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Open access peer-reviewed chapter

By Gabino Torres-Vega

Submitted: May 29th 2012Reviewed: September 19th 2012Published: April 3rd 2013

DOI: 10.5772/53598

Downloaded: 2630

There are many proposals for writing Classical and Quantum Mechanics in the same language. Some approaches use complex functions for classical probability densities [1] and other define functions of two variables from single variable quantum wave functions [2,3]. Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects. In this chapter, we derive many inter relationships between conjugate variables.

An important object in Quantum Mechanics is the eigenfunctions set

Classical motion takes place on the associated cotangent space

Where

Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton’s equations of motion, and the chain rule, we have that

Now, a point in cotangent space can be specified as the intersection of

Thus, in general, the

A point in this set will be denoted as an abstract bra

We can also have marginal representations of functions in phase space by using the eigensurfaces of only one of the functions,

A point in the set

It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables

A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface

Now, recall that any phase space function

where

Some relationships between a pair of conjugate variables are derived in this section. We will deal with general

The magnitude of the vector field

where

A unit density with the eigensurface

is the classical analogue of the corresponding quantum eigenstate in coordinate

The overlap between a probability density with an eigenfunction of

But, a complete description of a function in

In this way, we have the classical analogue of the quantum concepts of eigenfunctions of operators and the projection of vectors on them.

Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions *F*(*z*) and *G*(*z*) of points in cotangent space *z* ∈ *T*^{*}*Q* of a mechanical system, and a unit Poisson bracket between them,

valid on some domain *p* and *q*, and Eq. (10), suggests two ways of defining dynamical systems for functions *F* and *G* that comply with the unit Poisson bracket. One of these dynamical systems is

With these replacements, the Poisson bracket becomes the derivative of a function with respect to itself

Note that

We can also define other dynamical system as

Now,

The dynamical systems and vector fields for the motions just defined are

Then, the motion along one of the

If the motion of phase space points is governed by the vector field (15),

In contrast, when motion occurs in the

Hence, motion originated by the conjugate variables

The divergence of these vector fields is zero,

Thus, the motions associated to each of these conjugate variables preserve the phase space area.

A constant Poisson bracket is related to the constancy of a cross product because

where

The Jacobian for transformations from phase space coordinates to

and

We have seen some properties related to the motion of phase space points caused by conjugate variables.

We now consider the use of commutators in the classical realm.

The Poisson bracket can also be written in two ways involving a *commutator.* One form is

and the other is

With these, we have introduced the Liouville type operators

These are Lie derivatives in the directions of

Conserved motion of phase space functions moving along the

Indeed, with the help these definitions and of the chain rule, we have that the total derivative of functions vanishes, i.e. the total amount of a function is conserved,

and

Also, note that for any function

and

which are the evolution equations for functions along the conjugate directions

With these equations, we can now move a function

As in quantum theory, we have found commutators and there are many properties based on them, taking advantage of the fact that a commutator is a derivation.

Since the commutator is a derivation, for conjugate variables

Based on the above equalities, we can get translation relationships for functions on

In particular, we have that

Then,

From Eq. (32), we find that

But, if we multiply by

This is a generalized version of a shift of

This is a relationship that indicates how to translate the function

This implies that

i.e., up to an additive constant,

Continuing in a similar way, we can obtain the relationships shown in the following diagram

where the constant

Some of the things to note are:

The operator

The operators

But

The eigenfunction of

The variable

The steady state of

These comments involve the left hand side of the above diagram. There are similar conclusions that can be drawn by considering the right hand side of the diagram.

Remember that the above are results valid for classical systems. Below we derive the corresponding results for quantum systems.

We now derive the quantum analogues of the relationships found in previous section. We start with a Hilbert space

together with the domain

The eigenvectors of the position, momentum and energy operators have been used to provide a representation of wave functions and of operators. So, in general, the eigenvectors

With the help of the properties of commutators between operators, we can see that

Hence, for a holomorphic function

i.e., the commutators behave as derivations with respect to operators. In an abuse of notation, we have that

We can take advantage of this fact and derive the quantum versions of the equalities found in the classical realm.

A set of equalities is obtained from Eq. (43) by first writing them in expanded form as

Next, we multiply these equalities by the inverse operator to the right or to the left in order to obtain

These are a set of generalized shift relationships for the operators

Now, as in Classical Mechanics, the commutator between two operators can be seen as two different derivatives introducing quantum dynamical system as

where

These equations can be written in the form of a set of quantum dynamical systems

where

The inner product between the operator vector fields is

where

We can define many of the classical quantities but now in the quantum realm. Liouville type operators are

These operators will move functions of operators along the conjugate directions

There are many equalities that can be obtained as in the classical case. The following diagram shows some of them:

Note that the conclusions mentioned at the end of the previous section for classical systems also hold in the quantum realm.

Next, we illustrate the use of these ideas with a simple system.

As a brief application of the abovee ideas, we show how to use the energy-time coordinates and eigenfunctions in the reversible evolution of probability densities.

Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave functions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function

With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian given by (we will use dimensionless units)

Given and energy scaling parameter

We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve

With the choice of phase we have made,

These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is

At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momentum. One is to use half lines and

Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon

where we have made use of the equality

where we have made use of the result that

For a function of

This means that evolution in energy-time space also is quite simple, it is only a shift of the function along the

So, let us take a concrete probability density and let us evolve it in time. The probability density, in phase space, that we will consider is

with

Recall that the whole function

This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction

The projection of a wave function onto this vector is

Which is the time dependent wave function, in the coordinate representation, and evaluated at

The time component of a propagated wave function for a time

Then, time evolution is the translation in time representation, without a change in shape. Note that the variable

Now, assuming a discrete energy spectrum with energy eigenvalue

i.e. the wave function in energy space only changes its phase after evolution for a time

Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both theories.

Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as coordinates for representing dynamical quantities.

Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is found in Reference [15].

These were some of the properties and their consequences in which both conjugate variables participate, influencing each other.

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