What is quantization of AM (L^2)
Only one component of angular momentum can be known precisely along with L^2
The quantization means angular momentum takes discrete magnitudes and discrete orientations in space.
2x2 matrices for spin operators Sx, Sy and Sz
all x hbar/2
(0 1
1 0)
(0 -i
i 0)
(1 0
0 -1)
How do you find the eignvalue of a matrix?!
Diagonalise with -lambda, make it =0
How does the Stern-Gerlach experiment demonstrate quantisation of spin AM
A beam of silver atoms (with a single unpaired electron, so net spin-½) is passed through a non-uniform magnetic field.
Classically, the magnetic moments could point in any direction, so the beam should spread continuously along the magnetic-field gradient.
Instead, the beam splits into two discrete spots on the detector, indicating that the spin angular momentum along the field direction (S_z) can only take discrete values
Steps to separate schrodinger into radial and angular parts
- Express in spherical coords
- Express laplacian in spherical
- Assume it can be split
- divide by psi and multiply by 2mr^2/hbar^2
- Set each side equal to l(l+1)
Termination condition
v(w) terminates then k_max + l +1 =n
helps find energy eigenvalues
What is perturbation theory
Procedure for obtaining approximate solutions to the perturbed problem by building on the known exact solutions to the unperturbed case
First order energy correction
H0 psi1 + H’ psi0 = E0 psi1 + E’ psi0
Take inner product with psi0
…. = expectation value of UNperturbed state
what is degenerate perturbation theory and why does it break down non-degenerate PT
When two states send the same energy (are degenerate)
i.e. E0(n)=E0(m)
to solve, must diagonalize the perturbation Hamiltonian within the subspace of degenerate states
gives first-order energy corrections as the eigenvalues of the matrix
Why is the concept of J=L+S crucial for describing atoms
Molecular physics: how angular momenta combine helps predict energy levels.
Quantum entanglement: For multiple particles, AM explains entangled spin states (e.g., singlet and triplet states).
Possible total spin quantum numbers (s) for two spin 1/2 particles
s=0 (singlet) or s=1 (triplet)
