Atomic physics Part PHYS261

LECTURE NOTE 2013.09.05 to index 2013.09.03 - previous lecture note 2013.09.10 - next lecture note

There are 4 student tasks at the end of this page

Helium atom - Spin - Magnetic Moment - Angular Momentum

We start by reminder of "well known" apparatus of the angular momentum algebra.
That starts with the actual def. of angular momentum (classical) - where is the origin?
And what happens when we change the origin of coordinates?
Then quantum angular momentum, then generalization to "any type" of quantum angular momentum
Spherical harmonics - functions of theta, phi, covering the unit sphere

(From "smartboard" review of Angular momentum)
01_Angular_momentum_QM_basics.png

01_Angular_momentum_QM_basics.png

From generalized Quantum Angular Momentum Operators - to SPIN
Pauli Matrices   (Wolfgang Pauli: http://en.wikipedia.org/wiki/Wolfgang_Pauli )

Next step: addition of angular momenta
see the picture 3_angular_momentum_VILVITE_tech.jpg bellow
      02_mathematics_of_Pauli_spin.png

02_mathematics_of_Pauli_spin.png

Addition of angular momenta - and conservation of the TOTAL ANGULAR MOMENTUM are favourite demonstrations in classical mechanics
courses. Here is one local in the VilVite center (2009 - it is still there?) and two pictures from the web (image search).
Especially the drawing is illustrative.

In quantum mechanics courses this simple physics is usually not remembered. It should be, it is the same, but so different!

3_angular_momentum_VILVITE_tech.jpg

3_angular_momentum_VILVITE_tech.jpg

Rotating-platform-Bicycle-Wheel.png

Rotating-Stool-Bicycle-Wheel.jpg

ALLOWED VALUES - historically called SPACE quantization
Spin only has "two projections", general L has 2L + 1 ( S=1/2)
02_spin_and_angular_momentum.png

02_spin_and_angular_momentum.png

(Comment on symmetry - i.e. exchange - symmetry has become "more fashionable"
Based on very nice mathematics
Groups of permutations - also called Symmetrical group
Group theory and quantum mechanics (but all can be formulated with simple math and physics
intuition - as the Pauli, Einstein, Bohr did....
03_Addition_of_angular_momentum_spin_algebra.png

03_Addition_of_angular_momentum_spin_algebra.png
LITTLE REMINDER of How to multiply matrices (in colors above).
(take a column of the second member and make "scalar product"
with the row of the first member - that gives the (row, column) element of the resulting product

Magnetic Moment - Spin

Angular momentum (rotational motion -> "current loop" -> magnetic moment - like a tiny electromagnet ) -> orbital Magnetic moment
Spin Angular momentum    --> Spin Magnetic moment ??

No, it looks rather like this:
   Spin Magnetic moment      --> something like spin (angular) momentum DIRAC EQUATION
   Spin gyromagnetic ratio (electron) is TWICE the normal (orbital) gyromagnetic ratio
      04_magnetic_moment_spin.png

04_magnetic_moment_spin.png

Here we worked out the combinations of the states of two spins
There are 4 possible combinations of "up" and "down"
these 4 - two are symmetric; the remining two are combined into one more symmetric
and one asymmetric
05_addition_of_two_spins.png

05_addition_of_two_spins.png

There are thus two sets - a triplet of symmetric ones - and a single antisymmetric one

It turns out that the symmetrization also accomplishes the Addition of the two spins

06_Symmetrization_of_spins_and_addition.png

06_Symmetrization_of_spins_and_addition.png

It turns out that the symmetrization also accomplishes the Addition of the two spins

Tasks
1. Set up the steps to demostrate explicitely that the symmetric function gives S=1, using the sigma-matrices.
    An easier start might be to show that the antisymmetric product gives S=0

2. Antisymmetric function for 2 particles was relatively easy. How would you construct antisymmetric product states
    from three orbitals (a) (b) (c) for 3 particles ( Hint: how many permutations; and did we mention determinant?
    ... It is not necessary to do all the details - but it is a nice game)

3. Is it possible to make an antisymmetric function for three spins?

4. It would seem that the generalization of our combining 2 spins could be generalized to 3 spins. But there is
    task 3. Can you give arguments for which total spin values there will be possible for three spins? There
   are 2 . 2 . 2 = 8 totally of three spin functions. How does this fits the (2 S + 1) counts