Quantum Transitions And Its Evolution For Systems With Canonical And Noncanonical Symplectic Structures
1.
Quantum transitions and its evolution for systems with canonical and
noncanonical symplectic structures
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de M´ xico, M´ xico
o e e e
Abstract. I obtain from the beginning to the end the quantum transitions for the coordinates of the phase space and its evolution
for the one dimensional isotropic harmonic oscillator with canonical and noncanonical symplectic structures using the ket-bra
formalism and a formulation based on ascent (creation) and descent (destruction) operators. I study two classical and quantum
ˆ
solutions for a charged particle in a constant magnetic ﬁeld (B = B k) with mass m and charge e, at quantum level I map the
original problem to the one dimensional isotropic harmonic oscillator and with this I can obtain the quantum transitions for the
coordinates of the phase space and its evolution, too.
1. Introduction
Classical mechanics is one of the fundamental physical theories of our days, all this area must be based on experimental
facts, my own point of view and probably is not the same for all the people in the physical sciences is that all should be
shown with the erros bars, statistics and probably more for the physical experiments, in the opposite case, we are making
correct mathematical theories, wrong mathematical theories, philosophy and son on, or in the worst case, anything.
My interest in the present paper is to study the classical one dimensional isotropic harmonic oscillator with canonical
and noncanonical symplectic structures (see [1] for a treatment of noncanonical symplectic structures) and a charged
ˆ
particle with charge e, mass m inmerse in a constant magnetic ﬁeld B = B k, with B a constant. I use the ket and
bra formalism of quantum mechanics (see [2] and [3] for instance), I make use of the creation and destruction operators
and few mathematical methods (see [4] and [5]). My study does not use the wave function formalism or the coordinate
representation like a lot of texts (see [6] and [7] for instance).
In this work I ﬁnd hamiltonian formulations for all the systems, I write all the equations of motion and I ﬁnd the
classical solutions with the constants of motion associated with the initial conditions of the problem. Then, I write the
time evolution for the coordinate operators or the Heisenberg’s representations for these theories (see [8] for a detailed
study of symplectic quantization) and using the corresponding eigenkets for the number operator I ﬁnd the evolution of
the previous quantum coordinates, and later I ﬁnd the quantum transitions of states and its time evolution.
The same results can be reproduced if the reader make my procedure carefully, my proper interests are of
mathematical character, only. My physical science spirit is not covered because I do not make physical science
experiments.
2. One dimensional isotropic harmonic oscillator (canonical case)
In this section I will study one of the most important physical systems: the one dimensional isotropic harmonic oscillator,
the specialist in classical mechanics and quantum mechanics can understand its importance because it can be solved
completely at classical level, at quantum level this system can be quantized using the coordinates representation, the
momenta representation or equivalently using the bra and ket formalism, with the help of the previous studies the specialist
can quantize the electromagnetic ﬁeld and a lot of systems, in this section I study the system when it has the canonical
symplectic structure.
† vladimir.cuesta@nucleares.unam.mx
2.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 2
2.1. Classical case
Every undergraduate student have studied the one dimensional isotropic harmonic oscillator using Lagrange’s equations,
using Legendre’s transformations the same student can understand it and in fact the last formalism can be though like a
phase space with coordinates (xµ = (q, p)), µ = 1, 2, and a simplectic structure (ω µν ) = {xµ , xν }, where
0 1
(ω µν ) = ,
−1 0
I mean,
{q, p} = 1, {q, q} = 0, {p, p} = 0, (1)
with the previous ingredients the equations of motion for the system can be written like,
∂H
xµ = ω µν
˙ , (2)
∂xν
with the hamiltonian,
1 2 mω 2 2
H= p + q , (3)
2m 2
and we can obtain,
p
q=
˙ , p = −mω 2 q,
˙ (4)
m
using usual mathematical methods I obtain the solutions,
1
q(t) = cos(ωt)q0 + sin(ωt)p0 ,
mω
p(t) = − mω sin(ωt)q0 + cos(ωt)p0 , (5)
where q0 and p0 mean the coordinate and the momentum at time equal to zero.
2.2. Quantum case
In the previous section I have found the solution for the classical one dimensional isotropic harmonic oscillator, then for
the quantum theory I ﬁnd,
1
q (t) = cos(ωt)ˆ0 +
ˆ q sin(ωt)ˆ0 ,
p
mω
p(t) = − mω sin(ωt)ˆ0 + cos(ωt)ˆ0 ,
ˆ q p (6)
the following step is to solve the eigenvalue equation for the hamiltonian, I introduce the set of operators a and a† , they
ˆ ˆ
are known like the destruction and creation operators,
mω ı mω ı
a=
ˆ q0 +
ˆ p0 ,
ˆ a† =
ˆ q0 −
ˆ p0 ,
ˆ (7)
2¯
h mω 2¯
h mω
I can ﬁnd the inverse transformations,
h
¯ h
¯ mω †
q0 =
ˆ a + a† , p0 = ı
ˆ ˆ ˆ a −a ,
ˆ ˆ (8)
2mω 2
in such a way that the q (t) and q (t) operators can be written like,
ˆ ˆ
h
¯ h
¯
q (t) =
ˆ (cos(ωt) − ı sin(ωt)) a +
ˆ (cos(ωt) + ı sin(ωt)) a† ,
ˆ (9)
2mω 2mω
h
¯ mω h
¯ mω
p(t) = −
ˆ (sin(ωt) + ı cos(ωt)) a +
ˆ (ı cos(ωt) − sin(ωt)) a† ,
ˆ
2 2
3.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 3
√ √
using the mathematical expressions a|n = n|n − 1 and a† |n = n + 1|n + 1 follows,
ˆ ˆ
h
¯ √ √
q (t)|n =
ˆ cos(ωt) n|n − 1 + n + 1|n + 1
2mω
h
¯ √ √
+i sin(ωt) n + 1|n + 1 − n|n − 1 ,
2mω
h
¯ mω √ √
p(t)|n = −
ˆ sin(ωt) n|n − 1 + n + 1|n + 1
2
h
¯ mω √ √
+i cos(ωt) n + 1|n + 1 − n|n − 1 , (10)
2
the transition for the phase space coordinates are,
h
¯ (n + 1)
r|ˆ(t)|n =
q (cos(ωt) + ı sin(ωt)) δr,n+1
2mω
h
¯n
+ (cos(ωt) − ı sin(ωt)) δr,n−1 ,
2mω
h
¯ mωn
r|ˆ(t)|n = −
p (sin(ωt) + ı cos(ωt)) δr,n−1
2
h
¯ mω(n + 1)
+ (− sin(ωt) + ı cos(ωt)) δr,n+1 , (11)
2
and the time evolution for the transition between the r and the n states for the q (t) and p(t) operators are:
ˆ ˆ
d r|ˆ(t)|n
q h
¯ ω(n + 1)
= (− sin(ωt) + ı cos(ωt)) δr,n+1
dt 2m
h
¯ ωn
− (sin(ωt) + ı cos(ωt)) δr,n−1 ,
2m
d r|ˆ(t)|n
p h
¯ mωn
= −ω (cos(ωt) − ı sin(ωt)) δr,n−1
dt 2
h
¯ mω(n + 1)
−ω (cos(ωt) + ı sin(ωt)) δr,n+1 . (12)
2
3. One dimensional isotropic harmonic oscillator (noncanonical case)
3.1. Classical case
In this section I study the one dimensional isotropic harmonic oscillator, using a different hamiltonian formulation, in
this case the system can be described by a phase space with coordinates (xµ ) = (q, p), µ = 1, 2, in this case I have the
hamiltonian,
1 1 2 mω 2 2
H= p + q , (13)
α 2m 2
and the symplectic structure,
{q, p} = α, {q, q} = 0, {p, p} = 0, (14)
or equivalently,
0 α
(ω µν ) = ,
−α 0
4.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 4
an in this case I obtain the same equations of motion like the previous section, I mean,
∂H
xµ = ω µν
˙ , (15)
∂xν
or equivalently,
p
q=
˙ , p = −mω 2 q,
˙ (16)
m
and I obtain the solutions,
1
q(t) = cos(ωt)q0 + sin(ωt)p0 ,
mω
p(t) = − mω sin(ωt)q0 + cos(ωt)p0 , (17)
where q0 and p0 mean the coordinate and the momentum at time equal to zero.
3.2. Quantum case
I will follow the discussion of the previous section, I can make the change of variables and the resulting commutators are,
1
x0 = q0 , px0 = p0 , ˆx0 , px0 ] = ı¯ , [ˆ0 , x0 ] = 0, [ˆx0 , px0 ] = 0,
ˆ ˆ ˆ ˆ [ˆ ˆ h x ˆ p ˆ (18)
α
and the solutions with the original variables are,
1
q (t) = cos(ωt)ˆ0 +
ˆ q sin(ωt)ˆ0 ,
p
mω
p(t) = − mω sin(ωt)ˆ0 + cos(ωt)ˆ0 ,
ˆ q p (19)
and the solutions with the second set of variables are,
1
q (t) = α cos(ωt)ˆ0 +
ˆ x sin(ωt)ˆx0 ,
p
mω
p(t) = − mωα sin(ωt)ˆ0 + cos(ωt)ˆx0 ,
ˆ x p (20)
in this step I introduce the destruction operator a and the creation operator a† ,
ˆ ˆ
mω ı mω ı
a=
ˆ x0 +
ˆ px ,
ˆ a† =
ˆ x0 −
ˆ px ,
ˆ (21)
2¯
h mω 0 2¯
h mω 0
and the quantum solutions for the one dimensional isotropic harmonic oscillator are in terms of a and a† ,
ˆ ˆ
h
¯ h
¯
q (t) =
ˆ (α cos(ωt) − ı sin(ωt)) a +
ˆ (α cos(ωt) + ı sin(ωt)) a† ,
ˆ (22)
2mω 2mω
h
¯ mω h
¯ mω
p(t) = −
ˆ (α sin(ωt) + ı cos(ωt)) a +
ˆ (ı cos(ωt) − α sin(ωt)) a† ,
ˆ
2 2
√ √
and using the mathematical identities a|n = n|n − 1 and a† |n = n + 1|n + 1 I ﬁnd,
ˆ ˆ
h
¯ √ √
q (t)|n = α
ˆ cos(ωt) n|n − 1 + n + 1|n + 1
2mω
h
¯ √ √
+i sin(ωt) n + 1|n + 1 − n|n − 1 ,
2mω
h
¯ mω √ √
p(t)|n = − α
ˆ sin(ωt) n|n − 1 + n + 1|n + 1
2
h
¯ mω √ √
+i cos(ωt) n + 1|n + 1 − n|n − 1 , (23)
2
5.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 5
and the ﬁnal results are,
h
¯ (n + 1)
r|ˆ(t)|n =
q (α cos(ωt) + sin(ωt)) δr,n+1
2mω
h
¯n
+ (α cos(ωt) − ı sin(ωt)) δr,n−1 ,
2mω
h
¯ mωn
r|ˆ(t)|n = −
p (α sin(ωt) + ı cos(ωt)) δr,n−1
2
h
¯ mω(n + 1)
+ (−α sin(ωt) + ı cos(ωt)) δr,n+1 , (24)
2
so, the time evolution for the transition between the r and n states for the q (t) and p(t) operators are,
ˆ ˆ
d r|ˆ(t)|n
q h
¯ ω(n + 1)
= (−α sin(ωt) + cos(ωt)) δr,n+1
dt 2m
h
¯ ωn
− (α sin(ωt) + ı cos(ωt)) δr,n−1 ,
2m
d r|ˆ(t)|n
p h
¯ mωn
= −ω (α cos(ωt) − ı sin(ωt)) δr,n−1
dt 2
h
¯ mω(n + 1)
−ω (α cos(ωt) + ı sin(ωt)) δr,n+1 . (25)
2
4. Charged particle in a constant magnetic ﬁeld
4.1. Classical case I
In this section I will study a charged particle in a constant magnetic ﬁeld in the z direction. I mean, I ﬁnd the hamiltonian
ˆ
formalism for the system, when B = B k and B is a constant, the ﬁrst step to ﬁnd an electromagnetic potential A is to
write the equations B = × A,
∂Az ∂Ay ∂Ax ∂Az ∂Ay ∂Ax
0= − , 0= − , B= − , (26)
∂y ∂z ∂z ∂x ∂x ∂y
and I found as solution for the electromagnetic potential,
Ax = 0, Ay = Bx, Az = 0, (27)
and the hamiltonian will be,
1
H = π2 x + π2 y (28)
2m
e e
πx = px − Ax , πy = py − Ay , B= ×A
c c
2
1 eB 1 2eB e2 B 2
H= p2 x + p y − x = p2 + p 2 −
x y xpy + 2 x2 , (29)
2m c 2m c c
and the symplectic structures are,
{x, y} = 0, {x, px } = 1, {x, py } = 0, {y, px } = 0,
{y, py } = 1, {px , py } = 0,
{x, y} = 0, {x, πx } = 1, {x, πy } = 0, {y, πx } = 0,
eB
{y, πy } = 1, {πx , πy } = , (30)
c
6.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 6
I obtain the equations of motion:
px py eB eB e2 B 2
x=
˙ , y=˙ − x, px =
˙ py − x, py = 0,
˙ (31)
m m mc mc mc2
combining the ﬁrst, third and fourth equations I obtain,
e2 B 2 eB d2 c e2 B 2 c
x+
¨ 2 c2
x = 2 py0 , 2
x− py + 2 2 x − py = 0,
m m c dt eB m c eB
e2 B 2
px + 2 2 px = 0,
¨ (32)
m c
using the general solution for the previous equation, the initial conditions, inserting into the ﬁrst equation and so on, I ﬁnd
the classical solutions,
c eB c eB c
x(t) = px sin t + x0 − py cos t + py ,
eB 0 mc eB 0 mc eB 0
c eB mc py0 eB eB c
y(t) = px cos t + − x0 sin + y0 − px ,
eB 0 mc eB m mc mc eB 0
eB eB eB
px (t) = px0 cos t + py0 − x0 sin t ,
mc c mc
py (t) = py0 , (33)
4.2. Quantum case I
I will present the quantum case, for this purpose I choose the following variables:
c
x =x− py , y = y, px = px , py = py , (34)
eB
with the set of inverse transformations,
c
x=x + p , y = y, px = px , py = py , (35)
eB y
I can write the original hamiltonian like,
2
1 2 m eB
H= p + x 2, {x , px } = 1, (36)
2m x 2 mc
and like the reader can recognize, I have found the hamiltonian for the one dimensional isotropic harmonic oscillator with
eB
frequency ω = mc , I have studied this system in the previous section and I can obtain all the transitions between the n
and r states for all the phase space coordinates like in the previous section.
4.3. Classical case II
ˆ
The last example for this work is a charged particle in a constant magnetic ﬁeld B = B k, where B is a constant, I must
solve the set of equations B = × A,
∂Az ∂Ay ∂Ax ∂Az ∂Ay ∂Ax
0= − , 0= − , B= − , (37)
∂y ∂z ∂z ∂x ∂x ∂y
and I ﬁnd these solutions:
Ax = −By, Ay = 0, Az = 0, (38)
in this case the hamiltonian will be,
1
H = π 2 + πy2
(39)
2m x
e e
πx = px − Ax , πy = py − Ay , B= × A,
c c
7.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 7
or equivalently,
2
1 eB 1 2eB e2 B 2
H= px + y + p2 =
y p2 +
x ypx + 2 y 2 + p2 ,
y (40)
2m c 2m c c
with these symplectic structures,
{x, y} = 0, {x, px } = 1, {x, py } = 0, {y, px } = 0,
{y, py } = 1, {px , py } = 0,
{x, y} = 0, {x, πx } = 1, {x, πy } = 0, {y, πx } = 0,
eB
{y, πy } = 1, {πx , πy } = , (41)
c
the equations of motion are:
px eB py eB e2 B 2
x=
˙ + y, y =
˙ , px = 0, py = −
˙ ˙ px − y, (42)
m mc m mc mc2
combining the previous set of equations I can obtain the following:
2
eB
py +
¨ py = 0,
mc
2 2
eB eB d2 c eB c
y+
¨ y = − px , y+ px + y+ px = 0, (43)
mc m2 c 0 dt2 eB mc eB
I mean, a set of two one dimensional isotropic harmonic oscillators, using standard mathematical methods I found the
solutions:
c eB c c eB
x(t) = x0 + py + sin t y0 + px − cos t py 0 ,
eB 0 mc eB 0 eB mc
eB c c eB c
y(t) = cos t y0 + px + sin t py 0 − px ,
mc eB 0 eB mc eB 0
px (t) = px0 ,
eB eB eB
py (t) = − sin t y0 + px0 + cos t py 0 . (44)
mc c mc
4.4. Quantum case II
I will use the set of transformations,
c
x = x, y =y+ px , p x = px , p y = py , (45)
eB
with inverse transformations,
c
x=x, y=y − p , p x = px , p y = py , (46)
eB x
and the original hamiltonian can be written as,
2
1 2 m eB
H= p + y 2, {y , py } = 1, (47)
2m y 2 mc
and like the reader can recognize I have found the hamiltonian for the one dimensional isotropic harmonic oscillator with
eB
frequency ω = mc , and with a the new set of variables I can make the study of the previous section in a direct way.
8.
Quantum transitions and its evolution for systems with canonical and noncanonical symplectic structures 8
5. Conclusions and perspectives
In this work I have studied three systems: the one dimensional isotropic harmonic oscillator (with canonical and
noncanonical symplectic structures), a charged particle in a constant magnetic ﬁeld with two different solutions for the
electromagnetic potential. For the last two systems I found that are equivalent to the one dimensional isotropic harmonic
oscillator with a special set of variables, I make the quantum theory for all the cases if I undertand the ﬁrst of all the
systems. I made that carefully and I solve the eigenvalue problem and I ﬁnd a good base for that and I ﬁnd the transition
between two different quantum states and its temporal variation.
ˆ
I present a couple of two solutions for a charged particle in a constant magnetic ﬁeld (B = B k) with mass m, charge
equal to e these set of solutions are:
Ax = α(x + z), Ay = Bx, Az = α(x + z),
Ax = − By, Ay = β(y + z), Az = β(y + z), (48)
where α and β are constants.
At classical and quantum level, the interested reader can apply a similar procedure for solving this couple of systems,
but this is future work.
References
[1] V. Cuesta, M. Montesinos and J. D. Vergara, Gauge invariance of the action principle for gauge systems with noncanonical symplectic structures,
Phys. Rev. D 76, 025025 (2007).
[2] J. J. Sakurai, Modern Quantum Mechanics, Late, University of California, Los Angeles, Addison-Wesley Publishing Company, Inc. (1994).
[3] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, (1981).
[4] D. Hetenes and J. W. Holt, Crystallographic space groups in geometric algebra, J. Math. Phys. 48, 023514 (2007).
[5] R. Amorim, Tensor coordinates in noncommutative mechanics, J. Math. Phys. 50, 052103 (2009).
[6] W. Heisenberg, The Physical Principles of the Quantum Theory, University of Chicago Press, Chicago, (1930).
[7] G. Friesecke and M. Koppen, On the Ehrenfest theorem of quantum mechanics, J. Math. Phys. 50, 082102 (2009).
[8] M. Montesinos and G. F. Torres del Castillo, Symplectic quantization, inequivalent quantum theories, and Heisenberg’s principle of uncertainty,
Phys. Rev. A 70, 032104 (2004).
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