The harmonic oscillator
En = (n + ½)ħω = (n + ½)hf, n = 0. 1, 2, ...
Uncertainty principle: ΔxΔp ~ ħ
Δp ~ ħc/(c*10-15 m) ~ (1240 eV nm)/(c*10-6 nm *2π) ~ 200 MeV/c
The 3D infinite square well
E = (nx2 + ny2+ nz2)2ħ2/(2mL2),
nx, ny, nz = 1, 2, 3, ...
E1 = 3π2ħ2/(2mL2), degeneracy, = 1 (nx = ny = nz = 1)
E2 = 6π2ħ2/(2mL2), degeneracy = 3 (ni = 2, nj = nk = 2, i = 1, 2, 3)
E3 = 9π2ħ2/(2mL2), degeneracy = 3 (ni = 1, nj = nk = 2, i = 1, 2, 3)
Problem 6: (A)
The energy operator:
Problem 7: (E)
If an operator commutes with the Hamiltonian, it is called a "constant of motion". For each eigenvalue, the probability that a measurement will yield this eigenvalue does not change with time.
Hydrogenic atoms, scaling rule
a = a0/Z. (Exact: a = ħ2/(Zμe2))
Probability density: P(x)dx = |ψ(x)|2dx
P (x2 > x > x1) = (1/d)∫-x1x2
exp(-2x/d) dx = ½∫-2x1/d2x2/d exp(-y) dy
= ½(exp(-2x1/d) -exp(-2x2/d))
Addition of angular momenta
The possible values of l range from |l1 - l1| to l1 + l2 in integer steps. For each l, m can take on values between -l and +l in integer step. When adding angular momenta, the m values add algebraically, m = m1 + m2. Here m = m1 + m2 = 5, l must be greater or equal to 5.