Section 6: Angular Momentum Theory

Question 1

Electron orbitals introduced by Schrödinger are characterised by

Answer 1

quantum numbers

Question 2

principle q no, n

Answer 2

Indicates the energy level and relative size of the orbital. It can take positive integer values

Question 3

angular momentum qn, l

Answer 3

Defines the shape of the orbital and can take values from 0 to n-1 for each value of n

Each value of l corresponds to a specific type of orbital (s, p, d, f…).

Question 4

magnetic qn (ml)

Answer 4

Describes the orientation of the orbital in space and can take integer values from -l to +l, including 0

Question 5

spin qn (ms)

Answer 5

Specifies the electron’s spin direction, which can be either +1/2 or -1/2.

Question 6

L=

Answer 6

rxp

|i j k|
|x y z|
|px py pz|

Question 7

εijk

Answer 7

levi-civita tensor

Question 8

levi-civita tensor properties

Answer 8

has cyclic symmetry
is 0 if any ijk is repeated
is 1 if ijk are in xyz=123 cyclic order
is -1 if in the zyx=321 or any anticyclic order

Question 9

the operators Li are

Answer 9

Hermitian since components are all of the form ripj with i does not =j

Question 10

canonical commutator is [ri,pj]=

Answer 10

i h bar δij

Question 11

the canonical commutator can be used to show the commutation relations:

Answer 11

[ri,Lj]=i hbar εijk rk

[pi,Lj]=i hbar εijk pk

k does not = i does not =j

Question 12

important
commutation relations for components of angular momentum

Answer 12

[Li,Lj]=i h bar εijk Lk

Question 13

Different components of the angular momentum are

Answer 13

incompatible eigenstates

they cannot have simultaneously well-defined Li projections in different directions

Question 14

The ordering sensitivity between angular momentum in any two axial directions leads to

Answer 14

a difference that points in the third.

Question 15

we can’t have states that are simultaneously

Answer 15

eigenstates of two different components of the angular momentum
eg Lx and Ly

Question 16

The components satisfy an

Answer 16

uncertainty relation according to the generalised uncertainty principle

Question 17

the square of the angular momentum commutes with

Answer 17

each component

Question 18

The components of the angular momentum and its square, Li and L^2 are

Answer 18

compatible - we can have simultaneous eigenstates of full and projected angular momentum

Question 19

the full angular solution, with normalisation factors, are the

Answer 19

spherical harmonics

Question 20

spherical harmonics are

Answer 20

solid angle equivalent of Fourier series terms
based of the ALF and constitute an orthonormal function basis

Question 21

l determines the

Answer 21

polynomial order

Question 22

m controls

Answer 22

the orientation

Question 23

orbital ang mom and spher harm in dirac notation

Answer 23

define a ket state with the two quatum numbers l,m

Question 24

angular momentum ladder operators are defined as

Answer 24

L+/- = Lx +/- iLy

Question 25

function of angular momentum ladder operators

Answer 25

changing m by one unit without affecting l

Question 26

moves off the ladder (beyond the limits m=+/-l returns

Answer 26

zero (as with the QHO)

Question 27

spherical harmonics are not eigenstates of Lx or Ly as they are

Answer 27

composed of single ladder operators which change the state away from the original

Question 28

spherical harmonics expectation values must be zero as

Answer 28

the change of ket state forces the overlap integrals to zero

Question 29

expectation values are the

Answer 29

mean eigenvalues, averaged over wavefunction collapses

Question 30

Given a general angular wavefunction, first see if you can spot its

Answer 30

decomposition into spherical harmonic terms

Question 31

after decomposition into spherical harmonic terms, the

Answer 31

ci, m and l values are easily read off expectation calculations simplify

Question 32

decompose general operators into

Answer 32

ladder operators

Question 33

why are ladder operators much easier to work with

Answer 33

spherical harmonics are eigenfunctions of these operators

Question 34

the spin operator S, its components and its square have

Answer 34

the exact same properties and commutator algebra of the orbital angular momentum

Question 35

spin states are described by the

Answer 35

spin quantum number s and the spin magnetic quantum number ms |s,ms>

Question 36

The physical processes underlying orbital and intrinsic angular momentum are entirely separate, therefore

Answer 36

any of the components of one commutes with the components of the other

Question 37

We can define ladder operators also for the spin, exactly like for the orbital angular momentum:

Answer 37

S+/- = Sx +/- iSy

Question 38

There is one important difference between the spin eigenstates and the OAM ones

Answer 38

the spin eigenstates are not spherical harmonics

Question 39

pi mesons spin

Answer 39

s=0

Question 40

photons spin=

Answer 40

1

Question 41

delta baryons, s=

Answer 41

3/2

Question 42

total angular momentum is defined as

Answer 42

J=L+S

Question 43

total angular momentum quantum number

Answer 43

j=|l-s|, ...., l+s

Question 44

j takes all the integer values allowed between

Answer 44

completely antiparallel and completely aligned sources of angular momentum

Question 45

the components of the total angular momentum satisfy the commutation relation:

Answer 45

[Ji,Jj]=i hbar εijk Jk

Question 46

since [L,s]=0, L^2 and S^2 are both

Answer 46

compatible with J^2

Question 47

J^2=

Answer 47

(L+S)^2 =L^2+S^2+2L.S

Question 48

[J^2,L^2] =

Answer 48

[J^2,S^2]=0

Question 49

Angular momentum requires

Answer 49

3D wavefunction treatment.

Question 50

Wavefunction in a central potential is separable into r, theta and thi dependences: angular part given by

Answer 50

orthogonal spherical harmonics

Question 51

cannot simultaneously measure Lx,y,z but can measure

Answer 51

one projection plus L^2

Question 52

projection always smaller than

Answer 52

full to preserve uncertainty