Section 7

Question 1

Angular momentum L(v) =

Answer 1

r(v) x p(v)

v is vector

classically a conserved quantity, quantised in QM

Question 2

Stern-Gerlach experiment showed

Answer 2

existence of intrinsic spin S in addition to orbital L

Question 3

L(hat)i are

Answer 3

Hermitian , since [r(hat)i,p(hat)i] = 0

Question 4

incompatible eigenstates

Answer 4

can’t have simultaneously well-defined L projections in different directions

Question 5

[L(hat)i,L(hat)^2] =

Answer 5

0

Question 6

Compatible L(hat)i and L(hat)^2

Answer 6

can have simultaneous eigenstates of total and projected L

Question 7

angular momentum in spherical polar coords

φ solution

Answer 7

Φ(φ) = exp[imφ]

Question 8

angular momentum in spherical polar coords

θ solution

Answer 8

Θ(θ) = AP^m(l) cosθ

Question 9

P(l) (x) are

Answer 9

polynomials of positive integer degree l in x (=cosθ)

Question 10

The integer l is

Answer 10

the orbital angular momentum quantum number.

values l = 0,1,2… correspond to the named, s, p, d, f… etc. atomic orbitals

Question 11

The integer m is

Answer 11

the magnetic quantum number

quantised z-projections of the angular momentum

Question 12

spherical harmonics

Answer 12

the full angular solution, with normalisation factors

Question 13

<Y^m(l) | Y^m’(l’) > =

Answer 13

δ(l l’) δ(m m’)

Question 14

Under parity transform

total parity

Answer 14

P(hat)(Y^m(l)) = (-1)^l Y^m(l)

Question 15

under z-projection of angular momentum

Answer 15

L(hat)z = -iℏ ∂/∂φ

=> L(hat)z Y(^m(l)) = mℏY^m(l)

Question 16

under squared angular momentum

Answer 16

L(hat)^2 = -ℏ^2 [ 1/sinθ ∂/∂θ (sinθ ∂/∂θ) + 1/sin^2θ ∂^2/∂φ^2]

=> [L(hat)^2,H(hat)] = 0

Question 17

Discretised angular momenta:

Answer 17

l and m are the quantum numbers. L^max(z) = lh is less than L^max = √(l(l+1)ℏ: uncertainty principle

Question 18

L(hat) (±) =

Answer 18

L(hat) (x) ± iL(hat) (y)

Question 19

expectation values are the

Answer 19

mean eigenvalues, averaged over wave-function collapses

Question 20

< Ψ|O(hat)|Ψ> =

Answer 20

∫dΩ Ψ*O(hat)Ψ

= (0 ∫2π) dφ (0 ∫π) dθ sinθ Ψ(θ,φ)* O(hat)Ψ(θ,φ)

Question 21

atomic electrons have magnetic moment

Answer 21

µ(hat)(v) ~ g(l) qL(hat)(v) / 2m

= - g(l) e/2m(e) L(hat)(v)

Question 22

orbital g factor g(l) =

Answer 22

(1-m(e)/m(nucl)) ~ 1

Question 23

interaction with a magnetic field B along e.g. z shifts the energy by

Answer 23

ΔH(hat) = -µ(hat)(v) . B(hat)(v) = eB/2m(e) L(hat)(z)

Question 24

Bohr magneton:

Answer 24

µ(B) = eℏ/2m(e)

splits the energy levels: Zeeman effect

Question 25

Stern-Gerlach experiment & intrinsic spin

Answer 25

neutral atoms with magnetic moment: moment precesses in uniform B field, atom displaces in inhomogeneous B field

if l = 1/2 : identified spin angular momentum

Question 26

Total angular momentum :

Answer 26

J(hat)(v) = L(hat)(v) + S(hat)(v)

Question 27

[S(hat)(x),S(hat)(y)] =

Answer 27

iℏS(hat)(z)

Question 28

[J(hat)(x),J(hat)(y)] =

Answer 28

iℏJ(hat)(z)

Question 29

Independent orbital and spin physics :

Answer 29

[L(hat)(v),S(hat)(v)] = 0

Question 30

J(hat)^2 = (L(hat)(v)+S(hat)(v))^2

Answer 30

= 0

Question 31

Spin further splits states in the Zeeman effect,

Answer 31

µ = µ(B)g/ℏ J(v)

Question 32

Combine L and S into total momentum J :

Answer 32

further splitting of energy levels, the Zeeman effect