1. Spin and Magnetic Moments (skip sect. 10-3)
Orbital and intrinsic (spin) angular momentum produce magnetic moments
coupling between moments shift atomic energies
Look first at orbital (think of current in a loop)
the “g-factor” is 1 for orbital moments. The Bohr magneton is introduced as the natural unit and the “-” sign is due to the electron’s charge
P460 - Spin

2. Spin Spin particle postulated particle
Particles have an intrinsic angular momentum - called spin though nothing is “spinning”
probably a more fundamental quantity than mass integer spin  Bosons half-integer Fermions
Spin particle postulated particle
pion Higgs, selectron
1/ electron photino (neutralino)
photon
3/ D
graviton
relativistic QM Klein-Gordon and Dirac equations for spin 0 and 1/2.
Solve by substituting operators for E,p. The Dirac equation ends up with magnetic moment terms and an extra degree of freedom (the spin)
P460 - Spin

3. Spin 1/2 expectation values
similar eigenvalues as orbital angular momentum (but SU(2)). No 3D “function”
Dirac equation gives g-factor of 2
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4. Spin 1/2 expectation values
non-diagonal components (x,y) aren’t zero. Just indeterminate. Can sometimes use Pauli spin matrices to make calculations easier
with two eigenstates (eigenspinors)
P460 - Spin

5. Spin 1/2 expectation values
“total” spin direction not aligned with any component.
can get angle of spin with a component
P460 - Spin

6. Spin 1/2 expectation values
Let’s assume state in an arbitrary combination of spin-up and spin-down states.
expectation values. z-component
x-component
y-component
P460 - Spin

7. Spin 1/2 expectation values example
assume wavefunction is
expectation values. z-component
x-component
Can also ask what is the probability to have different components. As normalized, by inspection
or could rotate wavefunction to basis where x is diagonal
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8. Can also determine
and widths
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9. Widths- example
Can look at the widths of spin terms if in a given eigenstate
z picked as diagonal and so
for off-diagonal
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10. Components, directions, precession
Assume in a given eigenstate
the direction of the total spin can’t be in the same direction as the z-component (also true for l>0)
Example: external magnetic field. Added energy
puts electron in the +state. There is now a torque
which causes a precession about the “z-axis” (defined by the magnetic field) with Larmor frequency of S B z
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11. Precession - details Hamiltonian for an electron in a magnetic field
assume solution of form
If B direction defines z-axis have Scr.eq.
And can get eigenvalues and eigenfunctions
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12. Precession - details Assume at t=0 in the + eigenstate of Sx
Solve for the x and y expectation values. See how they precess around the z-axis
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13. Arbitrary Angles
can look at any direction (p 160 and problem 10-2 or see Griffiths problem 4.30)
Construct the matrix representing the component of spin angular momentum along an arbitrary radial direction r. Find the eigenvalues and eigenspinors.
Put components into Pauli spin matrices
and solve for its eigenvalues
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14. Go ahead and solve for eigenspinors.
Gives (phi phase is arbitrary)
if r in z,x,y directions
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15. Combining Angular Momentum
If have two or more angular momentum, the combination is also an eigenstate(s) of angular momentum. Group theory gives the rules:
representations of angular momentum have 2 quantum numbers:
combining angular momentum A+B+C…gives new states G+H+I….each of which satisfies “2 quantum number and number of states” rules
trivial example. Let J= total angular momentum
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16. Combining Angular Momentum
Non-trivial examples. add 2 spins. The z-components add “linearly” and the total adds “vectorally”. Really means add up z-component and then divide up states into SU(2) groups
4 terms. need to split up. The two 0 mix
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17. Combining Angular Momentum
add spin and orbital angular momentum
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18. Combining Angular Momentum
Get maximum J by maximum of L+S. Then all possible combinations of J (going down by 1) to get to minimum value |L-S|
number of states when combined equals number in each state “times” each other
the final states will be combinations of initial states. The “coefficients” (how they are made from the initial states) can be fairly easily determined using group theory (step-down operaters). Called Clebsch-Gordon coefficients
these give the “dot product” or rotation between the total and the individual terms.
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19. Combining Angular Momentum
Clebsch-Gordon coefficients
these give the “dot product” or rotation between the total and the individual terms. “easy” but need to remember what different quantum number labels refer to
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20. Combining Angular Momentum
example 2 spin 1/2
have 4 states with eigenvalues 1,0,0,-1. Two 0 states mix to form eigenstates of S2
step down from ++ state
Clebsch-Gordon coefficients
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21. Combining Ang. Momentum
check that eigenstates have right eigenvalue for S2
first write down
and then look at terms
putting it all together see eigenstates
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22. 2 terms L=1 + S=1/2 Example of how states “add”:
Note Clebsch-Gordon coefficients (used in PHYS 374 class for Mossbauer spectroscopy).
2 terms
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23. Clebsch-Gordon coefficients for different J,L,S
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