Atomic physics Part PHYS261

LECTURE NOTE 2013.10.15 to index 2013.10.10 - previous lecture note 2013.10.17 - next lecture note

Interaction of Atoms with Light - Part 1

The PDF-file of the presentation : Light_Atom-2013.10.16.pdf WORK WITH SCF-exercise at the end

Time developement of probability - Decaying States

This is the entry page of the presentation linking to the various parts of the topic. Below it is commented.
The part addressed today Time developement of probability - Decaying States is the first block to the left (up)
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This is the entry page of the presentation commented.
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Time developement of the probability density in Quantum Mechanics
Example: two adjecent potential wells (just like stability of molecules - electrons close to 2 nuclei etc)

The probability oscillates. This has been demonstrated e.g. in "quantum dots" in more recent experiments

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The anatomy of this animation
is displayed below.

See also the theory of the two state system below
in the continuation of the discussion

Theory is given below

This slide is more relevant below - time-dependent Schrödinger Eq. - we place it here to remind us
of EXPANSION
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A031-expansion_of_wavefunction.png

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the hamiltonian is not time-dependent. Thus each of the EIGENstates follows its time developement
The EIGENSTATES are not the states in one or the other well

but the linear combinations with plus and minus signs
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States in two identical wells
and
States in two very different wells - where one has "nearly continuous" levels.

The eigenstate picture used in the oscillating case will not realize.
It is not easy to see it without "trying" - but what will happen is that
"the probability" can not find its way to the original situation - so it will slowly
flow away - the original state will decay
(we have the solution of the TDSE below in the simulator ....)
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This picture is important. It shows the following:
The eigenstates of the problem are not "localized" in the narrow well or the broad (quasicontinuum energy levels) well.
When the system is placed in the narrow well (isolated) eigenstate, it will develop in time just like in the two state case.

Except that this will not lead to an oscillation, but to a flow of probability to the broad well (to the quasicontinuum).

This allows the understanding of the "energy conservation" problem.
The original state is not an eigenstate.
It thus have the probability amplitudes of being in any of the eigenstates with a certain probability
given by the expansion of the eigenstates in terms of the states of the isolated wells.

This will give us the understanding of the LINE WIDTH discussed in the next lecture.
The DELTA FUNCTION construction is thus only a mathematical trick.

Naturally, total energy is conserved, but that has nothing to do with the isolated well energy eigenstate.

The animated gif of two-well developement (to the right) - displayed in Mac program "Preview" (left panel)

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(See also the expansion slide above )
Here we have gone through the perturbation theory derivation of the Golden rule
following the presentation. This derivation can be found in many textbooks.
An essential part is the arrival at the DELTA FUNCTION in energy - and the
CONSTANT RATE of probability change
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( see also the expansion slide above)
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A104-coupling_matrix.png

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Here we shall arrive at the DELTA FUNCTION mentioned in the start of the discussion
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Golden Rule Simulators
A 1990s "picture book" about the simulators    ( http://web.ift.uib.no/AMOS/Pictures/Golden.html )
These simulators were written for different systems, including MATLAB (see below)
For about 4 years we have now BROWSER-based simulator    ( http://web.ift.uib.no/AMOS/golden/ )
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http://web.ift.uib.no/AMOS/golden/
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http://web.ift.uib.no/AMOS/golden/ for the two state problem

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http://web.ift.uib.no/AMOS/golden/ for the four state problem
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Golden Rule simulator in MATLAB. The circles show the energy positions and the amplitudes (square of expansion coefficients
for the original isolated state ) obtained by DIAGONALIZATION (eigenvalues and eigenstates) of the model problem.
See also the two asymmetric wells picture below this slide) (see also "picture book" about the simulators http://web.ift.uib.no/AMOS/Pictures/Golden.html
A204Golden_Rule_MATLAB.png

A204Golden_Rule_MATLAB.png

The lower part of this cut illustrates the eigenvalues. The eigenvector component corresponding to the
isolated left well energy (eigenstate) is the y-coordinate of the circles in the MATLAB snapshot above.
A204Golden_Rule_MATLAB-explain.png

A204Golden_Rule_MATLAB-explain.png

Work with the SCF numerical exercise

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Plotting potentials using matlab
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carbon_potentials_MATLAB.png

Plotting potentials using Office spreadsheet ( OpenOffice here )

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carbon_potentials_ODS.png