The Magnetic Top of Universe As a Model of Quantum Spin

The magnetic top of Universe as a model of quantum spin

Source file of A.O. Barut, M. Bozic and Z. Maric

Substitution, conversion and transformation by Dusan Stosic

Abstract

The magnetic top is defined by the property that the external magnetic field B coupled to the angular velocity as distinct from the top fhose magnetic moment is independent of angular velocity. This allows one to

construct a "gauge" theory of the top where the caninical angular momentum of the ooint particle and the B field plays the role of the gauge potential. Magnetic top has four constants of motion so that Lagrange equations for Euler angles ,, (wich define the orientation of the top) are solvable, and are solved here. Although the Euk=ler angles have comlicated motion.,the canonical angular momentum s, interpreted as spin , obeys precisely a simple precession equation. The Poisson brackets of allow us further to make an unambiguous quantization of spin , leading to the Pauli spin Hamiltonian. The use of canonical angular momentumalleviates the ambiguity in the ordering of the variables in the Hamiltonian. A detailed gauge theory of the asimmetric magnetic top is alsou given.

Euler angles - The xyz (fixed) system is shown in blue, the XYZ (rotated) system

is shown in red. The line of nodes, labelled N, is shown in green.

Contents

page

Introduction

II . Lagrangian and Hamiltonian of the symmetric magnetic top 6

III. Lagrange equation for the magnetic topand their solutions for

constant magnetic field1 10

IV. The torque equation and its equivalence with the Lagrange

equations 17

V. Hamilton's equations for the magnetic top 18

VI. Quantum magnetic top 21

VII. The states of the quantum magnetic top 26

VIII. The Asymmetric Magnetic top 29

Appendix A.Top with magnetic moment fixed in the body frame 36

I. Inroduction

References 41

Whereas the coordinates and momenta of quantum particles have a classical origin

or a classical

counterpart,the spin is generally thought to have no classical origin. It is, in Pauli

words,"a calassicay non

-explanable two-valuedness"{1} .Thus, the spin and coordinates are not on the same

footing as far as the

picture of the particles is concerned.

In atomic physics the role of spin is enormous due to the Pauli-principle and spin

statistics connection,althougt the numerical values of spin orbit terms are small.

In nuclear and particle physics and in very high energy physics, there spin hyperfine

terms turned out to play an essential role, whose theoretical understandig is still

lacking (2). Even in the interpretation and foundations of quantum theory, the nature

of spin seems to be rather crucial, and a need for a classical model of spin has long

been felt (3).

Our knowledge about the importance of spin in all these areas comes from the

widespread and succesfull applicability of Pauli and Dirac matrices and spin

representation of Galillei and Poincare groups. Although there is no mystery is

actually some mystery in the physical origin and in the visualization of spin.

(It cocerns the spin 1/2 as well as the higher spins). Because of all those reasons

there has been in the past many attempts to identify internal spin variables and to main

clssical models of spin, both of Pauli (4-12) as well as of Dirac spin (13-18)

But , none ofthe nonrelativistic spin models has been generally accepted, either

because none of the propsed models is without shorthcomings and difficulties or

because the prevailling attitude of physicists towards internal spin variables is, in

Schulman's words: a general unconfortablenes at the mention of internal spin variables

and a reliance on the more formal, but nevertheless completely adequate, spinor

wave functions which are labelled basis vectors for a representation of so*3)

but are endowed with no further properties"(10)

In this paper, we shall consider the nonrelativistic Pauli spin, and a minimal

classical model - in the sense of the smallest possible phase space dimension

- underlying the Pauli equation. Our classical model of quantum spin is based

on magnetic top , wich we define as a top whose mafnetic moment is

proportional to the angular velocity(Chapter II) By solving the classical

equation of motion of the magnetic top we shall show that it has, by virtue

of the special coupling to the magnetic field, a unique property that the motion

of its magnetic moment is one dimensional (i.e ptecessio around the magnetic field)

whereas the top itself performs a complicated three-dimensional motion

(Chapters III and IV).

The motion of the magnetic moment of the magnetic top is different in an essential

way from the motion of the top which carries magnetic moment fixed in the body

frame. Namely, a magnetic moment which is fixed to the top preform a

three-dimensional motion (precession with nutation) since it shares the motion

of the body to which it is attached (Apendix A). This distinction is the consecuence

of the differnce in the form of the two Lagrangian. The potential in the Lagrangian

of magnetic top (Chapter II) is angular velocity

dependent whereas the potential of the top which carries magnetic moment is velocity

indeoendent (Apendix A). Also, Hamiltonian of the latter top is simple sum of kinetic

and angular velocity independent potential wheras Hamiltonian of magnetic top is

not of this form(Chapter II).

It is necessery to relize those differences in order to understand the difference

between our work and previous works (8,9,10( on the classical models of spin

which were also based on the top.

In Rosen work, classical model of spin is in fact the top with angular velocity

independent potential (8). In our oppinion this model is unsatisfactory because

for quantum spin there exists the linear relation between magnetic

moment operator and spin angular momentum operator ,whereas, such

a relation does not characterize Rosen's classical model in which it is assumed

that Hamiltonian is a sum of kinetic energy and potential energy is

independent of angular velocity. But this is possible only if is independent

of spin angular momentum.

The Lagrangian of the magnetic top is identical with the Lagrangian of the Bopp

and Haag (9) model of spin. But the procedure of the construction of the Hamiltonian

and subsequent quantization procedures differ in our and in the Bopp and Haag

aproach (Chapter VI).

Certain authors have arged in the past that the top is not an appropriate model

of spin, because its configuration space (which is three dimensional ) is larger

than it is necessert. Namely, in Nielsen and Rohrlich words (11) "quantum-mechanical

perticle of definite spin is essentially one-dimension (since it is completelt by

the eigenstates of one coordinate) so Schulman's formulation seems over complicated".

It follous from our analysis that this remark is not applicable to the magnetic top

because although its configuration space is three-dimensional, the magnetic moment

of magnetic top precesses around constant magnetic field (Chapter III). Moreover, in

the light of this result it becomes understadable why Pauli theory of the spin motion in

a magnetic field has been so succseful despite the fact that it avoids to answer the

question as to what the internal spin variables are and what the variables conjugate

to spin are. The explanation is simple. It is a satisfactory theory for those phenomena

for which only thr motion of magnetic moment is relevant. But, are there phenomena

determined by the motion of the magnetic top itself. Our answer is positive. One

example is the phase change of spinors in magnetic fields (Chapter VII).

II Lagrangian and Hamiltonian of the symetric magnetic top

As stated in the Inroduction we shell use the word"top" to denote the mechanical

object whose orientation in the reference frame is discribed by Euler angles ,,.

Magnetic top by definition has a magnetic moment proportional to its angular

momentum 

(1)

The angular momentum  itself is proportional to the vector of angular velocity 

(2)

are unit vectors of the coordinates system attached to the body

and whose orientation iz the Laboratory frame are three Euler angles

are unit vectors along the axis of the Laboratory reference

frame. The components of in the Laboratory frame are :

The components of in the body-fixsed frame, on the other hand are:

The kinetic energy T.sv of the free symmetrical top is a simple function of (or )

According to classical electrodynamics the potential energy of the magnetic moment M

in a magnetic field B is:

Conseqently , the Lagrangian takes the form:

But , for our magnetic top we assume that the relation(1) is valid. By incorporating

this relation into the Lagrangian we get:

It is important to realize that this Lagrangian is different, in an essential way , from

the Lagrangian studied in classical electromagnetism, where is a fixed vector

in body frame and

where

Following general procedures we need now to express ,, and 

in terms of , and 

Bicause the dependence of the Lagrangian on , and  is

trough angular velocity it is usefull ro express angular velocity through

the cannonical moment , and 

We shallnow define a new vector quantity - cannonical angular

momentum s, by

It is seen ...take the form

The latter relation is analogous to the relation between the

kinetic momentum in the electromagnetic field of the vector potential A

Now we are ready to write the Hamiltonian of the magnetic top

according to

After some algebra we obtain

So ,again the form of the Hamiltonian ......

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III . Lagrange equations for the magnetic top and their

solutions for constant magnetic fields

We shell now write and solve Lagrange equations of motion for magnetic top in a constant

magnetic field, assumed to be directed along the z-axis of the space-fixed reference frame. This assumption does not reduce the generality of our solution, since the orientation of the Laboratory frame may be chosen convenniently. With this assumption the Lagrangian (8) takes the form :

Because this Lagrangian does not depende on f and c the momenta and are integrals of motions :

Hence the corresponding two Lagrange equations reduce to two first order differential equations :

The third Lagrange equation is a second order differential equation

In order to solve the latter equation we shall substitute into it the following expressions

obtained from eqs.(20)

Now we note the remarkable identities

With the aid of those identities we transforme equation (25) to any one of following two forms :

Now multiplying bots equations with = we find

So, we found two other integrals of motion. In order to find (t). It is sufficient to use of them

After some algebraic operations we recognize on the left hand site an integrable function ;x=sin()

where

The solution reads

where

Therefore cos( ) oscillates with the period between the two values determined by

Consequently,  oscillates with the same period between the corresponding values and : depending on the initial condition.

Now we are ready to determine and . By integrating the equation (23) we find :

38 a

38 b

In an analogous way we obtain

39 a

39 b

Not that only (t) depends on the magnetic field

Implicit assumption that sin( )=0 , there exist particular solution of Lagrange equation, which are characterised by : sin(t) for any value of t. Such solution exist for initial conditios : or , , Lagrange equation 20-22 are then equvalent to :

The solution of the latter equations are :

The fact that for those initial conditions the equations give only the dependence of (+) on t and do not give the dependence on t of each angle separately is understandable. When the z-axis of the body frame coincides with the z-axis of laboratory frame the rotation described by (t) and (t) are rotations about the same axis (z-axis) and consequently the angles (t) and (t) do not appear separately but together in a sum.

Having determined the solution of Lagrange equations of motion we may now determine the time dependence of the most important quantity for our purpouse, i.e. kinetic angular momentum (2) and cannonical (spin) - (12). By virtue of the equations (23) and (24) we find that is a constant of motion

Further, taking into account the relation (24) and (30) and introducing the angle such that :

we can write and in the form

Taking into account the solution given in (37) we obtain a simple dependence of -on t.

Cosequently, the dependence of and on tis simple too. The vector precesses around the z-axis with the frequency forming fixed angle with the x-axis.

Taking into account the relation (13) between and s we find that the canonical angular momentum also precesses around time-independent magnetic field

In the case of motion described by the solution (40a ) neither  nor s precess because for and

we have:

But , at the same time the body rotates around z with the frequency