Download Propagator of a Charged Particle with a Spin in Uniform Magnetic

Lett Math Phys (2008) 84:159–178
DOI 10.1007/s11005-008-0239-6
© Springer 2008
Propagator of a Charged Particle with a Spin
in Uniform Magnetic and Perpendicular
Electric Fields
RICARDO CORDERO-SOTO, RAQUEL M. LOPEZ, ERWIN SUAZO,
and SERGEI K. SUSLOV
Department of Mathematics and Statistics, Arizona State University, Tempe,
AZ 85287–1804, USA. e-mail: [email protected]; [email protected];
[email protected]; [email protected]
URL: http://hahn.la.asu.edu/∼suslov/index.html
Received: 12 February 2008 / Revised: 30 March 2008 / Accepted: 4 April 2008
Published online: 15 May 2008
Abstract. We construct an explicit solution of the Cauchy initial value problem for the
time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding
Green function (propagator) is given in terms of elementary functions and certain integrals
of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent
frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and
some special and limiting cases are outlined.
Mathematics Subject Classification (2000). Primary 81Q05, 35C05, 42A38.
Keywords. The Cauchy initial value problem, Schrödinger equation in electromagnetic
field, nonlinear Schrödinger equation, forced harmonic oscillator, Landau levels, Green
function, propagator, Riccati differential equation, Fourier transform, Bessel functions.
1. Introduction
The time-dependent Schrödinger equation
i
∂ψ
= H (t)ψ
∂t
(1.1)
can be solved using the time evolution operator given formally by
⎞⎞
⎛
⎛
t
i
U (t, t0 ) = T ⎝exp ⎝−
H (t ) dt ⎠⎠ ,
(1.2)
t0
We dedicate this paper to the memory of Professor Basil Nicolaenko for his significant contributions
to the area of nonlinear partial differential equations, applied mathematics, and related topics.
160
RICARDO CORDERO-SOTO ET AL.
where T is the time ordering operator which orders operators with larger times to
the left [4,13]. This unitary operator takes a state at time t0 to a state at time t,
so that
ψ (x, t) = U (t, t0 ) ψ (x, t0 ) .
(1.3)
The simplicity of these formulas is deceptive, since the time evolution operator
can be found explicitly as certain integral operator only in a few special cases. An
important example of this source is the forced harmonic oscillator originally considered by Richard Feynman in his path integrals approach to the nonrelativistic
quantum mechanics [8–12]; see also [23]. Since then this problem and its special
and limiting cases were discussed by many authors; see Refs. [5,14,17,24,26,44] for
the simple harmonic oscillator and Refs. [2,7,16,29,36] for the particle in a constant external field and references therein. It is worth noting that an exact solution
of the n-dimensional time-dependent Schrödinger equation for certain modified
oscillator is found in Ref. [25].
In this Letter we construct the time evolution operator explicitly in a general
case of the one-dimensional Schrödinger equation when the Hamiltonian is an
arbitrary quadratic form of the operator of coordinate and the operator of linear
momentum; see Equation (2.1) below. In this approach, all exactly solvable models, that have been cited above, are classified in terms of elementary solutions of a
characterization equation given by Equation (2.12) below. A particular solution of
the corresponding nonlinear Schrödinger equation is obtained in a similar fashion.
By separation of variables, we apply this method to another classical problem—
the motion of a charged particle with a spin in uniform magnetic and perpendicular electric fields that are changing with time. The corresponding Green function
(or Feynman’s propagator) is found as an elementary function of certain integrals
of our characteristic function and the electromagnetic fields; see Equation (6.14)
below. Special cases of constant and linear magnetic fields are discussed as examples. Moreover, these explicit solutions can also be useful when testing numerical
methods of solving the time-dependent Schrödinger equation.
2. Solution of a Cauchy Initial Value Problem
The fundamental solution of the time-dependent Schrödinger equation with the
quadratic Hamiltonian of the form
∂2 ψ
∂ψ
∂ψ
∂ψ
2
= −a(t) 2 + b(t)x ψ − i c(t)x
+ d(t)ψ − f (t)xψ + ig(t)
,
i
∂t
∂x
∂x
∂x
(2.1)
where a(t), b(t), c(t), d(t), f (t), and g(t) are given real-valued functions of time
t only, can be found with the help of a familiar substitution
ψ = Aei S = A(t)ei S(x,y,t) ,
1
A = A(t) = √
2πiµ(t)
(2.2)
161
PROPAGATOR OF A PARTICLE WITH A SPIN
with
S = S (x, y, t) = α(t)x 2 + β(t)x y + γ (t)y 2 + δ(t)x + ε(t)y + κ(t),
(2.3)
where α(t), β(t), γ (t), δ(t), ε(t), and κ(t) are differentiable real-valued functions
of time t only. Indeed,
2
∂S
∂S
∂S
− bx 2 + f x + (g − cx)
= −a
(2.4)
∂t
∂x
∂x
by choosing
∂2 S
µ
= a 2 + d = 2α(t)a(t) + d(t).
2µ
∂x
(2.5)
Equating the coefficients of all admissible powers of x m y n with 0 ≤ m + n ≤ 2, gives
the following system of ordinary differential equations
dα
+ b(t) + 2c(t)α + 4a(t)α 2 = 0,
(2.6)
dt
dβ
+ (c(t) + 4a(t)α(t)) β = 0,
(2.7)
dt
dγ
(2.8)
+ a(t)β 2 (t) = 0,
dt
dδ
+ (c(t) + 4a(t)α(t)) δ = f (t) + 2α(t)g(t),
(2.9)
dt
dε
= (g(t) − 2a(t)δ(t)) β(t),
(2.10)
dt
dκ
= g(t)δ(t) − a(t)δ 2 (t),
(2.11)
dt
where the first equation is the familiar Riccati nonlinear differential equation; see,
for example, [15,34,45] and references therein. Substitution of Equation (2.5) into
Equation (2.6) results in the second-order linear equation
µ − τ (t)µ + 4σ (t)µ = 0
(2.12)
with
τ (t) =
a
− 2c + 4d,
a
σ (t) = ab − cd + d 2 +
d
2
a d −
,
a
d
(2.13)
which must be solved subject to the initial data
µ(0) = 0,
µ (0) = 2a(0) = 0
(2.14)
in order to satisfy the initial condition for the corresponding Green function; see
the asymptotic formula (2.22) below. We shall refer to Equation (2.12) as the characteristic equation and its solution µ(t), subject to Equation (2.14), as the characteristic function. As the special case (2.12) contains the generalized equation of
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RICARDO CORDERO-SOTO ET AL.
hypergeometric type, whose solutions are studied in detail in [31]; see also [1,30,
43,45].
Thus, the Green function (fundamental solution or propagator) is explicitly
given in terms of the characteristic function
1
i
e
ψ = G (x, y, t) = √
2πiµ(t)
α(t)x 2 +β(t)x y+γ (t)y 2 +δ(t)x+ε(t)y+κ(t)
.
(2.15)
Here
d(t)
1 µ (t)
−
,
4a(t) µ(t) 2a(t)
⎛
⎞
t
1
exp ⎝− (c(τ ) − 2d(τ )) dτ ⎠ ,
β(t) = −
µ(t)
α(t) =
0
⎛
a(t)
exp ⎝−2
γ (t) =
µ(t)µ (t)
t
−4
0
⎛
(2.16)
(2.17)
⎞
t
(c(τ ) − 2d(τ ))dτ ⎠ −
0
⎛
⎞⎞
τ
a(τ )σ (τ ) ⎝
exp ⎝−2 (c (λ) − 2d (λ)) dλ⎠⎠ dτ,
(µ (τ ))2
⎛
1
exp ⎝−
δ(t) =
µ(t)
0
t
⎞
(c(τ ) − 2d(τ )) dτ ⎠
0
t
⎛
exp ⎝
0
τ
⎞
(c (λ) − 2d (λ)) dλ⎠ ×
0
g(τ ) d(τ )
g(τ ) µ(τ ) +
µ (τ ) dτ,
a(τ )
2a(τ )
⎛
⎞
t
2a(t)
ε(t) = − δ(t) exp ⎝− (c(τ ) − 2d(τ )) dτ ⎠ +
µ (t)
×
f (τ ) −
0
t
+8
0
t
+2
0
κ(t) =
⎛
a(τ )σ (τ )
(µ (τ ))2
exp ⎝−
⎛
a(τ )
exp ⎝−
µ (τ )
a(t)µ(t) 2
δ (t) − 4
µ (t)
t
−2
0
t
0
τ
0
(2.19)
⎞
(c (λ) − 2d (λ)) dλ⎠ (µ(τ )δ(τ )) dτ +
0
τ
(2.18)
⎞
d(τ )
g(τ ) dτ, (2.20)
(c (λ) − 2d (λ)) dλ⎠ f (τ ) −
a(τ )
a(τ )σ (τ )
(µ (τ ))2
a(τ )
(µ(τ )δ(τ ))
µ (τ )
(µ(τ )δ(τ ))2 dτ −
f (τ ) −
d(τ )
g(τ ) dτ
a(τ )
(2.21)
PROPAGATOR OF A PARTICLE WITH A SPIN
163
with δ(0) = g(0)/ (2a(0)), ε(0) = −δ(0), and κ(0) = 0. We have used integration by
parts to resolve the singularities of the initial data. Then the corresponding asymptotic formula is
1
g(0)
ei S(x,y,t)
(x − y)2
∼√
exp i
exp i
G (x, y, t) = √
(x − y)
4a(0)t
2a(0)
2πiµ(t)
4πia(0)t
(2.22)
as t → 0+ . Notice that the first term on the right-hand side is a familiar free particle propagator [cf. Equation (3.1) below].
By the superposition principle, we obtain an explicit solution of the Cauchy initial value problem
i
∂ψ
= H (t)ψ,
∂t
ψ (x, t)|t=0 = ψ0 (x)
(2.23)
on the infinite interval −∞ < x < ∞ with the general quadratic Hamiltonian as in
Equation (2.1) in the form
∞
G (x, y, t) ψ0 (y) dy.
ψ (x, t) =
(2.24)
−∞
This yields the time evolution operator (1.2) explicitly as an integral operator.
Properties of similar oscillatory integrals are discussed in [41].
3. Some Special Cases
Now let us consider several elementary solutions of the characteristic equation
(2.12); more complicated cases may include special functions, like Bessel, hypergeometric or elliptic functions [1,31,33,45]. Among special cases of our general expressions for the Green function (2.15)–(2.21) are the following:
When a = /2m, b = c = d = f = g = 0, and µ = 0, µ = t/m, one gets
G (x, y, t) =
im (x − y)2
m
exp
,
2πit
2t
(3.1)
which is the free particle propagator [12].
For a particle in a constant external field, where a = /2m, b = c = d = g = 0 and
f = F/ = constant, µ = t/m, the propagator is given by
G (x, y, t) =
im (x − y)2
i F (x + y)
i F2 3
m
exp
t−
t .
exp
2πit
2t
2
24m
(3.2)
164
RICARDO CORDERO-SOTO ET AL.
This case was studied in detail in [2,7,12,16,29,36]. Here we have corrected a typo
in [12]; see [42] for a complete list of known errata in the Feynman and Hibbs
book.
The simple harmonic oscillator with a = /2m, b = mω2 /2, c = d = f = g = 0 and
µ + ω2 µ = 0, µ = (/mω) sin ωt has the familiar propagator of the form
imω 2
mω
exp
x + y 2 cos ωt − 2x y ,
(3.3)
G (x, y, t) =
2πi sin ωt
2 sin ωt
which is studied in detail at [5,14,17,24,26,44]. For an extension to the case of the
forced harmonic oscillator including an extra velocity-dependent term and a timedependent frequency, see [8,9,12,23].
Furthermore, an exact solution of the n-dimensional time-dependent Schrödinger
equation for certain modified oscillator is found in [25]. In the one-dimensional
case we get functions a = cos2 t, b = sin2 t, c = 2d = sin 2t and our characteristic
Equation (2.12) takes the form
µ + 2 tan t µ − 2µ = 0,
(3.4)
whose elementary solution µ = cos t sinh t + sin t cosh t satisfies the initial conditions
(2.14). Further, the corresponding propagator is given by
1
×
G (x, y, t) = 2πi (cos t sinh t + sin t cosh t)
x 2 − y 2 sin t sinh t + 2x y − x 2 + y 2 cos t cosh t
× exp
,
2i (cos t sinh t + sin t cosh t)
(3.5)
which was found in [25] as the special case n = 1 of a general n-dimensional expansion of the Green function in hyperspherical harmonics. We have showed that
(3.5) is a generalization of the propagator for the simple harmonic oscillator; see
Ref. [25] for more details.
4. A Particular Solution of the Nonlinear Schrödinger Equation
The method of solving the Equation (2.1) can be extended to the nonlinear
Schrödinger equation with a general quadratic Hamiltonian of the form
∂2 ψ
∂ψ
∂ψ
2
= − a(t) 2 + b(t)x ψ − i c(t)x
+ d(t)ψ −
i
∂t
∂x
∂x
∂ψ
+ h(t) |ψ|2s ψ,
− f (t)xψ + ig(t)
s ≥ 0,
(4.1)
∂x
where a(t), b(t), c(t), d(t), f (t), g(t), and h(t) are certain functions of time t only.
A substitution of
eiφ
ei S(x,y,t) ,
ψ = ψ (x, t) = √
µ(t)
φ = constant,
(4.2)
165
PROPAGATOR OF A PARTICLE WITH A SPIN
where S = S (x, y, t) = α(t)x 2 + β(t)x y + γ (t)y 2 + δ(t)x + ε(t)y + κ(t) with the same
relation (2.5), results in a modified system (2.6)–(2.11) below:
dα
+ b(t) + 2c(t)α + 4a(t)α 2 = 0,
dt
dβ
+ (c(t) + 4a(t)α(t)) β = 0,
dt
dγ
+ a(t)β 2 (t) = 0,
dt
dδ
+ (c(t) + 4a(t)α(t)) δ = f (t) + 2α(t)g(t),
dt
dε
= (g(t) − 2a(t)δ(t)) β(t),
dt
dκ
h(t)
= g(t)δ(t) − a(t)δ 2 (t) − s ,
dt
µ (t)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
where only the last equation has an extra term which corresponds to the nonlinear
term in the original Schrödinger Equation (4.1). Therefore Equations (2.12)–(2.13)
and (2.16) solve once again the Riccati equation (4.3), however; in this case, we
would like to use nonsingular initial conditions
µ(0) = 0,
µ (0) = 2 (2α(0)a(0) + d(0)) µ(0).
(4.9)
With these conditions, our system (4.3)–(4.8) can be solved again in terms of the
characteristic function µ(t), thus giving us a particular solution of the nonlinear
Schrödinger equation (4.1) in the form (4.2), corresponding to the initial data
eiφ
i
e
ψ0 (x) = ψ (x, t)|t=0 = √
µ(0)
α(0)x 2 +β(0)x y+γ (0)y 2 +δ(0)x+ε(0)y+κ(0)
.
(4.10)
In the simplest case,
i
∂ψ
1 ∂2 ψ
=−
+ h |ψ|2s ψ,
∂t
2 ∂x 2
h = constant, s ≥ 0,
(4.11)
the solution of the characteristic equation µ = 0 is µ(t) = µ0 + tµ1 , µ0 > 0 and the
coefficients of the quadratic form are given by
µ0 β0
µ0 δ0
µ1
, β(t) =
, δ(t) =
,
2 (µ0 + tµ1 )
µ0 + tµ1
µ0 + tµ1
µ0 β02 t
µ0 β0 δ0 t
γ (t) = γ0 −
, ε(t) = ε0 −
,
2 (µ0 + tµ1 )
µ0 + tµ1
µ0 δ02 t
h
κ(t) = κ0 −
− ξs (t)
2 (µ0 + tµ1 ) µ1
α(t) =
(4.12)
(4.13)
(4.14)
166
with
RICARDO CORDERO-SOTO ET AL.
⎧
1 1−s
⎪
⎪
, when s = 1,
− µ1−s
⎨ (1 − s) (µ0 + tµ1 )
0
ξs (t) =
tµ
⎪
⎪
⎩ ln 1 + 1 ,
when s = 1.
µ0
(4.15)
Now, the limiting case where µ1 → 0 with µ0 > 0 is given by
1
i β x y+ γ0 −β02 t/2 y 2 +δ0 x+(ε0 −β0 δ0 t)y+κ0 −δ02 t/2−ht/µs0
e 0
.
ψ=√
µ0
√
Then |ψ| = 1/ µ0 is bounded at all times. Yet, when µ1 = 0, one gets
|ψ (x, t)| = √
1
,
µ0 + tµ1
t ≥ 0,
(4.16)
(4.17)
which is bounded if µ0 > 0 and µ1 > 0, and blows up at a finite time t0 = −µ0 /µ1
if µ1 < 0. See also [38] for another solution.
The same method shows that the Cauchy initial value problem
∂ψ 1 ∂2 ψ
+
= h |ψ|2s ψ,
h = constant, s ≥ 0,
∂t
2 ∂x 2
i (x − y)2
1
exp
ψ|t=0 = δε (x − y) = √
, ε>0
2ε
2πiε
i
(4.18)
has the classical solution of the form
i (x − y)2
ih
1
−
ψ = G ε (x, y, t) = √
exp
χs (t) ,
2 (t + ε)
(2π )s
2πi (t + ε)
(4.19)
where
⎧
(t + ε)1−s − ε1−s
⎪
⎪
, when s = 1,
⎨
1−s
χs (t) =
⎪
t
⎪
⎩ ln 1 +
,
when s = 1
ε
(4.20)
with χs (0) = 0. Here ψ|t=0 = G ε (x, y, 0) = δε (x − y) → δ (x − y) as ε → 0+ in the distributional sense (see also [3,21]).
In a similar fashion, the nonlinear Schrödinger equation
2
∂ψ
∂ψ
2
2 ∂ ψ
2
= − cos t
+ψ +
+ sin t x ψ − i sin t cos t 2x
i
∂t
∂x
∂x 2
+2 cos t sinh t |ψ|2s ψ
(4.21)
(cf. [25]) has a particular solution of the form
1
i
ψ = ψ (x, t) = √
e
µ(t)
α(t)x 2 +β(t)x y+γ (t)y 2 +κ(t)
,
(4.22)
167
PROPAGATOR OF A PARTICLE WITH A SPIN
where µ(t) = cos t cosh t + sin t sinh t and the coefficients are given by
α(t) =
cos t sinh t − sin t cosh t
,
2 (cos t cosh t + sin t sinh t)
(4.23)
β(t) =
1
,
cos t cosh t + sin t sinh t
(4.24)
γ (t) = −
⎧
⎨
cos t sinh t + sin t cosh t
,
2 (cos t cosh t + sin t sinh t)
(4.25)
(cos t cosh t + sin t sinh t)1−s − 1
,
κ(t) =
1−s
⎩
− ln (cos t cosh t + sin t sinh t) ,
−
when s = 1,
(4.26)
when s = 1.
Thus, initial function is the standing wave ψ0 (x) = ψ (x, t)|t=0 = ei x y . These exact
solutions may be of interest in a general treatment of the nonlinear Schrödinger
equation (see [18,20,28,35,37,39,40,46] and references therein).
5. Motion in Perpendicular Magnetic and Electric Fields
A particle with a spin s also has a magnetic momentum µ with the operator
µ = µ
s/s,
(5.1)
where s is the spin operator and µ is a constant characterizing the particle. For the
motion of a charged particle in a uniform magnetic field H and an electric field E,
both which are perpendicular to each other, the three-dimensional time-dependent
Schrödinger equation
i
∂ = H
∂t
(5.2)
has a Hamiltonian operator as in [22], namely,
eH 2
1 2
1 2 µ
= 1 H
px +
y +
p +
p − sz H − y F,
2m
c
2m y 2m z s
(5.3)
where p = −i∇ is the linear momentum operator. The corresponding vector and
scalar potentials are defined up to the gauge transformation. We use the original
choice [22] for the vector potential A = −y H (t) ex , H (t) > 0 and add a linear scalar
potential ϕ = − (F(t)/e) y (see also [23]). Then the uniform magnetic field H and
the corresponding perpendicular electric field E are given by
H = ∇ × A = H (t) ez ,
E = −∇ϕ −
1 ∂A y dH (t)
F(t)
=
ex +
ey ;
c ∂t
c dt
e
(5.4)
see [19] for more details.
Since (5.3) does not contain the other components of the spin, the operator sz
and the z-component of the spin is conserved.
commutes with the Hamiltonian H
168
RICARDO CORDERO-SOTO ET AL.
Thus the operator sz can be replaced by its eigenvalue sz = σ in the Hamiltonian
(5.3), so we have
= 1
H
2m
px +
eH
y
c
2
+
1 2
1 2 µσ
p +
p −
H − yF
2m y 2m z
s
(5.5)
with σ = −s, −s + 1, . . . , s − 1, s. Then the spin dependence of the wave function
becomes insignificant and the wave function in the Schrödinger equation (5.2) can
be taken as an ordinary coordinate function = (r, t, σ ) .
It should be noted that the Hamiltonian (5.5) does not contain the coordinates
pz also commute with the Hamx and z explicitly. Therefore, the operators px and iltonian and the x and z components of the linear momentum are conserved. The
corresponding eigenvalues px and pz take on all real values from −∞ to ∞. In
this Letter we consider the case when the magnetic field H and the electric force
F are arbitrary functions of time t only. Using the substitution
(r, t) = ei(x px +zpz −S F (t))/ ψ (y, t)
(5.6)
p 2 µσ
cpx F(t)
dS F
= z −
H (t) +
dt
2m
s
e H (t)
(5.7)
with
results in the one-dimensional Schrödinger equation of the harmonic oscillator
driven by an external force in the y-direction
i
∂ψ
2 ∂2 ψ mω2H
=−
+
(y − y0 )2 ψ − F(t) (y − y0 ) ψ
∂t
2m ∂y 2
2
(5.8)
with time-dependent quantities
|e| H (t)
,
ω H (t) =
mc
cpx
y0 (t) = −
,
eH (t)
a H (t) =
.
mω H (t)
(5.9)
When the magnetic field is a constant H ≡ 0, the solution is well known (see
[22,23]). In the absence of external electric force F ≡ 0, there exist discrete energy
values corresponding to motion in a plane perpendicular to the magnetic field,
namely, the Landau levels. See [22,23] and Section 7.1 for more details.
In general, the following substitution
ψ (y, t) = χ (η, t) ,
η=
y − y0 (t)
a H (t)
(5.10)
gives the time-dependent Schrödinger equation of the form (2.1) as follows,
∂χ ω H (t)
∂χ
∂2
2
i
=
,
(5.11)
− 2 + η χ − f (t)ηχ + i (g(t) + h(t)η)
∂t
2
∂η
∂η
169
PROPAGATOR OF A PARTICLE WITH A SPIN
where f (t) = a H (t)F(t)/, g(t) = y0 (t)/a H (t), and h(t) = a H (t)/a H (t), which can be
solved by the method discussed in the previous sections. In this case
τ (t) =
a (t) ωH (t)
+2 H
= ln ω H a 2H
= 0,
ω H (t)
a H (t)
1
σ (t) = ω2H (t)
4
and the corresponding characteristic Equation (2.12) coincides with the equation
of motion for the classical oscillator with a time-dependent frequency
µH + ω2H (t)µ H = 0,
µ H (0) = 0,
µH (0) = ω H (0).
(5.12)
The fundamental solution of (5.11) is given by
χ = G H η, η , t =
2
1
i α H (t)η2 +β H (t)ηη +γ H (t)(η ) +δ H (t)η+ε H (t)η +κ H (t)
e
,
=√
2πiµ H (t)
(5.13)
where
d
1
m
d
(ln µ H (t)) = a 2H (t) (ln µ H (t)) ,
2ω H (t) dt
2
dt
1
1
ω H (0)
H (0)
1 a H (t)
β H (t) = −
=−
=−
,
µ H (t) a H (0)
µ H (t) ω H (t)
µ H (t) H (t)
⎞
⎛
t ω H (τ ) 2 ⎠
1
ω H (0) ⎝
γ H (t) =
−
dτ ,
2
µ H (t)µH (t)
µH (τ )
α H (t) =
(5.14)
(5.15)
(5.16)
0
a H (t) (0)
(1)
δ F (t) + px δ H (t) ,
δ H (t) =
µ H (t)
(0)
δ F (t) =
(5.17)
t
µ H (τ )F(τ )dτ,
(5.18)
0
(1)
δ H (t) =
mc
e
t
0
µH (τ )H (τ )
dτ =
H 2 (τ )
mcµH (t)
e
e
−
−
=
|e|
eH (t)
mc
ε H (t) =
1
ma H (0)
t
µ H (τ )H (τ )dτ,
(5.19)
0
(0)
(1)
ε F (t) + px ε H (t) ,
(5.20)
170
RICARDO CORDERO-SOTO ET AL.
(0)
ε F (t) =
t t µ H (τ )µH (τ ) F(τ )
H (τ ) 2 (0)
e2
1−
dτ
+
δ F (t)dτ,
µ H (t)µH (t) µH (τ )
µH (τ )
m 2 c2
0
0
(5.21)
(1)
ε H (t) = −
(1)
δ H (t)
e2
+ 2 2
µ H (t)µ H (t) m c
t 0
H (τ )
µH (τ )
2
(1)
δ H (τ )dτ,
(5.22)
1 (0)
(1)
(2)
κ F (t) + px κ F (t) + px2 κ H (t) ,
κ H (t) =
2m
2 t ω (τ )δ (0) (t) 2
1
H
(0)
(0)
F
κ F (t) =
δ (t) −
dτ −
µ H (t)µH (t) F
µH (τ )
(5.23)
0
t
−2
0
(1)
F(τ )δ H (τ )
dτ,
µH (τ )
(1)
δ (0) (t)δ H (t)
1 (1)
κ F (t) = F
−
2
µ H (t)µH (t)
t 0
ω H (τ )
µH (τ )
(5.24)
2
(1)
δ (0)
F (t)δ H (t)dτ
t
−
0
(1)
F(τ )δ H (τ )
dτ,
µH (τ )
(5.25)
(2)
κ H (t) =
2
(1)
δ H (t)
µ H (t)µH (t)
t −
0
(1)
ω H (τ )δ H (τ )
µH (τ )
2
dτ
(5.26)
as a special case of (2.15)–(2.21).
6. The Propagator in Three Dimensions
By the superposition principle, the fundamental solution = G r, r , t of the
time-dependent Schrödinger equation of a particle with a spin in a uniform magnetic field
∂
G r, r , t = 0
i − H
(6.1)
∂t
is given by the double Fourier integral of the form
G r, r , t =
1
2
∞
ei ((x−x ) px +(z−z ) pz −S F (t, px , pz ))/ ×
(2π ) a H (0)
−∞
y − y0 (t) y − y0 (0)
,
, t d p x d pz .
×G H
a H (t)
a H (0)
(6.2)
171
PROPAGATOR OF A PARTICLE WITH A SPIN
Here we obtain from Equation (5.7) that
p2
cpx
S F (t, px , pz ) = z t +
2m
e
t
0
µσ
F(τ )
dτ −
H (τ )
s
t
H (τ )dτ.
(6.3)
0
By Equation (5.13) the corresponding Green function is represented as
G H η, η , t = √
1
ei S H (η,η ,t )
2πiµ H (t)
(6.4)
with
2
S H η, η , t = α H (t)η2 + β H (t)ηη + γ H (t) η + δ H (t)η + ε H (t)η + κ H (t)
(6.5)
and η = (y − y0 (t)) /a H (t), η = y − y0 (0) /a H (0), where y0 is a linear function of
px [see (5.9) for a definition of functions y0 (t) and a H (t)]. Next, we are given
lim G r, r , t = δ r − r = δ x − x δ y − y δ z − z (6.6)
t→0+
as our initial data by the asymptotic relation (2.22) and the integral
1
δ (α) =
2π
∞
eiαξ dξ
(6.7)
−∞
as the Dirac delta function.
The quadratic in Equation (6.5) is also a quadratic polynomial in px given as
(2) (1) (0)
S H η, η , t = S H y, y , t + px S H y, y , t + px2 S H (t)
(6.8)
with coefficients
c2 α H (t)
(0)
S H (t) =
e2 a 2H (t)H 2 (t)
(1)
+
(1) S H y, y , t =
+
c2 γ H (t)
c2 β H (t)
+
+
e2 a H (t)a H (0)H (t)H (0) e2 a 2H (0)H 2 (0)
(1)
(2)
cε H (t)
cδ H (t)
κ (t)
+
,
+ H
2
eµ H (t)H (t) mea H (0)H (0)
2m
(6.9)
(1)
δ H (t)
cβ H (t)
+
+
y+
ea 2H (t)H (t) ea H (t)a H (0)H (0) µ H (t)
(1)
ε H (t)
cβ H (t)
2cγ H (t)
+
+
y +
+
ea H (t)a H (0)H (t) ea 2H (0)H (0) ma 2H (0)
2cα H (t)
(1)
+
cε(0)
cδ (0)
κ (t)
F (t)
F (t)
+
,
+ F
eµ H (t)H (t) mea 2H (0)H (0)
2m
(6.10)
172
RICARDO CORDERO-SOTO ET AL.
and
(2) SH
α H (t) 2
β H (t)
γ H (t) 2
yy + 2
y, y , t = 2
y +
y +
a H (t)a H (0)
a H (t)
a H (0)
+
δ (0)
ε(0) (t) κ F(0) (t)
F (t)
y +
y + F2
.
µ H (t)
2m
ma H (0)
(6.11)
See Equations (5.9) and (5.14)–(5.26) for notation explanation.
Hence, the double Fourier integral (6.2) can be evaluated in terms of elementary
functions. So, integration over pz gives the free particle propagator of motion in
the direction of magnetic field in the following fashion,
∞
pz2
i 1
t
d pz
exp
z − z pz −
G0 z − z , t =
2π 2m
−∞
2 im z − z m
exp
=
(6.12)
2πit
2t
using the familiar elementary integral
∞
i az 2 +2bz
e
dz =
−∞
πi −ib2 /a
e
;
a
(6.13)
see Refs. [4] and [32].
Similarly, integration over px yields
⎛
⎞
t
, t
z
−
z
G
iµσ
0
G r, r , t =
exp ⎝
H (τ )dτ ⎠ ×
s
(0)
2π a H (0) 2µ H (t)S H (t)
0
⎛
⎞2 ⎞
⎛
t
1
F(τ ) ⎠ ⎟
⎜
⎝x − x − c
dτ ⎠ ×
× exp ⎝
(0)
2
e
H (τ )
4i S (t)
H
⎛
⎜
× exp ⎝
(1) SH
y, y , t
0
2
(2) (0)
− 4S H (t)S H
(0)
4i S H (t)
y, y , t
⎞
⎟
⎠×
⎞⎞
⎛
t
F(τ
)
c
⎝x − x −
dτ ⎠⎠
× exp ⎝
(0)
e
H (τ )
2iS (t)
⎛
(1) SH
y, y , t
H
(6.14)
0
with a H (0) = (/mω H (0))1/2 = (c/ |e| H (0))1/2 . Moreover, the quadratic form in
this expression for the 3D-propagator
(1) 2
(0)
(2) Q y, y , t = S H y, y , t
− 4S H (t)S H y, y , t
(6.15)
173
PROPAGATOR OF A PARTICLE WITH A SPIN
[or the discriminant of (6.8)] can be rewritten as
2
Q y, y , t = A(t)y 2 + B(t)yy + C(t) y + D(t)y + E(t)y + L(t),
(6.16)
where
1 ∂2 Q
∂2 Q
1 ∂2 Q
,
B
=
,
C
=
,
2 ∂y 2
∂y∂y 2 ∂y 2
∂Q ∂Q D=
, E = , L = Q| y=y =0 .
∂y y=y =0
∂y y=y =0
A=
(6.17)
Our final result is
2
(1)
c2 β H
(t) − 4α H (t)γ H (t)
2cβ H (t)δ H (t)
A(t) =
+
+
eµ H (t)a H (t)a H (0)H (0)
e2 a 2H (t)a 2H (0)H 2 (0)
+
B(t) = −
(1)
δ H (t)
2
2 µ2H (t)
(1)
−
mea 2H (t)a 2H (0)H (0)
2
(t) − 4α H (t)γ H (t)
2c2 β H
e2 a 2H (t)a 2H (0)H 2 (0)
(1)
+
4cα H (t)ε H (t)
mea 2H (t)a 2H (0)H (t)
C(t) =
2cβ H (t)ε H (t)
mea H (t)a 3H (0)H (0)
+
D(t) =
(1)
2
m 2 a 4H (0)
(6.18)
2cβ H (t)δ H (t)
+
eµ H (t)a H (t)a H (0)H (0)
(1)
eµ H (t)a 2H (0)H (0)
+
(1)
2δ H (t)ε H (t)
mµ H (t)a 2H (0)
2β H (t)κ H (t)
,
ma H (t)a H (0)
(6.19)
2cβ H (t)ε H (t)
mea H (t)a 3H (0)H (t)
+
(2)
eµ H (t)a 2H (0)H (t)
−
4γ H (t)κ H (t)
2ma 2H (0)
,
2c2 β H (t)ε(0)
F (t)
+
me2 a 2H (t)a 2H (0)H (t)H (0) me2 a H (t)a 3H (0)H 2 (0)
(1)
(0)
(0)
(1)
(1)
2c δ H (t)ε F (t) − 2δ F (t)ε H (t)
2cα H (t)κ F (t)
+
+
+
meµ H (t)a 2H (0)H (0)
mea 2H (t)H (t)
(1)
+
−
(1)
+
4cγ H (t)δ H (t)
4c2 α H (t)ε(0)
F (t)
,
(1)
−
(1)
−
ma 2H (t)
(2)
−
e2 a 2H (t)a 2H (0)H 2 (t)
ε H (t)
2α H (t)κ H (t)
4cγ H (t)δ H (t)
2
(t) − 4α H (t)γ H (t)
c2 β H
−
(1)
+
(1)
−
(2)
4cα H (t)ε H (t)
+
(1)
(1)
(0)
(2)
δ (t)κ F (t) − 2δ F (t)κ H (t)
cβ H (t)κ F (t)
+ H
−
mea H (t)a H (0)H (0)
2 mµ H (t)
(6.20)
174
RICARDO CORDERO-SOTO ET AL.
(0)
−
(0)
−
(0)
2c2 β H (t)δ F (t)
4c2 γ H (t)δ F (t)
−
−
e2 µ H (t)a H (t)a H (0)H (t)H (0) e2 µ H (t)a 2H (0)H 2 (0)
(1)
2cδ F (t)δ H (t)
2 eµ2H (t)H (t)
,
(6.21)
(0)
E(t) =
(0)
2c2 β H (t)δ F (t)
4c2 γ H (t)δ F (t)
+
+
e2 µ H (t)a H (t)a H (0)H 2 (t) e2 µ H (t)a 2H (0)H (t)H (0)
(0)
(1)
(1)
(0)
(1)
2c δ F (t)ε H (t) − 2δ H (t)ε F (t)
cβ H (t)κ F (t)
+
+
+
mea H (t)a H (0)H (t)
meµ H (t)a 2H (0)H (t)
(1)
+
2cγ H (t)κ F (t)
mea 2H (0)H (0)
(1)
+
(1)
(0)
m 2 a 2H (0)
(0)
−
L(t) =
me2 a 2H (t) a 2H (0) H 2 (t)
m 2 ea 4H (0)H (0)
2 e2 µ2H (t)H 2 (t)
me2 a H (t) a 3H (0) H (t) H (0)
−
(6.22)
2
(0)
c2 ε F (t)
+
m 2 e2 a 4H (0)H 2 (0)
(0)
me2 µ H (t)a 2H (0)H (t)H (0)
2c2 α H (t)κ F(0) (t)
me2 a 2H (t)H 2 (t)
−
+
(0)
2c2 δ F (t)ε F (t)
(1)
−
2c2 β H (t)ε F (t)
,
(0)
−
−
(1)
2cδ F (t)δ H (t)
2
(0)
c2 δ F (t)
+
−
(0)
4c2 α H (t)ε F (t)
(0)
−
(2)
ε H (t)κ F (t) − 2ε F (t)κ H (t)
+
2
(1)
κ F (t)
42 m 2
+
(1)
(0)
(1)
cε F (t)κ F (t)
cδ F (t)κ F (t)
+
−
2 meµ H (t)H (t) m 2 ea 2H (0) H (0)
2c2 β H (t)κ F(0) (t)
2c2 γ H (t)κ F(0) (t)
−
−
me2 a H (t)a H (0)H (t)H (0) me2 a 2H (0)H 2 (0)
(1)
(2)
2cδ H (t)κ F(0) (t)
κ F(0) (t)κ H (t)
2cε H (t)κ F(0) (t)
.
−
−
2 meµ H (t)H (t) m 2 ea 2H (0)H (0)
2 m 2
(6.23)
Finally, by the superposition principle, a general solution of the Cauchy initial
value problem in R3 subject to the initial data
(r, t)|t=0 = (r, 0) = φ (x, y, z)
has the form
(r, t) =
R3
G r, r , t r , 0 dx dy dz .
(6.24)
(6.25)
175
PROPAGATOR OF A PARTICLE WITH A SPIN
This gives us the time evolution operator (1.2) explicitly for a motion of a charged
particle in a uniform magnetic field and also a perpendicular electric field with a
given projection of the spin sz = σ in the direction of magnetic field.
7. Two Examples
7.1. MOTION IN A CONSTANT MAGNETIC FIELD
The simplest case occurs as H = F = 0, which implies µH + ω2H µ H = 0 with µ H =
sin (ω H t). Thus,
α H (t) = γ H (t) =
(0)
(1)
(0)
1
cot (ω H t) ,
2
(1)
(0)
β H (t) = −
(1)
1
sin (ω H t)
(7.1)
(2)
with δ F = δ H = ε F = ε H = κ F = κ F = κ H = 0 and
ωH t
1
e
ωH t (0)
(1) tan
, S H y, y , t = −
y + y ,
S H (t) = −
tan
mω
2
|e|
2
mω
2yy
2
(2)
,
cot (ω H t) y 2 −
+ cot (ω H t) y S H y, y , t =
2
sin (ω H t)
where the discriminant is
(1) 2
2
1 (2) (0)
Q y, y = S H y, y , t
− 4S H (t)S H y, y , t = 2 y − y .
Hence, the Green function is given by
mω H
iµσ H t
G r, r , t = G 0 z − z , t exp
×
s
4πi sin (ω H t/2)
2 2 imω H x − x + y − y
× exp
cot (ω H t/2) −
4
e .
−2
x − x y + y
|e|
(7.2)
(7.3)
See [23] for more details.
7.2. A LINEAR MAGNETIC FIELD
Now consider the case H (t) = H0 + t H1 , where H0 > 0 and H1 are constants. The
characteristic Equation (5.12) becomes a special case of the Lommel equation [1,
30,43], which can be solved in terms of Bessel functions [45] of orders ν = ±1/4.
It then follows that
3/2
|e| H02
π |e| H0 |e| H 2 (t)
H (t) J−1/4
J1/4
−
µ H (t) = 3/2
2mcH1
2mcH1
2 mcH1
|e| H02
|e| H 2 (t)
−J1/4
J−1/4
(7.4)
2mcH1
2mcH1
176
RICARDO CORDERO-SOTO ET AL.
with
|e| H02
|e| H 2 (t)
H0 H (t) 3/2
J1/4
J3/4
+
2
2mcH1
2mcH1
|e| H02
|e| H 2 (t)
+ J−1/4
J−3/4
.
2mcH1
2mcH1
dµ H (t)
π e2
= 2 2
dt
m c H1
(7.5)
A general expression of the propagator is given above by Equation (6.14). In the
(0)
(0)
(0)
(1)
case when F ≡ 0, one can simplify δ F = ε F = κ F = κ F ≡ 0. We shall elaborate on
these and some other interesting special cases elsewhere.
Acknowledgements
The authors are grateful to Professor Carlos Castillo-Chávez for support and
Ref. [6]. We thank Professors George Andrews, Slim Ibrahim, Hank Kuiper, Alex
Mahalov, Mizan Rahman, Christian Ringhofer, Svetlana Roudenko and Hal Smith
for valuable comments, and the referee for Refs. [35,38].
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