Quantum Mechanics II

Mind Map by , created over 5 years ago

Mind Map on Quantum Mechanics II, created by franz.sciortino on 04/02/2014.

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Created by franz.sciortino over 5 years ago
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Quantum Mechanics II
1 Ladder operators (not Hermitian) are "empirically" found to raise/lower energy states
1.1 Very useful: [H, a]= - h_bar *w*a and [H, a']=h_bar*w*a'
1.2 Use condition a*u_0=0 to find momentum eigenstates and multiply by a' to find energy eigenstates
1.3 Some results are representation-independent
2 Time-independent Perturbation Theory
2.1 find variation in eigenvalues by setting (u_n)'= u_n
2.2 Find eigenstates by letting (E_n)'=E_n and ignoring 2nd order terms
3 Degeneracy
3.1 Generally produced by symmetries
3.1.1 Individual states might not exhibit symmetry, but sums of prob. densities must always do
3.2 Superpositions of eigenstates are still eigenstates
3.3 Schmidt orthogonalization: procedure to make degenerate states orthogonal (always possible)
3.4 With degeneracy, if two operators commute, then there always exists a combination of them which is compatible
4 Orbital angular momentum
4.1 L_i components are given by (r x p)
4.2 In cyclic order, [Lx, Ly]= i h_bar *Lz
4.3 L^2 commutes with L_i components, but these do not commute between themselves
4.3.1 We can write eigenvalue equations: L^2 Y = alpha Y and L_z Y = beta Y
4.4 Define ladder operators L+ and L- to show many ang.momentum rotations for each length
4.4.1 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
4.5 Central potentials give [H, L^2]=0 (conservation of ang.mom.)
4.6 Obtain radial equation from TISE with central potential barrier and separation of variables
4.7 Measuring ang.mom. experiments: Zeeman, Stern-Gerlach, Uhlenbeck-Goudsmit spin proposition
5 Spin angular momentum
5.1 Analogies with orbital ang.mom. postulated, but only 2 states allowed
5.1.1 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

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