a small amount (a
perturbation) from a system that we know the
solution of. It is particularly useful when we
consider the interaction of atoms with external
electric and magentic fields.
Time-Independent Perturbation Theory
We know the eigenfunctions and eigenvalues of a
model system
The Hamiltonian for the perturbed, true system
can be written as
and the wavefunctions of the perturbed system
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The energy of the perturbed state also has
correction terms of various orders
We now need to solve
Normally the first order correction is already
good enough. We write the first order correction
to the wavefunction as a linear combination of
the unperturbed wavefunctions of the system
Can then show that (see compendium or text book)
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This is just what we did earlier (almost) for the
normal Zeeman effect!
with the H wavefunctions
We get
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Further Reading
http//www.nobel.se/physics/laureates/1902/ transl
ation of Zeemans original paper in Nature 55
(1897) p. 347http//dbhs.wvusd.k12.ca.us/Chem-His
tory/Zeeman-effect.html Molecular Quantum
Mechanics, Atkins and Friedman, 3rd Editions,
Chap. 6.