Different Quantum Spectra For The Same Classical System

Different quantum spectra for the same classical system
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de
o e
M´ xico, M´ xico
e e

Abstract. Using the bra, ket and ascent and descent formalism, I show that the usual one dimensional
isotropic harmonic oscillator admits two quantum spectra. Additionally, I find that for the case of the
two dimensional isotropic harmonic oscillator there are four different quantum spectra. The importance
of the result lies in the fact that at classical level the one and two dimensional isotropic harmonic
oscillator correspond to a pair of systems, only.

1. Introduction

Classical mechanics is one of the more important branches in modern physics, we can study
that subject using different approaches, using Newton’s equations, using Lagrange or Hamiltonian
formulations (see [1] for instance). In all these cases one key point is to obtain the equations of
motion for a given system, in the case of Newton’s equations or Lagrange point of view we work
with coordinates and time derivatives of coordinates (or velocities) and the equations of motion are of
second order for the time variable.
In the case of Hamiltonian formulations, the phase space is labeled by coordinates and
momentum variables and the set of differential equation are of first order in the time variable.
However, if we are interested in the equations of motion for a chosen system, these can be obtained
using variational principles, too. Particularly Hamilton’s equations need a Hamiltonian and the
canonical symplectic structure (see [1] for systems without constraints and [2], [3] for constrained
systems).
A lot of modern physical theories are formulated in terms of geometrical and advanced
mathematical theories, one of these is symplectic geometry and in fact there exist different
formulations of Hamilton’s equations with non canonical symplectic structures (see [4], [5] and [6]).
The present work will be focus in the study of the one and two dimensional isotropic harmonic
oscillator at classical and quantum level, at classical level a hamiltonian formulation for the first case
†
vladimir.cuesta@nucleares.unam.mx

Different quantum spectra for the same classical system 2

is the following: the phase space coordinates are (xµ ) = (x, px ) with Hamiltonian
1 2 mω 2 2
H= p + x, (1)
2m x 2
and equations of motion
px
x = , px = −mω 2 x,
˙ ˙ (2)
m
as second case we will study the two dimensional isotropic harmonic oscillator, at classical level the
Hamiltonian is
1 2 2 mω 2 2
H= px + py + x + y2 , (3)
2m 2
where the coordinates (xµ ) = (x, y, px , py ) label the phase space, the set of equations of motion are
px py
x = , y = , px = −mω 2 x, py = −mω 2 y,
˙ ˙ ˙ ˙ (4)
m m
at classical level we will study this pair of systems and like the reader can be note there exist two sets
of equations of motion, only.
Quantum mechanics is another branch of modern physics with great importance for the
understanding of nature, it was formulated in essentially two ways, the first one is in terms of a
Hilbert space of square-integrable functions, partial differential equations, wave functions and so on
(see [7] for instance). The second point of view is the formulation in terms of an abstract Hilbert
space of vectors (kets) and its duals (bras), abstract operators and so on (see [8] and [9] for a profound
review of the theory), with this in my mind I present my results following this second approach.
The purpose of this paper is to prove that the one and two dimensional isotropic harmonic
oscillators have not an unique spectrum, the importance of these systems is undeniable, it appears
like textbook exercises in basic classical mechanics, in quantum mechanics, in quantum field theory
appears when electromagnetism is quantized and in innumerable physical theories. To mention a
pair of different studies, in [10] is obtained a lot of different couples of Hamiltonians and symplectic
structures and using algebraic methods the spectrum of the system is obtained and is only one. In [11],
using different couples of Hamiltonians and symplectic structures Heisenberg’s uncertainty principle
is studied, this paper makes a different study and like I said, for the same classical system I find more
than one quantum spectra.

2. Quantum one dimensional isotropic harmonic oscillator

2.1. Case I: Usual case

The hamiltonian for the one dimensional isotropic harmonic oscillator is given by

ˆ 1 2 mω 2 2
H= p +
ˆ x,
ˆ (5)
2m x 2


now, following the procedure given in [9] we can define the following operators
mω ı mω ı
ax =
ˆ x+
ˆ p x , a† =
ˆ ˆx x−
ˆ px ,
ˆ (6)
2¯
h mω 2¯
h mω
ˆ
and the commutation relations between the number operator Nx = a† ax , ax and a† are
ˆx ˆ ˆ ˆx
ˆ ˆ
[Nx , ax ] = −ˆx ,
a ˆ ˆ
[ N x , a† ] = a† ,
ˆx [ˆx , a† ] = 1,
a ˆx (7)
x
ˆ
if I suppose that the number operator has as eigenkets |nx then we must have Nx |nx = nx |nx ,
using the previous equation we can obtain
ˆ ˆ
Nx ax |nx = (nx − 1)ˆx |nx ,
a ˆ ˆ
Nx a† |nx = (nx + 1)ˆ† |nx ,
ax (8)
x

and we can deduce ax |nx = α|nx −1 and a† |nx = β|nx +1 , where α and β are complex constants,
ˆ ˆx
choosing the constants suitably we find
√ √
ax |nx = nx |nx − 1 , a† |nx = nx + 1|nx + 1 ,
ˆ ˆx (9)
the ax operator is the destruction operator and a† is the creation operator and nx = 0, 1, 2, . . ., now
ˆ ˆx
inverting the set of equations (6) we obtain
h
¯ mω¯ †
h
x=
ˆ ax + a† , p x = ı
ˆ ˆx ˆ ax − ax ,
ˆ ˆ (10)
2mω 2
if I substitute (10) into (5) I obtain
ˆ ˆ 1 ˆ 1
H = Nx + hω, H|nx = nx +
¯ hω|nx ,
¯ (11)
2 2
and the spectrum of the one dimensional isotropic harmonic oscillator is
1
Enx = nx + hω,
¯ (12)
2
to finish, the vacuum ket is defined like
ax |0 = 0,
ˆ (13)
using the creation operator we can obtain
√ √
a† |0 = |1 , a† |1 = 2|2 ,
ˆ ˆ a† |2 =
ˆ 3|3 , a† |3 = 2|4 , . . . ,
ˆ (14)
and in general a state |mx can be obtained with the help of the vacuum ket as
1 mx
|mx = √ a†
ˆ |0 . (15)
mx !


2.2. Case II

I will be interested in the hamiltonian
2
Hˆ = 1 p2 + mω x2 ,
ˆ ˆ (16)
2m x 2
now, I define the operators
mω ı mω ı
ax = −
ˆ x−
ˆ px ,
ˆ x+
ˆ a† = −
ˆx px ,
ˆ (17)
2¯
h mω 2¯
h mω
ˆ
and the commutation relations between the new number operator Nx = a† ax , ax and a† are
ˆx ˆ ˆ ˆx
ˆ ˆ
[ N x , ax ] = ax ,
ˆ ˆ ˆx
[Nx , a† ] = −ˆ† ,
ax [ˆx , a† ] = −1,
a ˆx (18)
ˆ
if I suppose that the new number operator has as eigenkets |nx then we must have Nx |nx = nx |nx ,
after a straightforward calculation
ax |nx =
ˆ |nx ||nx + 1 , a† |nx =
ˆx |nx − 1||nx − 1 , (19)
where nx = . . . , −2, −1, 0., ax is the ascent operator and a† is the descent operator, if I invert the set
ˆ ˆx
of equations (17) I find
h
¯ mω¯ †
h
x=−
ˆ ax + a† , p x = ı
ˆ ˆx ˆ ax − ax ,
ˆ ˆ (20)
2mω 2
and the hamiltonian will be
ˆ ˆ 1 ˆ 1
H = Nx − hω, H|nx = nx −
¯ hω|nx ,
¯ (21)
2 2
and the spectrum of the one dimensional isotropic harmonic oscillator is
1
Enx = nx − hω,
¯ (22)
2
in this case the vacuum ket is |0 and is defined as a|0 = 0, using the ascent creation operator I find
ˆ
√ √
a† |0 = |−1 , a† |−1 = 2|−2 , a† |−2 = 3|−3 , a† |−3 = 2|−4 , . . .(23)
ˆ ˆ ˆ ˆ
and in general
1 |mx |
| − |mx | = a†
ˆ |0 , (24)
|mx |!
and with this I prove that the classical one dimensional isotropic oscillator has two different quantum
spectra.


3. Quantum two dimensional isotropic harmonic oscillator

3.1. Case I: Usual case

In this case the hamiltonian for the two dimensional isotropic harmonic oscillator is
ˆ 1 mω 2 2
H= p2 + p2 +
ˆx ˆy x + y2 ,
ˆ ˆ (25)
2m 2
the previous hamiltonian can be divided in a part that correspond with the x direction and another in
the y direction, both hamiltonians are independent, for the x direction I can choose
mω ı mω ı
ax =
ˆ x+
ˆ p x , a† =
ˆ ˆx x−
ˆ px ,
ˆ (26)
2¯
h mω 2¯
h mω
and for the y direction
mω ı mω ı
ay =
ˆ y+
ˆ p y , a† =
ˆ ˆy y−
ˆ py ,
ˆ (27)
2¯
h mω 2¯h mω
ˆ
with commutation relations between the number operator Nx = a† ax , ax and a†
ˆ ˆ ˆ ˆ
x x
ˆ ˆ
[Nx , ax ] = −ˆx ,
a ˆ ˆ
[ N x , a† ] = a† ,
ˆx [ˆx , a† ] = 1,
a ˆx (28)
x
ˆ
and for Ny = a† ay , ay and a†
ˆy ˆ ˆ ˆy
ˆ ˆ
[Ny , ay ] = −ˆy ,
a ˆ ˆ
[ N y , a† ] = a† ,
ˆy [ˆy , a† ] = 1,
a ˆy (29)
y

using the previous equations I ﬁnd
√ √
ax |nx , ny = nx |nx − 1, ny ,
ˆ a† |nx , ny =
ˆx nx + 1|nx + 1, ny ,
√
ay |nx , ny = ny |nx , ny − 1 ,
ˆ a† |nx , ny =
ˆy ny + 1|nx , ny + 1 , (30)
ˆ ˆ
where nx = 0, 1, 2, . . ., ny = 0, 1, 2, . . . and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny , with
inverse equations
h
¯ mω¯ †
h
x=
ˆ ax + a† ,
ˆ ˆx px = ı
ˆ ax − ax ,
ˆ ˆ (31)
2mω 2

h
¯ mω¯ †
h
y=
ˆ ay + a† , p y = ı
ˆ ˆy ˆ ay − ay ,
ˆ ˆ (32)
2mω 2
in such a way that the hamiltonian can be expressed as
ˆ ˆ ˆ
H = Nx + Ny + 1 hω,
¯ (33)
and the eigenvalue equation can be solved
ˆ
H|nx , ny = (nx + ny + 1) hω|nx , ny ,
¯ (34)
and the spectrum for this case is
Enx ,ny = (nx + ny + 1) hω,
¯ (35)


where nx = 0, 1, 2, . . ., ny = 0, 1, 2, . . ., the vacuum state is obtained as ax ay |0, 0 = 0 and a general
ˆ ˆ
state is given by
1 mx my
|mx , my = a†
ˆ a†
ˆ |0, 0 . (36)
(mx !)(my !)

3.2. Case II

The hamiltonian is
ˆ 1 mω 2 2
H= p2 + p2 +
ˆx ˆy x + y2 ,
ˆ ˆ (37)
2m 2
the previous hamiltonian can be divided in a part corresponding with to the x direction and another in
mω ı mω ı
ax =
ˆ x+
ˆ p x , a† =
ˆ ˆx x−
ˆ px ,
ˆ (38)
2¯
h mω 2¯
h mω
mω ı mω ı
ay = −
ˆ y−
ˆ p y , a† = −
ˆ ˆy y+
ˆ py ,
ˆ (39)
2¯h mω 2¯
h mω
ˆ
with commutation relations between the number operator Nx = a† ax , ax and a†
ˆx ˆ ˆ ˆx
ˆ ˆ
[Nx , ax ] = −ˆx ,
a ˆ ˆ
[ N x , a† ] = a† ,
ˆx [ˆx , a† ] = 1,
a ˆx (40)
x
ˆ
ˆy ˆ ˆ ˆy
ˆ ˆ
[Ny , ay ] = ay ,
ˆ ˆ ˆy
[Ny , a† ] = −ˆ† ,
ay [ˆy , a† ] = −1,
a ˆy (41)
√ √
ax |nx , ny = nx |nx − 1, ny ,
ˆ a† |nx , ny =
ˆx nx + 1|nx + 1, ny ,
ay |nx , ny =
ˆ |ny ||nx , ny + 1 , a† |nx , ny =
ˆy |ny − 1||nx , ny − 1 , (42)
ˆ ˆ
where nx = 0, 1, 2, . . ., ny = . . . , −2, −1, 0 and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny
with the inverse equations
h
¯ mω¯ †
h
x=
ˆ ax + a† ,
ˆ ˆx px = ı
ˆ ax − ax ,
ˆ ˆ (43)
2mω 2
and
h
¯ mω¯ †
h
y=−
ˆ ay + a† ,
ˆ ˆy py = ı
ˆ ay − ay ,
ˆ ˆ (44)
2mω 2
in such a way that the hamiltonian is
ˆ ˆ ˆ ¯
H = Nx + Ny hω, (45)


ˆ
H|nx , ny = (nx + ny ) hω|nx , ny ,
¯ (46)
Enx ,ny = (nx + ny ) hω,
¯ (47)
where nx = 0, 1, 2, . . ., ny = . . . , −2, −1, 0, the vacuum state is obtained as ax ay |0, 0 = 0 and a
ˆ ˆ
general state is
1 mx |my |
|mx , −|my | = a†
ˆ a†
ˆ |0, 0 . (48)
(mx !)(|my |!)

3.3. Case III

In this case the hamiltonian is
2
Hˆ = 1 p2 + p2 + mω x2 + y 2 ,
ˆ ˆy ˆ ˆ (49)
2m x 2
the previous hamiltonian can be divided into a part that correspond with a x direction and another in
mω ı mω ı
ax = −
ˆ x−
ˆ p x , a† = −
ˆ ˆx x+
ˆ px ,
ˆ (50)
2¯
h mω 2¯h mω
mω ı mω ı
ay =
ˆ y+
ˆ p y , a† =
ˆ ˆy y−
ˆ py ,
ˆ (51)
2¯
h mω 2¯
h mω
ˆ
with commutation relations between Nx = a† ax , ax and a†
ˆx ˆ ˆ ˆx
ˆ ˆ
[ N x , ax ] = ax ,
ˆ ˆ ˆx
[Nx , a† ] = −ˆ† ,
ax [ˆx , a† ] = −1,
a ˆx (52)
ˆ
ˆy ˆ ˆ ˆy
ˆ ˆ
[Ny , ay ] = −ˆy ,
a ˆ ˆ
[ N y , a† ] = a† ,
ˆy [ˆy , a† ] = 1,
a ˆy (53)
y

ax |nx , ny =
ˆ |nx ||nx + 1, ny , a† |nx , ny =
ˆx |nx − 1||nx − 1, ny ,
√
ay |nx , ny =
ˆ ny |nx , ny − 1 , a† |nx , ny =
ˆy ny + 1|nx , ny + 1 , (54)
ˆ ˆ
where nx = . . . , −2, −1, 0, ny = 0, 1, 2, . . . and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny
with the inverse equations
h
¯ mω¯ †
h
x=−
ˆ ax + a† ,
ˆ ˆx px = ı
ˆ ax − ax ,
ˆ ˆ (55)
2mω 2


and
h
¯ mω¯ †
h
y=
ˆ ay + a† , p y = ı
ˆ ˆ (56)
2mω 2
in such a way that the hamiltonian can be expressed as
ˆ ˆ ˆ ¯
H = Nx + Ny hω, (57)
and the eigenvalue equation is
ˆ
H|nx , ny = (nx + ny ) hω|nx , ny ,
¯ (58)
Enx ,ny = (nx + ny ) hω,
¯ (59)
where nx = . . . , −2, −1, 0, ny = 0, 1, 2, . . ., the vacuum state is obtained as ax ay |0, 0 = 0 and a
ˆ ˆ
general state is given by
1 |mx | my
| − |mx |, my = a†
ˆ a†
ˆ |0, 0 . (60)
(|mx |!)(my !)

3.4. Case IV

The usual hamiltonian for the two dimensional isotropic harmonic oscillator is
ˆ 1 mω 2 2
H= p2 + p2 +
ˆx ˆy x + y2 ,
ˆ ˆ (61)
2m 2
it can be divided in a part corresponding with the x direction and another in the y direction, both
hamiltonians are independent, for the x direction I can choose
mω ı mω ı
ax = −
ˆ x−
ˆ p x , a† = −
ˆ ˆx x+
ˆ px ,
ˆ (62)
2¯h mω 2¯
h mω
mω ı mω ı
ay = −
ˆ y−
ˆ p y , a† = −
ˆ ˆy y+
ˆ py ,
ˆ (63)
2¯h mω 2¯
h mω
ˆ
with commutation relations between the operators Nx = a† ax , ax and a†
ˆ ˆ ˆ ˆ
x x
ˆ ˆ
[ N x , ax ] = ax ,
ˆ ˆ ˆx
[Nx , a† ] = −ˆ† ,
ax [ˆx , a† ] = −1,
a ˆx (64)
ˆ
and for number operator Ny = a† ay , ay and a†
ˆy ˆ ˆ ˆy
ˆ ˆ
[Ny , ay ] = ay ,
ˆ ˆ ˆ
[Ny , a† ] = −ˆ† ,
ay [ˆy , a† ] = −1,
a ˆy (65)
y

ax |nx , ny =
ˆ |nx ||nx + 1, ny , a† |nx , ny =
ˆx |nx − 1||nx − 1, ny ,
ay |nx , ny =
ˆ |ny ||nx , ny + 1 , a† |nx , ny =
ˆy |ny − 1||nx , ny − 1 , (66)

ˆ
where nx = . . . , −2, −1, 0, ny = . . . , −2, −1, 0 and Nx |nx , ny ˆ
= nx |nx , ny , Ny |nx , ny =
ny |nx , ny with the inverse equations
h
¯ mω¯ †
h
x=−
ˆ ax + a† ,
ˆ ˆx px = ı
ˆ ax − ax ,
ˆ ˆ (67)
2mω 2
and
h
¯ mω¯ †
h
y=−
ˆ ay + a† , p y = ı
ˆ ˆ (68)
2mω 2
in such a way that the hamiltonian can be written as
ˆ ˆ ˆ
H = Nx + Ny − 1 hω,
¯ (69)
ˆ
H|nx , ny = (nx + ny − 1) hω|nx , ny ,
¯ (70)
Enx ,ny = (nx + ny − 1) hω,
¯ (71)
where nx = . . . , −2, −1, 0, ny = . . . , −2, −1, 0, the vacuum state is obtained as ax ay |0, 0 = 0, with
ˆ ˆ
the final expression for a general state
1 |mx | |my |
| − |mx |, −|my | = a†
ˆ a†
ˆ |0, 0 . (72)
(|mx |!)(|my |!)

4. Conclusions and perspectives

In the present paper I have deduced that two different classical systems have more than one quantum
spectra. This differs of my proper knowledge of quantum mechanics that I have learnt along my
physical science studies, it was a great surprise to find this result, I verified all these mathematical
results and all are correct. My personal point of view is that there are more systems with this
characteristic and in fact, using similar arguments for the case of n-dimensional isotropic oscillators
the number of different spectra is undeniable not unique. To finish, the possibility to make further
studies is open, for example, the study of the Kepler problem, a charged particle in a constant
gravitational field, problems in quantum field theory and so on. However, about the uniqueness of the
spectrum and the physical interpretation of my results I can not say more.

References

[1] V. I. Arnold, Mathematical methods of classical mechanics, Moscow, Springer-Verlag, New York, (1980).
[2] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University, Princeton, NJ (1992).
[3] K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 69, Springer-Verlag, Berlin, (1982).


[4] Ricardo Amorim, J. Math. Phys. 50, 052103 (2009).
[5] Vladimir Cuesta, Merced Montesinos and Jos´ David Vergara, Phys. Rev. D 76, 025025 (2007),
e
[6] David Hestenes and Jeremy W. Holt, J. Math. Phys. 48, 023514 (2007).
[7] W. Heisenberg, The Physical Principles of the Quantum Theory, University of Chicago Press, Chicago, (1930).
[8] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, (1981).
[9] J. J. Sakurai, Modern Quantum Mechanics, Late, University of California, Los Angeles, Addison-Wesley Publishing
Company, Inc. (1994).
[10] G. F. Torres del Castillo and M. P. Vel´ zquez Quesada, Rev. Mex. F´s. 50 (6) 608-613.
a ı
[11] Merced Montesinos and G. F. Torres del Castillo, Phys. Rev. A 70, 032104 (2004).

Different Quantum Spectra For The Same Classical System

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