Section 7 Flashcards
Angular momentum L(v) =
r(v) x p(v)
v is vector
classically a conserved quantity, quantised in QM
Stern-Gerlach experiment showed
existence of intrinsic spin S in addition to orbital L
L(hat)i are
Hermitian , since [r(hat)i,p(hat)i] = 0
incompatible eigenstates
can’t have simultaneously well-defined L projections in different directions
[L(hat)i,L(hat)^2] =
0
Compatible L(hat)i and L(hat)^2
can have simultaneous eigenstates of total and projected L
angular momentum in spherical polar coords
φ solution
Φ(φ) = exp[imφ]
angular momentum in spherical polar coords
θ solution
Θ(θ) = AP^m(l) cosθ
P(l) (x) are
polynomials of positive integer degree l in x (=cosθ)
The integer l is
the orbital angular momentum quantum number.
values l = 0,1,2… correspond to the named, s, p, d, f… etc. atomic orbitals
The integer m is
the magnetic quantum number
quantised z-projections of the angular momentum
spherical harmonics
the full angular solution, with normalisation factors
<Y^m(l) | Y^m’(l’) > =
δ(l l’) δ(m m’)
Under parity transform
total parity
P(hat)(Y^m(l)) = (-1)^l Y^m(l)
under z-projection of angular momentum
L(hat)z = -iℏ ∂/∂φ
=> L(hat)z Y(^m(l)) = mℏY^m(l)
under squared angular momentum
L(hat)^2 = -ℏ^2 [ 1/sinθ ∂/∂θ (sinθ ∂/∂θ) + 1/sin^2θ ∂^2/∂φ^2]
=> [L(hat)^2,H(hat)] = 0
Discretised angular momenta:
l and m are the quantum numbers. L^max(z) = lh is less than L^max = √(l(l+1)ℏ: uncertainty principle
L(hat) (±) =
L(hat) (x) ± iL(hat) (y)
expectation values are the
mean eigenvalues, averaged over wave-function collapses
< Ψ|O(hat)|Ψ> =
∫dΩ Ψ*O(hat)Ψ
= (0 ∫2π) dφ (0 ∫π) dθ sinθ Ψ(θ,φ)* O(hat)Ψ(θ,φ)
atomic electrons have magnetic moment
µ(hat)(v) ~ g(l) qL(hat)(v) / 2m
= - g(l) e/2m(e) L(hat)(v)
orbital g factor g(l) =
(1-m(e)/m(nucl)) ~ 1
interaction with a magnetic field B along e.g. z shifts the energy by
ΔH(hat) = -µ(hat)(v) . B(hat)(v) = eB/2m(e) L(hat)(z)
Bohr magneton:
µ(B) = eℏ/2m(e)
splits the energy levels: Zeeman effect
