introduction to quantum theory of angular momentum l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Introduction to Quantum Theory of Angular Momentum PowerPoint Presentation
Download Presentation
Introduction to Quantum Theory of Angular Momentum

Loading in 2 Seconds...

play fullscreen
1 / 57

Introduction to Quantum Theory of Angular Momentum - PowerPoint PPT Presentation


  • 324 Views
  • Uploaded on

Introduction to Quantum Theory of Angular Momentum. Angular Momentum. AM begins to permeate QM when you move from 1-d to 3-d This discussion is based on postulating rules for the components of AM Discussion is independent of whether spin, orbital angular momenta, or total momentum.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Introduction to Quantum Theory of Angular Momentum' - Patman


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
angular momentum
Angular Momentum
  • AM begins to permeate QM when you move from 1-d to 3-d
  • This discussion is based on postulating rules for the components of AM
  • Discussion is independent of whether spin, orbital angular momenta, or total momentum.
definition
Definition

An angular momentum, J, is a linear operator with 3 components (Jx, Jy,Jz) whose commutation properties are defined as

convention
Convention

Jz is diagonal

For example:

therefore
Therefore

Where |jm> is an eigenket

h-bar m is an eigenvalue

For a electron with spin up

Or spin down

definition7
Definition

These Simple Definitions have some major consequences!

slide8
THM

Proof:

QED

raising and lowering operators
Raising and Lowering Operators

Lowering Operator

Raising Operator

proof that j is the lowering operator
Proof that Jis the lowering operator

It is a lowering operator since it works on a state with an eigenvalue, m, and produces a new state with eigenvalue of m-1

j 2 j z 0 indicates j 2 and j z are simultaneous observables
[J2,Jz]=0 indicates J2 and Jz are simultaneous observables

Since Jx and Jy are Hermitian, they must have real eigenvalues so l-m2 must be positive!

l is both an upper and LOWER limit to m!

final relation
Final Relation

So the eigenvalue is mlarge*(mlarge +1) for any value of m

conclusions
Conclusions
  • As a result of property 2), m is called the projection of j on the z-axis
  • m is called the magnetic quantum number because of the its importance in the study of atoms in a magnetic field
  • Result 4) applies equally integer or half-integer values of spin, or orbital angular momentum
matrix elements of j
Matrix Elements of J

Indicates a diagonal matrix

theorems
Theorems

And we can make matrices of the eigenvalues, but these matrices are NOT diagonal

a matrix approach to eigenvalues
A matrix approach to Eigenvalues

If j=0, then all elements are zero! B-O-R-I-N-G!

Initial m

j= 1/2

final m

What does

J+ look like?

using our relations
Using our relations,

Answer:

Pauli Spin Matrices

rotation matices
Rotation Matices
  • We want to show how to rotate eigenstates of angular momentum
  • First, let’s look at translation
  • For a plane wave:
a translation by a distance a then looks like
A translation by a distance, A, then looks like

translation operator

Rotations about a given axis commute, so a finite rotation is a sequence of infinitesimal rotations

Now we need to define an operator for rotation that rotates by amount, q, in direction of q

slide27
So

Where n-hat points along the axis of rotation

Suppose we rotated through an angle f about the z-axis

what if f 2 p
What if f = 2p?

The naïve expectation is that thru 2p and no change.

This is true only if j= integer. This is called symmetric

BUT for ½ integer, this is not true and is called anti-symmetric

using the sine and cosine relation
Using the sine and cosine relation

And it should be no surprise, that a rotation of b around the y-axis is

consequences
Consequences
  • If one rotates around y-axis, all real numbers
  • Whenever possible, try to rotate around z-axis since operator is a scalar
  • If not possible, try to arrange all non-diagonal efforts on the y-axis
  • Matrix elements of a rotation about the y-axis are referred to by
slide33
And

Wigner’s Formula (without proof)

slide34
Certain symmetry properties of d functions are useful in reducing labor and calculating rotation matrix
coupling of angular momenta
Coupling of Angular Momenta
  • We wish to couple J1 and J2
  • From Physics 320 and 321, we know
  • But since Jz is diagonal, m3=m1+m2
coupling cont d
Coupling cont’d
  • The resulting eigenstate is called
  • And is assumed to be capable of expansion of series of terms each of with is the product of 2 angular momentum eigenstates conceived of riding in 2 different vector spaces
  • Such products are called “direct products”
coupling cont d37
Coupling cont’d
  • The separateness of spaces is most apparent when 1 term is orbital angular momentum and the other is spin
  • Because of the separateness of spaces, the direct product is commutative
  • The product is sometimes written as
the expansion is written as
The expansion is written as

Is called the Clebsch-Gordan coefficient

Or Wigner coefficient

Or vector coupling coefficient

Some make the C-G coefficient look like an inner product, thus

a simple formula for c g coefficients
A simple formula for C-G coefficients
  • Proceeds over all integer values of k
  • Begin sum with k=0 or (j1-j2-m3) (which ever is larger)
  • Ends with k=(j3-j1-j2) or k=j3+m3 (which ever is smaller)
  • Always use Stirling’s formula log (n!)= n*log(n)

Best approach: use a table!!!

slide41
What if I don’t have a table?

And I’m afraid of the “simple” formula?

Well, there is another path… a 9-step path!

9 steps to success
9 Steps to Success
  • Get your values of j1 and j2
  • Identify possible values of j3
  • Begin with the “stretched cases” where j1+j2=j3 and m1=j1, m2=j2 , and m3=j3, thus |j3 m3>=|j1 m1>|j2 m2>
  • From J3=J1+J2,, it follows that the lowering operator can be written as J3=J1+J2
9 steps to success cont d
9 Steps to Success, cont’d
  • Operate J3|j3 m3>=(J1+J2 )|j1 m1>|j2 m2>
  • Use
  • Continue to lower until j3=|j1-j2|, where m1=-j1 , m2= -j2, and m3= -j3
  • Construct |j3 m3 > = |j1+j2 -1 j1+j2-1> so that it is orthogonal to |j1+j2 j1+j2-1>

Adopt convention of Condon and Shortley,

if j1 > j2 and m1 > m2 then

Cm1 m2j1 j2 j3 > 0

(or if m1 =j1 then coefficient positive!)

9 steps to success cont d44
9 Steps to Success, cont’d
  • Continue lowering and orthogonalizin’ until complete!

Now isn’t that easier?

And much simpler…

You don’t believe me… I’m hurt.

I know! How about an example?

step 7 keep lowering
Step 7—Keep lowering

As low as we go

an aside to simplify notation
An aside to simplify notation

Now we have derived 3 symmetric states

Note these are also symmetric from the standpoint that we can permute space 1 and space 2

Which is 1? Which is 2?

“I am not a number; I am a free man!”

the infamous step 8
The infamous step 8
  • “Construct |j3 m3 > = |j1+j2 -1 j1+j2-1> so that it is orthogonal to |j1+j2 j1+j2-1>”
  • j1+j2=1 and j1+j2-1=0 for this case so we want to construct a vector orthogonal to |1 0>
  • The new vector will be |0 0>
performing step 8
Performing Step 8

An orthogonal vector to this could be

or

Must obey Condon and Shortley: if m1=j1,, then positive value

j1=1/2 and |+> represents m= ½ , so only choice is

step 9 the end
Step 9– The End

This state is anti-symmetric and is called the “singlet” state. If we permute space 1 and space 2, we get a wave function that is the negative of the original state.

These three symmetric states are called the “triplet” states. They are symmetric to any permutation of the spaces

a cg table look up problem
A CG Table look up Problem

Part 1—

Two particles of spin 1 are at rest in a configuration where the total spin is 1 and the m-component is 0. If you measure the z-component of the second particle, what values of might you get and what is the probability of each z-component?

cg helper diagram
CG Helper Diagram

j3

m3

C

m1 m2

It is understood that a “C” means square root of “C” (i.e. all radicals omitted)

solution to part 1
Solution to Part 1
  • Look at 1 x 1 table
  • Find j3 = 1 and m3 = 0
  • There 3 values under these
part 2
Part 2

An electron is spin up in a state, y5 2 1, where 5 is the principle quantum number, 2 is orbital angular momentum, and 1 is the z-component.

If you could measure the angular momentum of the electron alone, what values of j could you get and their probabilities?

solution
Solution
  • Look at the 2 x ½ table since electron is spin ½ and orbital angular momentum is 2
  • Now find the values for m1=1 and m2=1/2
  • There are two values across from these:
  • 4/5 which has j3 = 5/2
  • -1/5 which has j3 = 3/2
  • So j3=5/2 has probability of 4/5
  • So j3 = 3/2 has probability of 1/5