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7.1 Angular momentum
Slides: Video 7.1.1 Angular momentum operators
Text reference: Quantum Mechanics for Scientists and Engineers
Chapter 9 introduction and Section 9.1 (first part)
Angular momentum
Angular momentum operators
Quantum mechanics for scientists and engineers David Miller
Angular momentum operators - preview
We will have operators corresponding to angular momentum about different orthogonal axes
, , and though they will not commute with
one another in contrast to the linear momentum operators
for the different coordinate directions, , andwhich do commute
ˆxL ˆ
yL ˆzL
ˆ xp ˆ yp ˆ zp
Angular momentum operators - preview
We will, however, find another useful angular momentum operator,
which does commute separately with each of , , and
The eigenfunctions for , , and are simple Those for
the spherical harmonics, are more complicated but can be understood relatively simply
and form the angular shapes of the hydrogen atom orbitals
2L̂
ˆxL ˆ
yL ˆzL
ˆxL ˆ
yL ˆzL
2L̂
x
y
r
p q
origin
position of object
momentum
Classical angular momentum
The classical angular momentum of a small object
of (vector) linear momentum pcentered at a point given
by the vector displacement r relative to some originis L r p
Vector cross product
As usual
where i, j, and k are unit vectors in x, y, and z directionsand Ax is the component of A in the x direction
and similarly for the y and z directions and the components of B
sin
( ) ( ) ( )
x y z
x y z
y z z y x z z x x y y x
AB A A AB B B
A B A B A B A B A B A B
i j kC A B c
i j k
Vector cross product
In
C is perpendicular to the plane of A and B just as the z axis is perpendicular to the plane containing the x and y axes in right-handed axes
is the angle between the vectors A and Bc is a unit vector in the direction of the vector C
sin
( ) ( ) ( )
x y z
x y z
y z z y x z z x x y y x
AB A A AB B B
A B A B A B A B A B A B
i j kC A B c
i j k
Vector cross product
Note that, in
the ordering of the multiplications in the second line is chosen to work also for operators instead of numbers for one or other vector
the sequence of multiplications in each term is always in the sequence of the rows from top to bottom
sin
( ) ( ) ( )
x y z
x y z
y z z y x z z x x y y x
AB A A AB B B
A B A B A B A B A B A B
i j kC A B c
i j k
Angular momentum operators
With classical angular momentum
we can explicitly write out the various components
Now we can propose a quantum mechanical angular momentum operator
based on substituting the position and momentum operators
and similarly write out component operators
x z yL yp zp y x zL zp xp z y xL xp yp
L r p
ˆ ˆ ˆ i L r p r
L̂
Angular momentum operators
Analogously, we obtain three operators
which are each Hermitian and so, correspondingly, is the operator itself
ˆ ˆ ˆ ˆˆx z yL yp zp i y zz y
ˆ ˆ ˆˆˆy x zL zp xp i z xx z
ˆ ˆˆ ˆ ˆz y xL xp yp i x yy x
L̂
Commutation relations
The operators corresponding to individual coordinate directions obey commutation relations
These individual commutation relations can be written in a more compact form
ˆ ˆ ˆ ˆ ˆ ˆ ˆ,x y y x x y zL L L L L L i L
ˆ ˆ ˆ ˆ ˆ ˆ ˆ,y z z y y z xL L L L L L i L
ˆ ˆ ˆ ˆ ˆ ˆ ˆ,z x x z z x yL L L L L L i L
ˆ ˆ ˆi L L L
Commutation relations
Unlike operators for position and for linear momentum the different components of this angular momentum operator do not commute with one another
Though a particle can have simultaneously a well-defined position in both the x and y directions
or have simultaneously a well-defined momentum in both the x and y directions
a particle cannot in general simultaneously have a well-defined angular momentum component in more than one direction
7.1 Angular momentum
Slides: Video 7.1.3 Angular momentum eigenfunctions
Text reference: Quantum Mechanics for Scientists and Engineers
Section 9.1 (remainder)
Angular momentum
Angular momentum eigenfunctions
Quantum mechanics for scientists and engineers David Miller
Spherical polar coordinates
The relation between spherical polar and
Cartesian coordinates is
is the polar angle, and is the azimuthal angle
x
y
z
r
(x, y, z)
f
qsin cosx r
sin siny r
cosz r
Spherical polar coordinates
In inverse form
x
y
z
r
(x, y, z)
f
q
2 2 2r x y z
2 21
2 2 2sin
x y
x y z
1tan yx
Angular momentum in spherical polar coordinates
With these definitions of spherical polar coordinates and with standard partial derivative relations of the form
for each of the Cartesian coordinate directionswe can rewrite the angular momentum operator
components in spherical polar coordinates
rx x r x x
Angular momentum in spherical polar coordinates
From
and
we obtain
ˆ ˆ ˆ ˆ ˆ ˆ ˆ,x y y x x y zL L L L L L i L ˆ ˆ ˆ ˆ ˆ ˆ ˆ,y z z y y z xL L L L L L i L
ˆ ˆ ˆ ˆ ˆ ˆ ˆ,z x x z z x yL L L L L L i L
ˆ sin cot cosxL i
ˆ cos cot sinyL i
ˆzL i
Lz eigenfunctions and eigenvalues
Using
we solve for the eigenfunctions and eigenvalues of
The eigen equation is
where is the eigenvalue to be determined The solution of this equation is
ˆzL i
ˆzL
ˆzL m
m
exp im
Lz eigenfunctions and eigenvalues
The requirements that the wavefunction and its derivative are
continuous when we return to where we started i.e., for
mean that m must be an integer positive or negative or zero
Hence we find that the angular momentum around the z axis is quantized
with units of angular momentum of
2