A Charged Particle in an Electromagnetic Field
A Charged Particle in an Electromagnetic Field
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Charged Particle in an Electromagnetic Field
We will apply quantum mechanics to treat the motion of a charged particle
in an external electromagnetic field.
The electromagnetic field is assumed to be produced by charges and
currents other than the one that we are considering; the field produced by
the charge that we are studying is also neglected.
Although the motion of the charged particle will be quantum mechanically
given, the treatment is inherently a semiclassical approach because the
external electromagnetic field is treated classically.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Brief Review of Electrodynamics
The classical theory of the electromagnetic field, including Maxwell’s
equations, the formalism of the scalar and vector potentials, and the gague
transformation:
The vacuum Maxwell’s equations are given by
1 ∂B
c ∂t
∇ ⋅ E = 4πρc
∇×E = −
∇ ⋅B = 0
∇ × B = 4π jc +
1 ∂E
c ∂t
The continuity equation is given by
∂ ρc
∂t
+ ∇ ⋅ jc = 0
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Brief Review of Electrodynamics
The Maxwell’s equations reveal that the most general electric and magnetic
fields can be expressed in terms of the scalar and vector potential, φ and A,
in the following way:
1 ∂A
B = ∇× A
c ∂t
These equations are not unique in defining electric and magnetic fields.
E = −∇ϕ −
If an arbitrary vector, ∇ f, is added to the vector potential, the magnetic field
is unchanged because of ∇ × ∇f = 0 .
If the quantity (−1 / c)(∂f ∂t ) is added to the scalar potential, the electric field
is also not changed.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Brief Review of Electrodynamics
The electric and magnetic fields remain invariant under the following
transformation of the potentials:
1∂ f
A′ = A + ∇ f
ϕ′ = ϕ −
c ∂t
This transformation is called the gauge transformation.
A common choice is to make: ∇ ⋅ A = 0
(Coulomb gauge )
suppose that an arbitrary set of potentials is given by φ and A, the gauge
transformation is used to obtain a new set of potentialsφ’and A’.
In order to obtain ∇ ⋅ A′ = 0 , the following condition needs to be satisfied :
∇ ⋅ A + ∇2 f = 0
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Brief Review of Electrodynamics
∇ ⋅ A + ∇ 2 f = 0 is just Poisson’s equation defining f in terms of the
specified function
∇⋅ A
The function f is given by f =
1
4π
∫∫∫
∇⋅ A
dx′ dy ′ dz ′
| r - r′ |
∇ ⋅ A′ = 0 can be always obtained by a gauge transformation.
In empty space the choice ∇ ⋅ A = 0 brings about
ϕ = 0.
1
∂A
∇ ⋅ E = −∇ ϕ − ∇ ⋅
=0
c
∂t
2
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
A Brief Review of Electrodynamics
The fact ∇ ⋅ A = 0 leads to ∇ 2ϕ = 0 that is simply Laplace’s equation.
It is well known that the only solution of this equation that is regular over all
of space is
ϕ = 0.
In empty space the following expressions for the fields can be obtained:
E=−
1 ∂A
c ∂t
B = ∇× A
subject to the condition that
2006 Quantum Mechanics
∇⋅A = 0
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Hamiltonian for a Charged Particle in an Electromagnetic Field
Considering a particle of charge q and mass m and in terms of φand A, the
classical Hamiltonian function is given by
2
q ⎞
1 ⎛
H=
⎜ p − A ⎟ + V (r ) + qϕ
2m ⎝
c ⎠
where V (r ) is the part of the potential energy which is of nonelectromagnetic
origin.
q
p → p− A
c
q& i =
∂H
∂ pi
;
2006 Quantum Mechanics
p& i = −
∂H
∂ qi
1⎛
q ⎞
r& = ⎜ p − A ⎟
m⎝
c ⎠
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Hamiltonian for a Charged Particle in an Electromagnetic Field
When there is a vector potential, the canonical momentum, p, is no longer
equal to its conventional value of
m r&
.
⎤
d 2r d ⎛
q ⎞
q ⎡∂ A
=
−∇
−
+
⋅
∇
m 2 =
p
−
A
H
(
v
)
A
⎟
⎜
⎢
⎥
dt
dt⎝
c ⎠
c ⎣ ∂t
⎦
2
⎡ 1 ⎛
q ⎞ ⎤
q ∂A q
− [− v × (∇ × A ) + ∇(v ⋅ A )]
= −∇ ⎢
⎜ p − A ⎟ ⎥ − ∇V (r ) − q∇ϕ −
c ⎠ ⎦⎥
c ∂t c
⎣⎢ 2m ⎝
⎛
q
1 ∂A⎞ q
⎜
⎟⎟ − [− v × (∇ × A ) + ∇(v ⋅ A )]
= ∇(v ⋅ A ) − ∇V (r ) + q⎜ − ∇ϕ −
c
c ∂t ⎠ c
⎝
q
v×B
c
It represents exactly the classical equations of motion.
= −∇V (r ) + q E +
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Hamiltonian for a Charged Particle in an Electromagnetic Field
The quantum-mechanical Hamiltonian operator can be obtained by
replacing p by the operator − ih∇
:
2
q ⎞
1 ⎛
H=
⎜ − ih∇ − A ⎟ + V (r) + qϕ
c ⎠
2m ⎝
− h2 2
ihq
q2
2
=
∇ + V (r ) + qϕ +
A
( A ⋅ ∇ + ∇ ⋅ A) +
2m
2m c
2m c 2
It is useful to derive the expression of the probability current density for a
charged particle under the influence of a static magnetic field.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Hamiltonian for a Charged Particle in an Electromagnetic Field
Using the time-dependent Schrödinger equation
ih(∂ψ ∂ t ) = Hψ
, we
obtain
q ⎞ ∗ ψ∗ ⎛
q ⎞
∂ρ ∂
∂ψ ∗
ψ ⎛
∗
∗ ∂ψ
=−
= (ψ ψ ) = ψ
+ψ
⎜ − ih∇ − A ⎟ ψ
⎜ ih∇ − A ⎟ ψ +
c ⎠
c ⎠
∂t
∂t
∂t ∂t
2mih ⎝
2mih ⎝
2
2
2
2
∗
⎡ 2 2 ih q
⎛ q ⎞ 2 ⎤ ∗ ψ ⎡ 2 2 ih q
⎛ q ⎞ 2⎤
=−
(∇ ⋅ A + A ⋅ ∇) + ⎜ ⎟ A ⎥ψ +
(∇ ⋅ A + A ⋅ ∇) + ⎜ ⎟ A ⎥ψ
⎢− h ∇ −
⎢− h ∇ +
h
c
c
mi
c
2mih ⎢⎣
2
⎝ ⎠
⎝c⎠
⎥⎦
⎢⎣
⎥⎦
h
q
=−
∇ ⋅ ψ ∇ψ ∗ −ψ ∗∇ψ +
∇ ⋅ ( Aψ ∗ψ )
mc
2mi
ψ
(
∂ρ
+∇⋅ j = 0
∂t
)
the probability current density is given by j =
2006 Quantum Mechanics
h
2mi
(ψ ∇ψ ∗ −ψ ∗∇ψ ) −
q
mc
( Aψ ∗ψ )
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Hamiltonian for a Charged Particle in an Electromagnetic Field
The complete Schrödinger equation is expressed as
∂ψ
2
q ⎞
1 ⎛
ih
i
h
A ⎟ ψ + (V (r ) + qϕ )ψ
=
−
∇
−
⎜
c ⎠
∂ t 2m ⎝
Making a gauge transformation, the Schrödinger equation becomes
∂ψ
q
q ⎞
q∂f
1 ⎛
′
′
(
)
ih
=
−
i
∇
−
+
∇
f
+
V
+
q
+
(
)
h
A
ψ
r
ϕ
ψ
ψ
⎟
⎜
∂ t 2m ⎝
c
c ⎠
c ∂t
2
After some algebra
∂ iq f / hc
1 ⎛
q ⎞ iq f / hc
ih
e
ψ =
ψ + (V (r) + qϕ ′) e iq f / hcψ
⎜ − ih∇ − A′ ⎟ e
∂t
2m ⎝
c ⎠
(
)
2
(
)
(
)
in terms of the new potentials,φ’and A’ , a new solution is given by
iq f / hc
′
ψ =e
ψ
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Aharonov-Bohm Effect
Imagine a particle constrained to move in a circle of radius b. There is a
solenoid of radius a<b along the center of the circle, carrying a steady
electric current I. If the solenoid is extremely long, the magnetic filed inside it
is uniform, and the field outside is zero. But the vector potential outside the
solenoid is not zero. Adopting the convention gauge condition ∇ ⋅ A = 0 ,
the vector potential is given by
1
a2 ΦB
A = (B × r ) 2 =
aˆ φ
2
2π r
r
where Φ B = π a 2 B is the magnetic flux through the solenoid.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Aharonov-Bohm Effect
Consider a charged particle that is moving through a region where B is zero
( ∇ × A = 0 ), but A itself is not.
solve the Schrödinger equation without the vector potential A, and find the
eigenfunction to be ψ
the eigenfuction for the presence of a vector potential is given by
iδ
ψ′=e ψ
q
δ =
hc
∫
s
A ⋅ dl
where s denotes the path of integration
moving a path through a region where the field is zero, but not the vector
potential, the wave function of the charged particle acquires an additional
phase.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Aharonov-Bohm Effect
Aharonov and Bohm proposed an experiment in which a beam of electrons
is split in two, and passed either side of a long solenoid, before being
recombined.
The phase difference accruing from traveling the different
paths is given by
q
δ 2 − δ1 =
hc
[∫
1
]
q
A ⋅ dl − ∫ A ⋅ dl =
A ⋅ dl
∫
2
hc
The combination of path 1 and back by path 2 makes a closed loop. With
the Stoke’s theorem
∫ A ⋅ dl = ∫∫ (∇ × A) ⋅ dS = ∫∫ B ⋅ dS = Φ
2006 Quantum Mechanics
B
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
The Aharonov-Bohm Effect
δ=
q
ΦB
hc
This phase shift leads to measurable interference, which has been
confirmed experimentally by Chambers (Phys. Rev. Lett. 5, 3, 1960) and
others.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Considering a charged particle moving in a constant magnetic field, the
classical trajectory is a helical path that is uniform translational motion in the
direction parallel to the magnetic field and uniform rotational motion in the
plane perpendicular to B .
For a positive charge q, the circular motion is in the counterclockwise
direction when viewed from the leaving direction of translational motion.
The Coulomb gauge
∇ ⋅ A = 0 leads to (∇ ⋅ A) ψ = (A ⋅ ∇ )ψ .
Therefore,
the quantum Hamiltonian is given by
− h2 2 i h q
q2
H=
∇ +
A ⋅∇ +
A2
2
2m
mc
2m c
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
One of the convenient expressions for the vector potential of a uniform
constant magnetic field is given by
1
A = − r×B
2
As a result, we find that
1
1
1
1
A ⋅ ∇ = − ( r × B ) ⋅ ∇ = B ⋅ (r × ∇) = −
B ⋅ (r × p ) = −
B⋅L
2
2
2ih
2ih
In addition, we use equation mentioned above to obtain
A2 =
[
1
1
(r × B ) 2 = r 2 B 2 − ( r × B ) 2
4
4
2006 Quantum Mechanics
]
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
After some algebra, the Hamiltonian for a charged particle in a constant
magnetic field is given by
[
− h2 2
q
q2
H=
∇ −
B⋅L +
r 2 B 2 − (r × B) 2
2
2m
2m c
8m c
]
Assuming B to be in the z-direction, the Eq. can be expressed as
− h2 2
1
H=
∇ − ω L Lz + mω 2L ( x 2 + y 2 )
2m
2
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
where ωL = qB /(2mc) is
the
Lamor
precession
frequency.
The
corresponding Schrödinger equation in cylindrical coordinates is given by
⎡ − h2 ⎛ ∂2 1 ∂
⎤
∂2 ⎞
∂ 1
1 ∂2
2 2
⎟
⎜
h
+
+
+
+
+
i
ω
m
ω
r
⎢
⎥ ψ ( r,θ , z ) = Eψ ( r,θ , z )
L
L
2
2
2
2 ⎟
⎜
∂θ 2
r ∂ r r ∂θ
∂z ⎠
⎣ 2m ⎝ ∂ r
⎦
„
The Schrödinger equation is separable in this case and its solution can be
expressed as
ψ ( r,θ , z ) = R( r ) e i l θ e ik z
z
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
This solution is an eigenfunction of both the
L̂z
and
z
kinetic
energy
terms which together have contributions to the total eigenenergy given by
− l hωL and h 2 k z2 /(2m ) , respectively. The remaining 2D problem is exactly
the same as the 2D isotropic harmonic oscillator in polar coordinates:
⎡ − h2 ⎛ d 2 1 d
⎤
l2 ⎞ 1
2 2
⎜
⎟
m
ω
r
+
−
+
⎢
⎥ ψ ( r ) = E ′ψ ( r )
L
2
2 ⎟
⎜
rdr r ⎠ 2
⎣ 2m ⎝ d r
⎦
where
E′
is related to the total energy as
h 2 k z2
E = E ′ − l hω L +
2m
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
In terms of the dimensionless variable ξ = 2 r / w , where w = 2h /(mωL )
⎡ − h2 ⎛ d 2 1 d
⎤
l2 ⎞ 1
2 2
⎟
⎜
+
− 2 ⎟ + mωL r ⎥ ψ ( r ) = E ′ψ ( r )
⎢
2
⎜
rdr r ⎠ 2
⎣ 2m ⎝ d r
⎦
can be reduced to
⎛ d2
l2
1 d
2⎞ ~
⎜⎜
⎟⎟ ψ (ξ ) = 0
+
+
−
−
ε
ξ
2
2
ξ dξ
ξ
⎝d ξ
⎠
with the dimensionless eigenvalue ε = 2 E ′ /( hω L ) and ψ~(ξ ) = ψ ( r ) .
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Since the behavior of
ψ~(ξ )
at infinity is important the existence of
normalizable solutions, the Eqs. above can be examined for large ξ:
⎛ d2
2⎞ ~
⎜⎜
⎟⎟ ψ (ξ ) = 0
−
ξ
2
⎝d ξ
⎠
,
−ξ 2 / 2
~
which has approximation solution, ψ (ξ ) ∝ e
.
On the other hand, the behavior near the origin is guaranteed to be the form
ξ | l | . Therefore, the eigenfunction ψ~(ξ ) can be expressed as
ψ~(ξ ) = ξ
|l |
e −ξ
2
/2
2006 Quantum Mechanics
g (ξ )
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Substituting Eq. above into
⎛ d2
l2
1 d
2⎞ ~
⎜⎜
⎟⎟ ψ (ξ ) = 0
+
+
−
−
ε
ξ
2
2
ξ dξ
ξ
⎝d ξ
⎠
,
we
have
⎛ d2
l2
1 d
2⎞ ~
⎜⎜
⎟⎟ ψ (ξ )
ε
ξ
+
+
−
−
2
2
ξ dξ
ξ
⎝d ξ
⎠
⎞ d g (ξ )
d 2 g (ξ ) ⎛ 2 | l | +1
ξ
=
+
−
2
+ (ε − 2 | l | −2) g (ξ ) = 0
⎜
⎟⎟
2
⎜
dξ
⎝ ξ
⎠ dξ
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Here we have used the following identities
[
|l |
e −ξ
2
/2
g (ξ ) = ξ
[
|l |
e −ξ
2
/2
g (ξ )
1 d
ξ
ξ dξ
]
|l |
e −ξ
2
/2
⎛|l |
1 d ⎞
⎜⎜ 2 − 1 +
⎟⎟ g (ξ )
ξ dξ⎠
⎝ξ
and
d2
ξ
2
dξ
=
d
dξ
=ξ
|l |
⎡
⎢ξ
⎣
e
|l |
−ξ 2 / 2
e −ξ
2
/2
]
⎤
⎛|l |
d ⎞
⎟⎟ g (ξ )⎥
⎜⎜ − ξ +
dξ⎠
⎝ξ
⎦
⎡l 2 − | l |
⎛2|l |
⎞ d
d2 ⎤
2
g (ξ )
− 2 | l | −1 + ξ + ⎜⎜
− 2ξ ⎟⎟
+
⎢
2
2⎥
ξ
ξ
d
ξ
d
ξ
⎝
⎠
⎦
⎣
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Next, a change of variable,
η =ξ2
, leads Eq. to be
⎞ d g (ξ )
d 2 g (ξ ) ⎛ 2 | l | +1
+
−
+ (ε − 2 | l | −2) g (ξ ) = 0
ξ
2
⎜
⎟⎟
⎜ ξ
ξ
d
d ξ2
⎝
⎠
⎛ 2 d 2 G (η )
⎞ dG (η )
dG (η ) ⎞ ⎛ 2 | l | +1
⎟
+
−
⇒ ⎜⎜ 4ξ
+
ξ
2
2
+ (ε − 2 | l | −2) G (η ) = 0
⎜
⎟⎟2ξ
2
⎜
⎟
η
ξ
η
d
d
dη
⎠
⎠ ⎝
⎝
d 2 G (η )
dG (η ) (ε − 2 | l | −2)
(
)
|
|
1
⇒ η
+
l
+
−
η
+
G (η ) = 0
4
dη
d η2
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
Here we have used
g (ξ ) = G (η )
d g (ξ )
dG(η )
= 2ξ
dξ
dη
2
d 2 g (ξ )
dG (η )
2 d G (η )
ξ
4
2
=
+
dη
d ξ2
d η2
The Eq. from previous page is the differential equation for Laguerre
polynomials, and, consequently the solution is given by G(η ) = Llp (η ) with the
dimensionless eigenvalue
(ε − 2 | l | −2)
= p ⇒ ε = 4 p + 2 | l | +2
4
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Quantum Eigenstates of a Charged Particle in a Constant Magnetic Field
where
p = 0, 1, 2...... n. To sum up, the eigenfunctions and eigenvalues
can be given by
ψ ( r,θ , z ) = C p ,l e −η / 2 η | l |/ 2 Llp (η ) e i l θ e ik z
z
= C p ,l
⎡ ⎛ r ⎞2 ⎤ ⎛ 2 r ⎞
⎟
exp ⎢− ⎜ ⎟ ⎥ ⎜⎜
⎟
⎣⎢ ⎝ w ⎠ ⎦⎥ ⎝ w ⎠
|l |
⎡⎛ 2 r ⎞ 2 ⎤
⎟ ⎥ e i l θ e ik z z
Llp ⎢⎜⎜
⎢⎣⎝ w ⎟⎠ ⎥⎦
and
ε = 4 p + 2 | l | +2 ⇒ E ′ = ( 2 p + | l | +1)hω L
⇒
where
C p ,l
E p ,l , k z
h 2 k z2
= ( 2 p + | l | +1)hω L − l hω L +
2m
are the normalization constants to be determined later.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Llp (x)
The Laguerre polynomials,
x −l d p
L ( x) = e
(e − x x p +l ) ,
p
p! d x
l
p
x
for arbitrary real
l > −1
, are defined by the formula
p = 0, 1, 2.....
.
From above, the first few Laguerre polynomials are
L l0 ( x ) = 1 ,
L l2 ( x ) =
L 1l ( x ) = 1 + l − x ,
1
[(1 + l )( 2 + l ) − 2( 2 + l ) x + x 2 ]
2
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
In general, Leibniz’z formula enable us to have
Γ( p + l + 1) ( − x ) k
L ( x) = ∑
k =0 Γ( k + l + 1) k! ( p − k )!
p
l
p
where for all k < p the ratio of gamma functions can be replaced by the
product
( p + l )( p + l − 1) L (l + k + 1)
2006 Quantum Mechanics
.
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
The generating function for Laguerre polynomials is given by
W ( x, t ) = (1 − t )
−l −1
e
− x t / (1−t )
∞
= ∑ L lp ( x ) t p
, | t |< 1
p =0
With the generating function, the following identity can be easily verified:
∂W
+ [ x − (1 − t )(1 + l )] W = 0
∂t
After algebra, we find that
(1 − t ) 2
(1 − t )
∞
2
∑pL
p =0
l
p
( x) t
p −1
∞
+ [ x − (1 − t )(1 + l )] ∑ L lp ( x ) t p = 0
p =0
which gives
( p + 1) L lp +1( x ) + ( x − l − 2 p − 1) L lp ( x ) + ( p + l ) L lp −1( x ) = 0
when the coefficient of
2006 Quantum Mechanics
tp
is set equal to zero.
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Similarly, substituting W ( x, t ) = (1 − t )
−l −1
e
− x t / (1−t )
∞
= ∑ L lp ( x ) t p
, | t |< 1
p =0
into the identity
(1 − t )
∂W
+tW =0
∂x
we obtain
∞
(1 − t ) ∑
p =0
dL lp ( x )
dx
∞
t + ∑ L lp ( x ) t p +1 = 0
p
p =0
which indicates
dL lp ( x )
dx
−
dL lp −1( x )
dx
2006 Quantum Mechanics
+ L lp −1( x ) = 0
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
From
( p + 1) L lp +1( x ) + ( x − l − 2 p − 1) L lp ( x ) + ( p + l ) L lp −1( x ) = 0
we
obtain
( x − p − 1)
dL lp ( x )
+ ( p + 1)
dL lp +1( x )
dx
− ( p + 1) L lp +1( x ) = 0
dx
+ ( 2 p + 2 + l − x ) L lp ( x )
Replacing p by p-1 and using equation from previous page to eliminate ,
dL lp −1( x ) / dx
x
dL lp ( x )
dx
then we get
= pL lp ( x ) − ( p + l ) L lp −1( x )
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
The last equation indicates that the derivative of a Laguerre polynomial can
be in terms of another Laguerre polynomial. Using the recurrence relations
Eqs. before, we can derive a differential equation satisfied by the Laguerre
polynomials.
The step is to differentiate
x
dL lp ( x )
respect to x and then to eliminate
2006 Quantum Mechanics
dx
= pL lp ( x ) − ( p + l ) L lp −1( x )
dL lp −1( x ) / dx
and L lp −1( x )
with
.
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Consequently, we obtain
x
d 2 L lp ( x )
dx 2
+ (l + 1 − x )
dL lp ( x )
dx
+ pL lp ( x ) = 0
,
l
which indicates that u = L p (x ) is a particular solution of the second-order
linear differential equation
xu ′′ + (l + 1 − x )u ′ + pu = 0
By making changes of variables, it can be derived other differential equations
whose integrals can be expressed in terms of Laguerre polynomials.
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
For instance, it can be shown that the differential equations
l + 1 x ν (ν − l ) ⎤
⎡
− +
u=0
x u ′′ + (l + 1 − 2ν )u ′ + ⎢ p +
⎥
x
2
4
⎣
⎦
have the particular solutions
u( x ) = xν e − x / 2 L lp ( x )
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Next we demonstrate one of the most important properties of the Laguerre
l/2
e − x / 2 on the interval
polynomials, i.e., their orthogonality with weight x
0≤ x<∞
.
Setting
u p ( x) = x l / 2 e − x / 2 L lp ( x)
and recalling eqution from previous page, we find that u p ( x ) and u p ( x )
satisfy the differential equations
⎡
l +1 x l2 ⎤
( x u ′p )′ + ⎢ p +
− − ⎥ up = 0
2
4 4x ⎦
⎣
⎡
l +1 x l2 ⎤
( x u q′ )′ + ⎢ q +
− − ⎥ uq = 0
2
4 4x ⎦
⎣
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
After subtracting, and integrating from 0 to ∞, we obtain
x ( u′p uq − uq′ u p )
∞
0
∞
+ ( p − q ) ∫ uq u p dx = 0
0
[
]
1 d
| l | −ξ 2 / 2
ξ
e
g (ξ ) = ξ
For l > −1 the first term of
ξ dξ
vanishes at both limits, and therefore
∞
uq u p dx = 0
if
p≠q
x l e − x L lp ( x) L lq ( x ) dx = 0
if
p ≠ q , l > −1
∫
0
|l |
e −ξ
2
/2
⎛|l |
1 d ⎞
⎜⎜ 2 − 1 +
⎟⎟ g (ξ )
ξ dξ⎠
⎝ξ
In other words,
∫
∞
0
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Now we evaluate the value of the integral for p = q .
Recall the recurrence
relation
( p + 1) L lp +1( x ) + ( x − l − 2 p − 1) L lp ( x ) + ( p + l ) L lp −1( x ) = 0
We can replace the index p by p-1 in the Eq. above and multiply the result by
L lp (x) . Then from this equation we subtract above Eq. multiplied by L l ( x) ,
p −1
obtaining
[
]
2
[
]
2
p L lp ( x) − ( p + l ) L lp−1( x) − ( p + 1) L lp+1( x) L lp−1( x)
+ 2 L lp ( x) L lp−1( x) + ( p + l − 1) L lp ( x) L lp−2( x) = 0 ,
2006 Quantum Mechanics
p = 2,3...
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Multiplying this equation by x l e − x , integrating from 0 to ∞, using the
orthogonality property ( p + 1) L lp +1( x) + ( x − l − 2 p − 1) L lp ( x) + ( p + l ) L lp −1( x) = 0 ,
we obtain
∞
p∫ x e
l
−x
0
[L
l
p
]
∞
[
]
( x ) dx = ( p + l ) ∫ x l e − x L lp −1 ( x ) dx ,
2
0
2
p = 2,3...
Repeated application of this formula gives
∫
∞
0
[
]
2
x l e − x L lp ( x) dx =
=
2006 Quantum Mechanics
[
]
2
( p + l )( p + l − 1)L(l + 2) ∞ l − x l
x
e
L
(
x
)
dx
1
∫0
p( p − 1)L3 ⋅ 2
Γ( p + l + 1)
,
p!
p = 2,3...
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
It follows by direct substitution that this formula is also valid for
p = 0,1
,
and hence
∫
∞
0
[
]
2
x l e − x L lp ( x) dx =
Γ( p + l + 1)
, l > −1,
p!
p = 0,1,2,3...
As a consequence, the functions
⎤
⎡
p!
ψ p ,l ( x) = ⎢
⎥
⎣ Γ( p + l + 1) ⎦
1/ 2
x l / 2 e − x / 2 L lp ( x ) ,
form an orthonormal system on the interval
2006 Quantum Mechanics
p = 0,1,2, L
0≤ x<∞
.
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
π
*
Y
∫ l ,−l (θ ,φ ) Yl ,−l (θ ,φ ) dΩ = ∫ sin θ dθ
0
∫
2π
0
dφ | C |2 sin 2l θ = 1
The θ integral is conveniently done using x = cosθ , hence
2π | C |2
∫
1
−1
(1 − x 2 ) l dx = 1
1 − x 2 = (1 − x)(1 + x)
Writing
, and further change the variable to
x = 2t − 1 , the right side of the above equation becomes
2π | C |
2
∫
1
0
[ 2t ( 2 − 2t )] dt = 2π | C | 2
2006 Quantum Mechanics
l
2
2 l +1
∫
1
0
t l (1 − t ) l dt
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
From the property of the Beta function
1
B( p, q) = ∫ t p −1 (1 − t ) q −1 dt =
0
Γ( p ) Γ( q )
Γ( p + q )
With the last three equations, we obtain
2π | C | 2
2
⇒ | C |=
2 l +1
Γ(l + 1) Γ(l + 1)
Γ(2l + 2)
1
(2l + 1)!
2l l !
4π
2006 Quantum Mechanics
= 4π | C | 2
2
2l
(l !) 2
1
(2l + 1)!
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
The commonly used convention is not to have an additional phase factor,
hence
Yl , −l (θ ,φ ) =
1 (2l + 1)! l
sin θ e −i lφ
l
2 l!
4π
From Eqs. mentioned before, we have
Lˆ + l , m = (l − m)(l + m + 1) l , m + 1
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
Eq. of last page indicates that all Yl ,m (θ , φ ) can be obtained by successively
acting
L̂+
on
Yl , − l (θ , φ )
.
With the equation mentioned before
∫
∞
0
uq u p dx = 0
if
p≠q
,
the spherical harmonic Yl ,m (θ , φ ) can be expressed as
Yl ,m (θ , φ ) =
1
1
1
1
L
h (l − m + 1)(l + m) h (l − m + 2)(l + m − 1) h (l + l − 1)(2) h (l + l )(1)
⎡ iφ ⎛ ∂
∂ ⎞⎤
× ⎢ihe ⎜ −i
+ cot θ
⎟⎥
∂
θ
∂φ ⎠ ⎦
⎝
⎣
l +m
Yl ,− l (θ , φ )
(l - m)! ⎡ iφ ⎛ ∂
∂ ⎞⎤
e
i
=
cot
θ
+
⎢ ⎜
⎟⎥
(2l )! (l + m)! ⎣ ⎝ ∂θ
∂φ ⎠ ⎦
2006 Quantum Mechanics
l +m
Yl ,−l (θ , φ )
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
The φ derivative always give eigenvalues. But each time
L̂+
acts, there is
a factor of eiφ and makes the eigenvalue of −i ∂ ∂φ increase one by one.
As a result, the last equation can be expressed as
Yl ,m (θ , φ ) =
1 (2l + 1)!
(l - m)!
⎞
imφ ⎛ d
θ
e
−
(
m
−
1)
cot
⎜
⎟
2l l !
4π
(2l )! (l + m)!
⎝ dθ
⎠
⎛ d
⎞ ⎛ d
⎞⎛ d
⎞
×⎜
− (m − 2) cot θ ⎟ L ⎜
− (−l + 1) cot θ ⎟⎜
− (−l ) cot θ ⎟ sin l θ
⎝ dθ
⎠ ⎝ dθ
⎠⎝ dθ
⎠
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Definition and Generating Function of the Laguerre Polynomials
With the property that
d
1 d
n
+ n cot θ =
θ
sin
n
dθ
sin θ dθ
,
equation from last page can be written as
Yl ,m (θ , φ ) =
1 (2l + 1)
2l l !
4π
(l - m)! imφ ⎛
1
d
⎞
e ⎜ − ( m −1)
sin − ( m −1) θ ⎟
θ dθ
(l + m)!
⎝ sin
⎠
1
d
d
⎛
⎞ ⎛ 1
⎞⎛ 1 d
⎞
× ⎜ − ( m − 2)
sin − ( m − 2) θ ⎟ L ⎜ l −1
sin l −1 θ ⎟ ⎜ l
sin l θ ⎟ sin l θ
θ dθ
⎝ sin
⎠ ⎝ sin θ dθ
⎠ ⎝ sin θ dθ
⎠
l +m
1 (2l + 1)
= l
2 l!
4π
(l - m)! imφ
⎛ 1 d ⎞
e sin m θ ⎜
⎟
(l + m)!
⎝ sin θ dθ ⎠
1 (2l + 1)
= l
2 l!
4π
(l - m)! imφ
d ⎞
⎛
e (1 − cos 2 θ ) m / 2 ⎜ −
⎟
(l + m)!
⎝ d cos θ ⎠
(−1)l + m
=
2l l !
(2l + 1)
4π
(l - m)! imφ
⎛d ⎞
e (1 − x 2 ) m / 2 ⎜ ⎟
(l + m)!
⎝ dx ⎠
sin 2l θ
l +m
(1 − cos 2 θ )l
l +m
(1 − x 2 )l
where x = cos θ
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Graphical Visualization of Spherical Harmonics
It is of significant importance to visualize the distribution patterns of spherical
harmonics in a pedagogical way. Before implementing the visualization of
spherical harmonics, an explicit polynomial expression for the associated
Legendre functions is essential.
From
⎡ − h2 ⎛ ∂2 1 ∂
⎤
∂2 ⎞
1 ∂2
∂ 1
2 2
⎜
⎟
h
+
+
+
+
i
ω
+
m
ω
r
⎢
⎥ ψ ( r,θ , z ) = Eψ ( r,θ , z ) ,
L
L
⎜ ∂ r2 r ∂ r r2 ∂ θ 2 ∂ z2 ⎟
m
2
θ
2
∂
⎝
⎠
⎣
⎦
the expression of the associated Legendre functions for is given by
(−1) l
Pl ( x) = l 1 − x 2
2 l!
m
(
2006 Quantum Mechanics
)
m/2
d l +m
(1 − x 2 ) l
l +m
dx
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Graphical Visualization of Spherical Harmonics
To obtain an explicit expression, we expand
d l +m
d l +m
2 l
(1 − x ) = l + m (1 − x) l (1 + x) l
l +m
dx
dx
l +m
=∑
k =0
l + m− k
⎤
⎛ l + m ⎞⎡ d k
l ⎤⎡ d
⎟⎟ ⎢ k (1 − x) ⎥ ⎢ l + m−k (1 + x) l ⎥
⎜⎜
⎝ k ⎠ ⎣ dx
⎦ ⎣ dx
⎦
Although the sum extends for 0 ≤ k ≤ l + m , the term in the first square
parentheses vanishes for k > l and
parentheses vanishes for
l +m−k >l
the
term
in
the
second
square
. Therefore, the sum is taken only
for m ≤ k ≤ l .
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Graphical Visualization of Spherical Harmonics
Then
⎛l + m⎞
⎟⎟
⎜⎜
⎝ k ⎠
l + m−k
⎡ dk
l⎤ ⎡ d
l⎤
⎢ k (1 − x) ⎥ ⎢ l + m −k (1 + x) ⎥
⎣ dx
⎦ ⎣ dx
⎦
l
⎤ ⎡ l!
⎤ ⎡ l!
⎤ l − m ⎛ l + m ⎞ ⎡ (−1) m + s l!
⎛ l + m ⎞ ⎡ (−1) k l!
⎤
⎟⎟ ⎢
⎟⎟ ⎢
(1 − x)l − k ⎥ ⎢
(1 + x) k − m ⎥ = ∑ ⎜⎜
(1 − x)l −m − s ⎥ ⎢ (1 + x) s ⎥
= ∑ ⎜⎜
⎦
k = m ⎝ k ⎠ ⎣ (l − k )!
⎦ s =0 ⎝ m + s ⎠ ⎣ (l − m − s )!
⎦ ⎣ s!
⎦ ⎣ (k − m)!
l −m
l −m
⎡ (−1) m+ s l!
⎛ l + m ⎞ ⎡ (−1) m+ s l!
1
1
(l + m)!
l − s ⎤ ⎡ l!
s+m ⎤
l − s ⎤ ⎡ l!
s+m ⎤
⎜
⎟
(
1
)
(
1
)
(
1
)
x
x
−
+
=
x
=
−
∑
∑
⎢
⎥
⎢
⎥ ⎢ (1 + x) ⎥
m
m
⎜ m + s ⎟ (l − m − s )!
⎢ s!
⎥
2
2
⎦ 1− x
⎦
s =0 (l − s )!( m + s )! ⎣ (l − m − s )!
s =0 ⎝
1− x
⎠⎣
⎦⎣
⎦ ⎣ s!
l +m
d l +m
2 l
(
1
)
x
−
=
∑
dx l + m
k =0
(
=
)
(
l −m
1
(1 − x ) ∑
2 m
s =0
)
⎡ (−1) m+ s l!
⎤ ⎡ l!
(l + m)!
⎤
(1 − x)l − s ⎥ ⎢ (1 + x) s + m ⎥
⎢
(l − s )!(m + s )! ⎣ (l − m − s)!
⎦
⎦ ⎣ s!
(−1) m (l + m)! l − m
(l − m)! ⎡ (−1) s l!
l!
l −s ⎤ ⎡
s+m ⎤
(
1
)
(
1
)
=
x
+
−
x
∑
⎥
⎢
⎢
⎥
m
1 − x 2 (l − m)! s =0 (l − m − s)! s! ⎣ (l − s )!
⎦
⎦ ⎣ (m + s )!
(
=
=
)
(−1) m (l + m)! l − m
∑
m
1 − x 2 (l − m)! s =0
(
)
⎤ ⎡ d l −m− s
⎤
⎛l − m⎞ ⎡ d s
⎜⎜
⎟⎟ ⎢ s (1 − x)l ⎥ ⎢ l − m− s (1 + x)l ⎥
⎝ s ⎠ ⎣ dx
⎦ ⎣ dx
⎦
(−1) m (l + m)! d l − m
(1 − x 2 )l
2 m (l − m)! dx l − m
1− x
(
)
2006 Quantum Mechanics
Prof. Y. F. Chen
A Charged Particle in an Electromagnetic Field
Graphical Visualization of Spherical Harmonics
With previous page, the alternative definition of the associated Legendre
functions is given by
(−1) l
Pl ( x) = l 1 − x 2
2 l!
m
(
)
|m|/ 2
d l +|m|
(1 − x 2 ) l
l +|m|
dx
(−1) l + m (l + | m |)!
1− x2
=
l
2 l! (l − | m |)!
(
)
−|m|/ 2
d l −|m|
(1 − x 2 ) l
l −|m|
dx
With above equation, the alternative expression for the spherical harmonics
with
m≥0
Yl m (θ , φ ) = (−1) m
is given by
2l + 1 (l − m)! i mφ m
e Pl (cosθ )
4π (l + m)!
(−1)l +m
=
2l l!
(−1)l
= l
2 l!
2l + 1 (l − m)! i mφ m
d l +m
l
2
θ
(
1
cos
)
−
e sin θ
4π (l + m)!
d (cosθ )l +m
2l + 1 (l + m)! i mφ −m
d l −m
(1 − cos 2 θ )l
e sin θ
l −m
4π (l − m)!
d (cosθ )
2006 Quantum Mechanics
Prof. Y. F. Chen