Quantum Mechanics II

Quantum Mechanics II

1. 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

2. 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

3. Degeneracy

   • Generally produced by symmetries
     • 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

4. 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
   • 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
   • 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

5. 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