Atomic physics Part PHYS261

LECTURE NOTE 2013.08.22 to index 2013.08.20 - previous lecture note 2013.09.03 - next lecture note

More on Hydrogen-like atoms - Stationary States.

The first step is to look at problems which have total energy (Hamiltonian) TIME-INDEPENDENT.
Thenm the time dependence can be separated - assuming solutions as a product
( separation of variables; allways the same trick - leading to a "constant" )
01_Time_separation_-classical_turning_points.png

01_Time_separation_-classical_turning_points.png
CLASSICAL TURNING POINTS - and their meaning in Quantum Systems

When the classical particle moves towards "the turning point" - i.e. the point when tht total energy "cuts" the
potential energy curve - the kinetic energy at this point is zero, i.e. the velocity goes to zero - and must chnge sign,
i.e. the motion turns the other way - thus "turning point".
Classically, the particle cannot pass the turning point, it can not be beyond it.
In quantum theory, the wave function does not turn. But the secons derivative, proportional to the
kinetic energy operator - becomes zero.
This results in the inflection point for one-dimensional motions (including the radial motion, see later)

In quantum mech
02_turning_points_QM_discrete_energy_values.png

02_turning_points_QM_discrete_energy_values.png

Probability density plots for harmonic oscillator states
(Coherent Glabuer States, Tide-dependent quantum motion; made using http://web.ift.uib.no/AMOS/glauber/ )
This shows that the behaviour at classical turning point is allways of the same type - inflection point,
second derivative is equal to zero)

02_turning_pointz_harmonic_oscillator.png

02_turning_pointz_harmonic_oscillator.png

02_wiki_INFLECTION_POINT.png

Inflection point at Wikipedia

SEPARATION OF VARIABLES: The potential energy is only function of the radial distance.
Does not depend on angles.
Again assuming solution in the form of a product of functions of the variables to be "separated"
03_Separation_spherical_coordinates_spherical_harmonics.png

03_Separation_spherical_coordinates_spherical_harmonics.png

Spherical Harmonics (in many textbooks; for orientation web materials are sufficient)
http://en.wikipedia.org/wiki/Spherical_harmonics
see also tables at
http://en.wikipedia.org/wiki/Table_of_spherical_harmonics
04_spherical_harmonics_polynomials_x_y_z.png

04_spherical_harmonics_polynomials_x_y_z.png

Radial wavefunctions ( Note also a short comment on how to make polar plots)
MATLAB demonstration of polar plots of "density" for spherical harmonics
and the radial functions: hydro_YLM

05_how_to_plot_radial_Angular_POLAR_PLOTS.png

05_typical_plot_hydrogen.png

Most of the information we need is contained in vast ammounts of text on the web
Again for quick overview and orientation:
http://en.wikipedia.org/wiki/Hydrogen-like_atom   ( basic facts )
http://en.wikipedia.org/wiki/Hydrogen_atom   (wavefunctions; but not the separation in variables; wikipedians are loosing focus here)
http://en.wikipedia.org/wiki/Atomic_orbital    - visualizations, some details
At citizendum, we get presentation more to our needs:
http://en.citizendium.org/wiki/Hydrogen-like_atom

2 Schrödinger equation

(This seems stable)

The spectrum of hydrogen atom (not hydrogen molecule! )
Selection Rules (tasks in previous years of this course)

06__l_quantum_numbers_spectra_selection_rules_parity.png

06__l_quantum_numbers_spectra_selection_rules_parity.png

Atomic Units - the "everyday", i.e. every atomic unit of time "life in the atomic world"

We need measurement units to describe the happenings on 0.05 nano-meter scale with times
of the arder of 0.1 femto-seconds - what determines these units?
... And the big problem of the SI-electromagnetic philosophy confusing the physically clear
"Gaussian approach" (This is in a confusing way referred to as SI units vs Gaussian units; it
is not a problem of units but of confusion of certain engineers)
07_atomic_units.png

07_atomic_units.png

08_fine_structure_constant_velocity_unit.png

08_fine_structure_constant_velocity_unit.png

( This board is from 2011)
09_Bohr_radius_atomic_units_1.png

09_Bohr_radius_atomic_units_1.png

( This board is from 2011)
10_Bohr_radius_atomic_units_2.png

10_Bohr_radius_atomic_units_2.png