710078
Description
Mind Map by franz.sciortino, updated more than 1 year ago
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Created by franz.sciortino
over 10 years ago
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Quantum Mechanics II
- Ladder operators (not Hermitian) are
"empirically" found to raise/lower energy states
- Very useful: [H, a]= - h_bar *w*a
and [H, a']=h_bar*w*a'
- Use condition a*u_0=0 to find momentum eigenstates
and multiply by a' to find energy eigenstates
- Some results are
representation-independent
- Very useful: [H, a]= - h_bar *w*a
and [H, a']=h_bar*w*a'
- Time-independent Perturbation Theory
- find variation in eigenvalues by setting (u_n)'= u_n
- Find eigenstates by letting (E_n)'=E_n
and ignoring 2nd order terms
- find variation in eigenvalues by setting (u_n)'= u_n
- Degeneracy
- Generally produced by symmetries
- Individual states might not exhibit symmetry,
but sums of prob. densities must always do
- Individual states might not exhibit symmetry,
but sums of prob. densities must always do
- Superpositions of eigenstates are still eigenstates
- Schmidt orthogonalization: procedure to make
degenerate states orthogonal (always possible)
- With degeneracy, if two operators commute, then there
always exists a combination of them which is compatible
- Generally produced by symmetries
- Orbital angular momentum
- L_i components are given by (r x p)
- In cyclic order, [Lx, Ly]= i h_bar *Lz
- L^2 commutes with L_i components, but these do
not commute between themselves
- We can write eigenvalue equations:
L^2 Y = alpha Y and L_z Y = beta Y
- We can write eigenvalue equations:
L^2 Y = alpha Y and L_z Y = beta Y
- Define ladder operators L+ and L- to show many
ang.momentum rotations for each length
- Use conditions of ladder operators to find eigenvalues:
alpha=l(l+1) h_bar and beta= m_l h_bar
- Find ang. momentum eigenstates using L_z and L^2
spherical components --> Legendre equations --> spherical harmonics
- Find ang. momentum eigenstates using L_z and L^2
spherical components --> Legendre equations --> spherical harmonics
- Use conditions of ladder operators to find eigenvalues:
alpha=l(l+1) h_bar and beta= m_l h_bar
- Central potentials give [H, L^2]=0
(conservation of ang.mom.)
- Obtain radial equation from TISE with central
potential barrier and separation of variables
- Measuring ang.mom. experiments: Zeeman,
Stern-Gerlach, Uhlenbeck-Goudsmit spin proposition
- L_i components are given by (r x p)
- Spin angular momentum
- Analogies with orbital ang.mom.
postulated, but only 2 states allowed
- Knowing needed eigenvalues, deduce eigenstates (matrices)
- Pauli matrices, up/down spin states
- Find shifts in energies in uniform magnetic fields by mu_B *B
- Larmor precession of S_x and S_y, with constant S_z over time
- Larmor precession of S_x and S_y, with constant S_z over time
- Find shifts in energies in uniform magnetic fields by mu_B *B
- Pauli matrices, up/down spin states
- Knowing needed eigenvalues, deduce eigenstates (matrices)
- Analogies with orbital ang.mom.
postulated, but only 2 states allowed
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