Unformatted text preview: V. Angular
Momentum
Grifﬁths Chapter 4 1 A. Orbital Angular Momentum
The principal quantum number n of the
hydrogen atom gives the energy of the state.
The azimuthal and magnetic quantum numbers l
and m are related to the orbital angular
momentum. 2 A. Orbital Angular Momentum In classical physics, the orbital angular
momentum is given by L = r × p.
Lx = ypz − zpy Ly = zpx − xpz In cartesian coordinates Lz = xpy − ypx We use the same deﬁnition in quantum physics,
but these are now operators.
Lx = Y Pz − ZPy L=R×P Ly = ZPx − XPz Lz = XPy − Y Px 3 A. Orbital Angular Momentum 1. Some properties
[Lx , Ly ] = iLz The orbital angular
momentum operators
don’t commute. L × L = iL It’s useful to deﬁne
the square of the
magnitude of the
orbital angular
momentum. L2 ≡ L2 + L2 + L2
x
y
z L2 DOES commute
with the orbital
angular momentum
operator
components. [Ly , Lz ] = iLx
[Lz , Lx ] = iLy [L2 , Lx ] = 0
[L2 , L] = 0 [L2 , Ly ] = 0
[L2 , Lz ] = 0 See board V.A.1
4 A. Orbital Angular Momentum 2. Eigenvalues
Operators that don’t commute obey uncertainty
relations. Operators that DO commute share
eigenfunctions.
Since L2 commutes with each angular momentum
component, we can ﬁnd simultaneous
eigenfunctions with one of them. Let’s choose Lz.
We can also deﬁne ladder operators based on
our choice of Lz as a reference.
L± ≡ Lx ± iLy From this we get the eigenvalue equations
Lz flm = mflm L2 flm = 2 l(l + 1)flm
5 A. Orbital Angular Momentum
integer STEPS Lz flm = mflm L2 flm = 2 l(l + 1)flm some, as of yet, unknown
f unction of position positive integer and
HALF integer values
l = 0, 1/2, 1, 3/2, ... See board V.A.2 6 m = l, l+1, ..., l1, l A. Orbital Angular Momentum 3. Eigenfunctions
Lz flm = mflm L2 flm = 2 l(l + 1)flm We have found the eigenvalues for the orbital
angular momentum operator, but what are these
eigenfunctions, flm ?
We shall see that they are the spherical
harmonics, so that the quantum numbers l and m
of the hydrogen atom correspond to angular
momentum values.
BUT... what about the half integer values? 7 A. Orbital Angular Momentum We have found the eigenfunctions or the
orbital angular momentum operator, but what
are these eigenfunctions, flm ?
The angular momentum operator is L = (ħ/i)(r×∇).
Write this in spherical coordinates and we get
Lx =
i ∂
∂
− sin φ
− cos φ cot θ
∂θ
∂φ
∂
∂
Ly =
+ cos φ
− sin φ cot θ
i
∂θ
∂φ ∂
Lz =
i ∂φ See board V.A.3.a 8 A. Orbital Angular Momentum
∂
Lz =
i ∂φ The raising and lowering operators in spherical
coordinates are
L± = ±e ±iφ ∂
∂
± i cot θ
∂θ
∂φ These can be used to ﬁnd L2.
L = −
2 2 1∂
sin θ ∂θ ∂
sin θ
∂θ 1∂
+
sin2 θ ∂φ2
2 See Grifﬁths problem 4.21 9 A. Orbital Angular Momentum
∂
Lz =
i ∂φ
2
1∂
∂
1∂
2
2
L = −
sin θ
+
sin θ ∂θ
∂θ
sin2 θ ∂φ2 flm is an eigenfunction of Lz and L2.
m
Lz fL = ∂ m
fl = mflm
i ∂φ Equivalent to the azimuthal equation, Griffiths 4.21 L2 flm = − 2 1∂
sin θ ∂θ ∂
sin θ
∂θ 1∂
+
flm = 2 l(l + 1)flm
sin2 θ ∂φ2
2 Angular equation, Griffiths 4.18
Solution: spherical harmonics
10 A. Orbital Angular Momentum
So the eigenfunctions of Lz and L2 are the spherical harmonics.
The quantum numbers l and m enumerate the values of the
orbital angular momentum.
But what about the half integer values we found using the
ladder operators? Why did separation of variables not ﬁnd
them?
Separation of variables looks for solutions as functions of r, θ
and φ.
The half integer solutions are apparently not functions of
position.
They, therefore, are not related to ORBITAL angular momentum,
but are solutions corresponding to a type of angular momentum
intrinsic to a quantum mechanical particle.
The classical analogue to such a quantity is SPIN.
Grifﬁths warns us not to take this analogy too far.
11 B. Spin
To write the mathematical properties of this
“spin” type of angular momentum, we just
copy the mathematical properties of the
orbital angular momentum, which is
summarized by the commutation relations.
[Sx , Sy ] = iSz S h as the cartesian
c omponents Sx, Sy a nd Sz. [Sy , Sz ] = iSx
[Sz , Sx ] = iSy Therefore, many of our previous derivations
for orbital angular momentum apply also
for spin, just replacing L with S.
12 B. Spin
Therefore, many of our previous derivations
for orbital angular momentum apply also
for spin, just replacing L with S.
Lz flm = mflm L2 flm = 2 l(l + 1)flm whereas flm was a function of position, fsm i s
n ot, so it’s more convenient just to write it in
D irac notation. flm c ould have been written
t his way too > lm>. S 2 sm = 2 s(s + 1)sm L± = Lx ± iLy
S± = Sx ± iSy Sz sm = msm Griffiths Problem 4.18 S± sm = s(s + 1) − m(m ± 1)s(m ± 1) l = 0, 1, 2, · · ·
m = −l, −l + 1, · · · , l − 1, l
1
3
s = 0, , 1, , · · · m = −s, −s + 1, · · · , s − 1, s
2
2
13 B. Spin
1
3
s = 0, , 1, , · · · m = −s, −s + 1, · · · , s − 1, s
2
2 Spin can have both integer and half integer values
since there are position independent solutions to the
angular momentum operators for both these cases.
The hydrogen atom electron, for example, can take on
different l orbital angular momentum values since
orbital angular momentum is a function of position.
Since spin angular momentum is not a function of
position, a quantum mechanical particle can only ever
have ONE value s for spin.
Spin is therefore an intrinsic property of a quantum
mechanical particle, like its mass or charge.
An electron has spin 1/2, for example, or a photon has
spin 1.
14 B. Spin 1. Spin 1/2
Spin 1/2 systems are important because protons,
neutrons, electrons, and other particles in quantum
physics all have spin 1/2.
It is also one of the simplest quantum systems to study.
If s=1/2, then m can only have two values: 1/2 or 1/2.
There are therefore only two spin states,
½,½> or ½,½>.
“spin up”
↑,+ “spin down”
↓,  We can write these states as orthonormal column vectors 15 B. Spin
We can write these states as orthonormal column vectors
1
0
χ+ =
χ− =
0
1
“spin up”
↑,+ “spin down”
↓, Any general spin state of a spin 1/2 particle is then a
superposition of these.
a
χ=
= aχ+ + bχ−
b Spin operators can also be written in terms of this basis.
Sx = σ x
32 1 0
2
S2 =
01
4
spin operator components
Sy = σ y
2
total spin (squared )
Sz = σ z
2
16 B. Spin
Spin operators can also be written in terms of this basis.
32 1 0
2
S=
01
4
01
σx ≡
Sx = σ x
10
2
0 −i
Pauli Spin Matrices
σy ≡
Sy = σ y
i0
2
10
Sz = σ z
σz ≡
2
0 −1
See board V.B.1 From this, we see that χ+ and χ are eigenfunctions
(eigenspinors) of S2 and Sz.
Eigenvalue 3/4 ħ2 for S2 for both χ+ and χ.
Eigenvalues +ħ/2 and ħ/2 for Sz for χ+ and χ, respectively.
17 B. Spin
The total wave function of a quantum mechanical particle
might depend on, for example, the coordinates x, y, z as
well as s and m.
Unlike x, y and z, s and m are not continuous.
The spinor space is a complex, two dimensional space,
unlike the continuous Hilbert spaces that we’ve discussed
so far.
We will ignore these complications and not look, in this
course, at complete wave functions including spin.
For spin problems, the wave function is a two component
spinor, as in Grifﬁths example of Larmor precession.
See Grifﬁths Example 4.3 B. Spin 2. Addition of Angular Momentum
What if the electron in the hydrogen atom is not in the
ground state and has some orbital angular momentum?
The means there is both spin and orbital angular
momentum, so we have to add the angular momentum.
The TOTAL ANGULAR MOMENTUM of a system is denoted
by J=L+S
total
a ngular
momentum orbital
a ngular
momentum spin
a ngular
momentum The total angular momentum obeys the same form of
commutation relations as the orbital and spin angular
momenta. B. Spin J=L+S
The total angular momentum obeys the same form of
commutation relations as the orbital and spin angular
momenta.
[Jx , Jy ] = iJz J h as the cartesian
c omponents Jx, Jy a nd Jz. [Jy , Jz ] = iJx
[Jz , Jx ] = iJy To add the momenta, one uses PRODUCT STATES and
CLEBSCHGORDAN coefﬁcients.
C oefficients of the tensor
p roduct expansion Tensor products: ⊗ The same is true if we add spin to spin – for example
proton spin to electron spin to get the total spin angular
momentum of the hydrogen atom.
We will skip studying the mathematics of all this. ...
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 Spring '08
 RESCH
 Angular Momentum, Energy, Momentum, orbital angular momentum

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