Download Gauge invariance and the Aharonov-Bohm effect

Gauge invariance and the Aharonov-Bohm effect∗
Shreyas Patankar
Chennai Mathematical Institute
1
Introduction
The scalar and vector potentials are introduced in classical mechanics as solutions to the Maxwell’s equations. However,
the potentials (unlike the fields) are not completely deterministic, they are fixed only up to a “gauge”, and all electromagnetic observables, (the Lagrangian, momentum, etc.) that depend explicitly on the potentials satisfy conditions for
Gauge invariance.
In quantum mechanics, it is possible to get transformations between gauges for state kets by means of a unitary operator,
that is, we can define for each gauge transformation a unitary operator that acts on state kets. We shall try to obtain
a form for this unitary operator, and by suitably transforming the Hamiltonian, show that the Schrödinger equation is
gauge invariant.
In electrodynamics, there is no way to detect potentials in the absence of fields. However, due to the potential dependance
of quantum operators, we can design experimental setups in which we shall observe physical effects of the presence of
potentials, in the setup of the Aharonov - Bohm experiment of shift in interference pattern. We shall also show that the
vector potential can cause energy eigenvalues in bounded potential wells to shift.
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Gauge invariance in classical electromagnetism
We know that[1, p315] the potential solutions to the Maxwell’s equations can be written as
~
~ = −∇φ − 1 ∂ A
E
c ∂t
(1)
~ =∇×A
~
B
(2)
~ are the scalar and vector potentials respectively. Now, it can be easily shown that the fields remain invariant
Where φ, A
if the potentials undergo transformation
∂γ
φ→φ−
(3)
∂t
A → A + ∇γ
(4)
where γ is an arbitrary scalar field. γ is called the electromagnetic gauge parameter or simply as the electromagnetic gauge
The electromagnetic field Lagrangian is given by[1, p317]
L(r, v, t) =
h
i
1
~ t)
mv2 − q U (r, t) − v · A(r,
2
(5)
we can write the momentum conjugate to the coördinate xi as
~ t)
p = ∇v L = mv + q A(r,
(6)
which is evidently, not the same as the mechanical momentum π = mv
Thus we can write,
r0 (t)
0
p (t)
∗
=
r(t)
=
p(t) + q∇γ(r, t)
Term paper for Quantum Mechanics II
1
(7)
2.1
Definition
1. A true physical quantity associated with the system is a quantity whose value at any time does not depend (for given
motion of system) on the electromagnetic gauge γ
2. A non-physical quantity is a quantity whose value is a function of the electromagnetic gauge γ
Thus, we can see that the conjugate momentum p is a non-physical quantity whereas r, π are true physical quantities.
Now suppose F(r, p, t) denotes a physical quantity such that the form of F is independant of the gauge γ. Then, if
we shift to a different gauge γ 0 , F(r, p, t) → F(r0 , p0 , t). As p0 explicitly depends on the gauge transformation γ → γ 0 ,
F(r, p, t) 6= F(r, p0 , t), and hence, F is not a true physical quantity.
Thus, for a function to describe a true physical quantity, its form Gγ should explicitly depend on the electromagnetic
gauge, such that Gγ (r, p, t) = Gγ 0 (r, p0 , t)
Using (4),(6), we get
Gγ (r, p, t) = Gγ 0 (r, p + ∇γ, t)
(8)
which must be satified for all time t.
2.2
Examples
We shall give some examples of the (gauge-dependant) function Gγ (r, p, t)
~ Evidently, πγ (r, p, t) = πγ 0 (r, p0 , t) making π a true
1. Consider the mechanical momentum πγ (r, p, t) = p − q A.
physical quantity.
2. Similarly, the kinetic energy T , which is a function of the mechanical momentum π is also a true physical quantity,
πγ20
πγ2
=
= Tγ
for Tγ 0 =
2m
2m
~ always defines a true physical
3. In general, any physical quantity tha takes the functional form Gγ (r, p, t) = F (p − q A)
quantity.
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Gauge invariance in quantum mechanics
In classical mechanics, we saw that the value of the dynamical variables change with respect to the employed electromagnetic gauge. We shall now show that in quantum mechanics, the state vector |ψi of a system is transformed under a
unitary transformation that corresponds to the change of gauge.
If R and P are the operators corresponding to position and (conjugate) momentum, then they satisfy
[X, Px ] = [Y, Py ] = [Z, Pz ] = i~
(9)
∂
|Xi i. Evidently, these functional forms are gauge independant.
∂xi
Now consider the operator associated with the mechanical momentum, Πγ = P − qA(R, t). Obviously, if the gauge is
changed, we have Πγ 0 = P − qA0 (R, t) where Πγ 6= Πγ 0
In the position basis, Xi |Xi i = xi |Xi i, and Pi |Xi i = −i~
Using (7), and writing expectation values, we have,
hψ 0 (t)|R0 |ψ 0 (t)i = hψ(t)|R|ψ(t)i
hψ 0 (t)|P0 |ψ 0 (t)i = hψ(t)|P + q∇γ(R, t)|ψ(t)i
(10)
This is possible if and only if |ψi, |ψ 0 i are distinct. Thus, suppose we have a unitary transformation Tγ satisfying,
|ψ 0 (t)i = Tγ (t)|ψ(t)i
(11)
From (9) we get,
Tγ† (t) R Tγ (t)
= R
Tㆠ(t)
= P + q∇γ(R, t)
P Tγ (t)
2
(12)
and hence, Tγ , R commute
From (12), we have [P, Tγ ] = qTγ ∇γ, or −i~∇Tγ = qTγ ∇γ
that is, −i~
∂Tγ
∂γ
∂Tγ
∂γ
∂Tγ
= qTγ
, and using chain rule, −i~
∂γ∂xi = qTγ
implying −i~
= qTγ
∂xi
∂xi
∂γ
∂xi
∂γ
Which has most general solution
q
Tγ (t) = ei ~ γ(R,t)
3.1
(13)
Time evolution
Let us now consider the Schrödinger’s equation under gauge transformation. Let us write
d
|ψ(t)i = H(t)|ψ(t)i
dt
(14)
d 0
|ψ (t)i = Hγ (t)|ψ 0 (t)i
dt
(15)
i~
Now suppose we shift to a new gauge γ. Then,
i~
Expanding the left hand side of the equation we get (using (11))
i~
d 0
d
|ψ (t)i = i~ (Tγ |ψ(t)i)
dt
dt
dTγ
d|ψ(t)i
= i~
|ψ(t)i + i~Tγ
dt
dt
Now, using (13), (14) we have,
d
i~ |ψ 0 (t)i =
dt
−q
∂
γ(R, t) Tγ (t)|ψ(t)i + Tγ (t)H(t)|ψ(t)i
∂t
(16)
We know that Tγ is unitary. Thus, let us write |ψ(t)i = Tγ† |ψ 0 (t)i, and let
H̃(t) = Tγ H(t)Tγ†
(17)
Then,
i~
d 0
|ψ (t)i = −q
dt
∂
γ(R, t) − H̃ |ψ 0 (t)i
∂t
(18)
and hence, we get the gauge - transformed Hamiltonian Hγ as
Hγ = H̃ −
4
∂
γ(R, t)
∂t
(19)
The Aharonov - Bohm effect
In classical electrodynamics, both the Maxwell’s equations and the Lorentz force law depend only on the electric fields
~ B
~ and the potentials A,
~ φ appear only while writing the solutions to the Maxwell’s equations, in the form
E,
~ = −∇φ,
E
~ =∇×A
~
B
(20)
Thus, classically, we do not expect the potentials to affect observables themselves, that is, we do not expect a potential
in absence of a field to affect the motion of a charged particle. However, in the Hamiltonian formalism, the Hamiltonian
of a charged particle explicitly depends on the scalar and vector potentials, and we shall see that this leads to observable
effects of potentials in quantum mechanics[2].
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4.1
Interference Pattern
Consider current flowing through a closely wound solenoid of radius R cenetered at the origin and with axis along z-axis[3].
Thus,
have a situation
Z we
I where the magnetic field B0 is confined to a circular region of space. However, using symmetry
~
~ · d~s, we can select a gauge in which, A
~ ∼ B0 × r̂
and
∇ × A dS =
A
2πr
S
∂S
Now suppose we have a coherent beam of electrons that is split into two parts that goes around two sides of the solenoid.
The solenoid can be shielded by a plate casting a “shadow”. The beams are then made to interfere at a point F beyond
the solenoid.
1
The Hamiltonian for a free particle in magnetic field is H =
2m
~
A
p−e
c
~
A
p−e
c
1
∂ψ(x, t)
=
i~
∂t
2m
!2
, we can write the Schrödinger equation as
!2
ψ(x, t)
(21)
Let us consider a suitably localized wave packet away from the solenoid. Suppose that initially the magnetic field is
switched off. Then we can write the free particle wave function[4]
ψ0 (x, t) = ψl (x, t) + ψr (x, t)
(22)
where ψl , ψr are functions such that they vanish on the left and right half planes respectively, that is, they describe
particles travelling on either side of the solenoid toward F .
Now suppose we turn on the magnetic field B0 . Then, it can be shown[3] that the corresponding solutions for the new
Sl
Schrödinger equation are just ψl (x, t)ei ~ and ψr (x, t)ei
Sl (x)
Sr
~
=
in either region respectively. Here,
Z x
~ 0 ) · dx0
A(x
(23)
lef t
x
Z
Sr (x)
~ 0 ) · dx0
A(x
=
(24)
right
Sl
Sr
Thus, the total wave function can be given by ψ(x, t) = ψl (x, t)ei ~ + ψr (x, t)ei ~
Z x
Z x
I
0
0
0
0
~
~
~ · d~s = ΦB , the flux of the magnetic field. Thus we may write
Now, Sl − Sr =
A(x ) · dx −
A(x ) · dx = A
lef t
the wave function
right
Sr
ΦB
ψ(x, t) = ψl (x, t)ei ~ + ψr (x, t) ei ~
(25)
Thus, the magnetic flux inside the solenoid causes an additional phase difference in the interfering wave functions, which
can be observed in the interference pattern.
4.2
Cylindrical shell
Consider particle in a cylindrical shell of inner and outer radii ρa , ρb respectively and height l
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Then, Schrödinger’s equation is
~2 2
∇ ψ = Eψ
(26)
2m
Writing in cylindrical coördinates and assuming variable seperability, we can write three seperate equations:
∂R
R ∂2F
1 ∂
= ER(ρ)F (φ)
(27)
ρ
+ 2
F
ρ ∂ρ
∂ρ
ρ ∂φ2
∂2Y
− 2 = EY
(28)
∂z
Subject to boundary conditions R(ρa ) = R(ρb ) = 0, Y (0) = Y (l) = 0 and F (θ) = F (θ + 2nπ) for all θ.
∂2F
Putting
= −m2 F (φ), we get Bessel equation for R:
2
∂φ
2
1 ∂
m
∂R
ρ
+R
−
E
= 0. Only solutions with integer m are meaningful, and hence, quantization condition can
ρ ∂ρ
∂ρ
ρ2
be got from Jm (Eρa ) = 0 = Jm (Eρb )
−
Now suppose we turn on a constant magnetic
is
B0 in the inner cavity. Thus, although magnetic field inside well
2field
Bρa
eBρ2a
0
~
φ̂, which leads to change in conjugate momentum pφ = pφ −
, or
still zero, there is a vector potential A =
2ρ
2c
∂
iC
∂
+
=
0
∂φ
∂φ
~
Now, the transformed Schrödinger equation is
C ∂F
C2
1 ∂
∂R
R ∂2F
+
2i
−
F
F
ρ
+ 2
= ER(ρ)F (φ)
ρ ∂ρ
∂ρ
ρ
∂φ2
~ ∂φ
~2
(29)
and we can put
∂2F
C ∂F
C2
+
2i
−
F
= −m2 F (φ)
(30)
∂φ2
~ ∂φ
~2
r
C
C2
Which has solutions F (φ) = eiζφ , where ζ = ± 2 2 + m2 . For meaningful solutions, we need ζ to be integer valued.
~
~
Thus, the new quantization condition is Jm0 (Eρa ) = 0 = Jm0 (Eρb ) where, m0 is not integral but satisfies the condtion
that ζ is integral. Thus, we have new zeroes for the Bessel functions and hence, we have a shift in the energy spectrum.
We emphasize that we have a shift in the energy spectrum although the electrons are never in “contact” with the actual
magnetic field.
References
[1] Claude Cohen-Tannoudji (1977), Quantum Mechanics, vol. 1, John Wiley & Sons
[2] J. J. Sakurai (1994), Modern Quantum Mechanics, Pearson Education, Inc.
[3] Y. Aharonov, D. Bohm (1959), Significance of electromagnetic potentials in quantum theory, Physical Review 115
[4] Frank Porter’s Notes on Quantum Mechanics, Caltech Physics Department
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