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
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
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 s i 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      P  P P in the Hamiltonian. A detailed
gauge theory of the asimmetric magnetic top is alsou given.
si
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
3
Introduction
I.
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
2
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    s between magnetic
3


moment operator  and spin angular momentum operator s ,whereas, such
a relation does not characterize Rosen's classical model in which it is assumed
2

that Hamiltonian is a sum of kinetic energy I   and potential energy   B is
2

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


Mtopsv  gsv   sv
 sv  1.171 
10
gsv  5.166 
 4 1 0 -1
10 cm gm sec
(1)
Mtopsv
95
erg  sec


gsv   sv
Mtopsv  6.05 
91
10
erg
stattesla
4


g sv   sv  6.05 
(2)


 sv


Isv  sv
erg
91
10
stattesla
Isv 
 sv..i  ei


Isv  x.sv  i  y.sv  j  z.sv  k
i
ei
are unit vectors of the coordinates system attached to the body
and whose orientation iz the Laboratory frame are three Euler angles
x.sv  sv
i  1  1
ei  1  1
y.sv  sv
sv..i  sv
z.sv  sv
k  1  1
j  1  1
Isv 
 sv  ei 
i
1.171·1095 gm cm2 sec-1
Isv  x.sv  i  y.sv  j  z.sv  k

3
1.171·1095 gm cm2 sec-1
 sv  1.171  1095 gm cm2 sec-1


 I     1.171  1095 gm cm2 sec-1
sv
sv 

i j k
   
are unit vectors along the axis of the Laboratory reference

frame. The components of  in the Laboratory frame are :
x
cos    '  sin   sin    '
y
sin   '  cos    sin    '
 3
z
'  cos     '

The components of  in the body-fixsed frame, on the other hand are:
1
sin    sin    cos     '
2
sin    cos     '  sin    '
 4
5
cos     '  '
3


The kinetic energy T.sv of the free symmetrical top is a simple function of  (or  )
  
Ha  Hb
1
  
  
 5

 2
Isv   sv 


Tsv
2
2
' 
d

d Ha
' 
d

d Ha
' 
d

d Ha

 

 sv 
2
1
Isv
2
   '     '   sin    
2
2
2
  '    '   cos     
2

 

 sv   1.031  1077 gm cm2 sec-2
2
2
Isv
gm  cm
2
   '     '   sin    
2
2
2
2
  '    '   cos       0 gm cm2 sec-2
2

 

gm  cm
sv 
2
2
2
2

   '     '   sin       '    '   cos      
 1.031  1077 gm cm2 sec-2
2
2
2
2
1
2
2
  d  2  d  2
2
 d   d 

           sin              cos     
  dx    dz 
  dz   dy 

Tsv 
Isv

 

 sv 
2
2
Isv

 2
Isv   sv 


2
According to classical electrodynamics the potential energy of the magnetic moment M
in a magnetic field B is:
Bsv  6.816 
 15
10
stattesla
 6
Vsv Mtopsv  Bsv
Vsv  Mtopsv  Bsv
Vsv  4.123 
77
2 -2
10 gm cm sec
Conseqently , the Lagrangian takes the form:
G  Msv
Rg sv
2
 2.062 
1077 gm cm2 sec-2
6




sv 

Isv
 Mtopsv  Bsv  5.154  1077 gm cm2 sec-2
2
2
Conseqently , the Lagrangian takes the form:
 7
Tsv  Vsv
Lsv

 2
Isv   sv 


2
 Mtopsv  Bsv
Tsv  Vsv  5.154  1077 gm cm2 sec-2

 2
Isv   sv 

 M
topsv  Bsv  5.154 
2
1077 gm cm2 sec-2
But , for our magnetic top we assume that the relation(1) is valid. By incorporating
this relation into the Lagrangian we get:

 2
Isv   sv 



 g I 
L
B
sv
2

 2
Isv   sv 


Lsv 
2
sv
sv
sv


 gsv  Isv  sv  Bsv
Lsv  5.154  1077 gm cm2 sec-2

 2
Isv   sv 



  g I 
sv sv sv  Bsv  5.154 
2
1077 gm cm2 sec-2
It is important to realize that this Lagrangian is different, in an essential way , from
the Lagrangian studied in classical electromagnetism, where M is a fixed vector
in body frame and
  90
  45
  
P
 d   I  B1  B 

 dx

I 
  130
Hb  1.76 
Ha  Hb
' 
 18 -1
10
sec
1
d

d Ha
 d L  I  '  I  g  B
 sv
sv sv sv
 d' 
p  
7
Isv  '  Isv  gsv  Bsv  2.342 
95
2 -1
10 gm cm sec
 d   cos     d    I  g1B

z
dx 
 dy
 d
d 

P I     cos        I  g1Bz
dz 
 dz

where
P
I 
sin  
cos   
B
Bx  cos    By  sin 
 cos    sin   0
R1   sin    cos    0

0
0
 0
 cos    sin   0  


 sin   cos    0  
0
0  
 0
  cos    sin   0 


0
sin     sin    cos    0 
 

0
cos     
0
0
1 
1
0
0
  cos    sin   0   0.044

   0.554
0 cos    sin   
  sin   cos    0  
 0
0 sin    cos     
0
0
1 
Bx  sin   sin    By  cos    sin    Bz  cos   
Bz1
 

 
 
1
0
0
cos   
sin   
0.61 0.791
0.674 0.489
0
0



Fo
in terms of P , P and P

0.044


0.554
R1  
d

dx
d

dy
d

dz
0
0.61 0.791
0.674 0.489
0
0



 P

   g1  B
 I 
 P  P  cos     I  g1  Bz  cos   Bz
I  sin   
2
 P  P  cos     I  g1  Bz1  cos   Bz
I  sin   
2


trough angular velocity  it is usefull ro express angular velocity  through
the cannonical moment P , P and P
cos    P 
 sv.x
Isv  sv.x
 sv.y
I  
 sv sv.y

sin  
 P 
sin   
cos  
sin    P 
sin   
sin   cos   
 P 
sin   
 P  gsv  Isv  Bsv.x
cos    cos   
sin  


 P  gsv  Isv  Bsv.y 
sv.x  sv
Isv  sv.x  1.171 
95
2 -1
10 gm cm sec
sv.y  sv
P  Isv   '  cos     '   Isv  gsv  Bsv
8
 
 

2

2
P  P
Bsv.y  Bsv
P  P
cos  
cos    cos   
I  

 P  gsv  Isv  Bsv.y 
 sv sv.y sin   P     P 


sin

sin



sin    P 
P  2.342 
cos  
sin   
 P 
cos    cos   
sin   
 P  gsv  Isv  Bsv.y  0 gm cm2 sec-1
95
2 -1
10 gm cm sec
sin  
 x  cos    P 
sin   
 11
cos  
 y  sin    P 
sin   
sin   cos   
 P 
 P  gsv  Isv  Bsv
sin   
cos    cos   
 P 
sin   
 P  gsv  Isv  Bsv
 z  P  gsv  Isv  Bsv
 z  0 gm cm2 sec-1
s x  cos    P 
sin 
sin  
 P 
sin   cos   
sin  
 P
 x  0 gm cm2 sec-1
 y  0 gm cm2 sec-1
s y  sin   P 
cos  
sin  
 P 
cos    cos   
sin  
 P
s z  P
s x  2.342 
95
2 -1
10 gm cm sec
Isv  gsv  Bsv  2.342 
Isv  sv.y  1.171 
95
2 -1
10 gm cm sec
95
2 -1
10 gm cm sec
s x  gsv  Isv  Bsv  0 gm cm2 sec-1
z
I  x
P  g1  IBz
We shallnow define a new vector quantity - cannonical angular
momentum s, by
sx
cos    P 
sy
sin   P 
sz
P
It is seen
sin 
sin  
 P 
sin  
 P 
cos  
sin   cos   
sin  
 P
sin  
 P
cos    cos   
...take the form
9


s  g1  I  B
The latter relation is analogous to the relation between the
kinetic momentum in the electromagnetic field of the vector potential A
1
L
 m q  e  A  q
2
2
1
m  q1
P
d
L
dq1
1
m q p  e  A
p m q  e  A

m  q  exp 1  A
Now we are ready to write the Hamiltonian of the magnetic top
according to
 

2
  
P  2.342 
95
2 -1
10 gm cm sec

  2

    g1  I  
H   P    P   I 
B 
2


P  3.513 


95
2 -1
10 gm cm sec
 2

 

sv 











 
 I 
 g sv  Isv 
  sv  Bsv   
2
sec
sec  sv
Msv  c
1.428  1019
2
P

P
sec
 
P
P
sec

  Isv

 2
 

  sv 
2







 gsvIsv
   svBsv  
19
1.42810
1.947
2
10 gm cm sec-2

 5.293 
10 9 cm
85
10
 2
 
sv 


   g  I    

Isv 
sv sv   sv  Bsv  
2
 Msv  c2 


 2 
P

sec
 
P
sec
  P  1.104 
 
 1.472 
 3
1096 gm cm2 sec-2
96
2 -1
10 gm cm sec

2
After some algebra we obtain
 
y  a0
1
1
x  a0
1
z  a0
s 
 
s
2

2
2
 g1  s  B  g1  I  B
2
2
s
 sv
I
2
 g1  s  B  g1  I  B
2
2
I

 
2
2
 2
 
H I
gsv  5.166 
s
2
2
2
M
2
I
( s  g1  I  B)
2
I  g1
2
2
I
2
s
2
2
 g1  s  B  g1 I  B
2
I
2
 4 1 0 -1
10 cm gm sec
2
2
 g1  s  B  g1  I  B 
2
I
 
2

2
11
 2
 
sv 


Isv 
2
























 2

 
 sv 

2I

sv

2
M topsv


2
2  Isv  g sv


2
s  gsv  Isv  Bsv


2I
sv

2
s
2
2 
 g sv  s  Bsv  g sv  Isv  Bsv

2  Isv

5

Mtopsv
2
2
gsv  5.166 
2
 1.031 

1.031
10
77

10
77
1.031

10
77
2.319

10
77
1.289

10
77



 erg



1077 gm cm2 sec-2
 4 1 0 -1
10 cm gm sec
2
Isv
 g sv  s  Bsv  g sv  Isv  Bsv
2
6.25




















1.031
2
Isv  g sv
s







2
 1.031 
1077 gm cm2 sec-2
 2
 
sv 


Isv 
2





 2

 
 sv 


2I
sv

2
M topsv


2
2I

sv  g sv

 sgsvIsvBsv 2

Isv


 4.5

2
s
2
2 
 g sv  s  Bsv  g sv  Isv  Bsv

2I
sv

6.25








1.031

1077
1.031

10
77
1.031

10
77
1.031

10
77
1.031

10
77



 erg



12
2


.50  I
sv  sv


2


 sv


.50 

Isv



2
M topsv


.50 


2
Isv  g sv




2
s  1.  gsv  Isv  Bsv


.50 


Isv


 .10  s 2  .20  g  s  B  .20  g 2  I  B 2 
sv
sv
sv sv sv 

Isv

4










1
4
g sv  Bsv
2
g sv  Bsv
2
Msv  c

2
2
2

2
2
2
2
  2  g sv  s  Bsv  5  Msv  c  4  g sv  Bsv  s 



1
4

2

M sv  c

  2  g sv  s  Bsv  5  Msv  c  4  g sv  Bsv  s 

2
2
2
2
20 
20 
g sv  s  Bsv  Msv  c 
2
g sv  s  Bsv  Msv  c 
2
25 
25 
Msv  c
2
Msv  c
2
2
2
 1.031 
5.572  10
1077 gm cm2 sec-2
Isv
112
 gm  cm 
2
 1.194
 Mtopsv 


  sv   5.166  10 4 
cm1 gm0 sec-1


4 
Mtopsv

5.166  10




  sv 


s
2
2
 g sv  s  Bsv  g sv  Isv  Bsv
2
Isv
 1.289 
5
2
2
1077 gm cm2 sec-2
  6.283
So ,again the form of the Hamiltonian ......
1
2
2
  d  2  d  2
2
 d   d 

           sin              cos     
  dx    dz 
  dz   dy 

1
2
H
me
 (p  e  A)
1
2
m
2
 ( p  e  A)
 
2
me  c
2
2
2
I
2
2
2
2
  d  2  d  2
 me  c  1 
2


d


d


           sin              cos      
 2.18  10 11 erg
2
2
2  sec
  dx    dz 
  dz   dy 

gm
13

4

4
 1 
 2 

 1 
 2 











1
2
L
me
1
2
 ( p  e  A)
2
( s  g1  I  B)
2
2I
 me  q  e  A  q
2
 1

m  g  h1 2 
 1  2 e
c
 me

 1

 2
 1  2  me  g  h1
 me
c

Ovde dodje tekst


  1.589  10 9 

 cm0 sec0

1.589  10 9 





g  h1  1.034  10 24 gm cm3 sec-3
h1
2g
 4.322  10 41 gm cm sec-1
2
2
c  1


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 :
18
 Isv

2
2
2
Lsv1  
  '  '  '  2  '  '  cos      gsv  Bsv  Isv   '  '  cos       sec  gsv  Bsv  Isv
 2

Lsv1  2.342 
95
2 -1
10 gm cm sec
Because this Lagrangian does not depende on f and c the momenta P and
P
integrals of motions :
d d
d
Lsv1  Lsv1
dHa d'
d
d
P  sec
d Ha
19
d d
d
Lsv1 
Lsv1
dHa d'
d
d
P  sec
dHa
Hence the corresponding two Lagrange equations reduce to two first order differential equations :
20
'  '  cos     g sv  Bsv
P
Isv
21
'  cos     '  g sv  Bsv  cos   
P
Isv
The third Lagrange equation is a second order differential equation
22
14
d2
'' 
d Ha
2

d d
d
Lsv 
Lsv  0 gm cm2 sec-2
dHa d'
d
''  '  '  sin   gsv  Bsv  '  sin   gm  cm  0 gm cm2 sec-2
2
In order to solve the latter equation we shall substitute into it the following expressions
23
P  P  cos   
1' 
Isv  sin   
 g sv  Bsv
2
24
P  P  cos   
Isv  sin   
1' 
2
 g sv  Bsv  0 sec-1
P  P  cos   
Isv  sin   
P  P  cos   
Isv  sin   
2
2
 3.521 
10 18 sec-1
obtained from eqs.(20)
P  P  cos   
Isv  sin   
25
'' 
2

P  P  cos   
Isv  sin   
P  P  cos   
2
Isv  sin   

 1.24 
P  P  cos   
Isv  sin   
10 35 sec-2
 1.24 
10 35 sec-2
Now we note the remarkable identities
P  P  cos   
sin   
26
d
d'
P  P  cos   
sin   
P  P  cos   
sin   
27
d
d '
 2.342 
2
P  P  cos   
2
sin   
1
 2.342 
P  P  cos   
sin  
 sec
1095 gm cm2 sec-1
 0 gm cm2 sec-1
1095 gm cm2 sec-1
 0 gm cm2
 2.342 
1095 gm cm2 sec-1
With the aid of those identities we transforme equation (25) to any one of following two forms :
15
'' 
P  P  cos   
P  P  cos   
d

Isv sin   
Isv sin   
d '
P  P  cos   
P  P  cos   
d

 0 sec-1


Isv sin   
I
sin

d '
sv
''  0 sec-1
'' 
P  P  cos   
P  P  cos   
d

Isv sin   
Isv sin   
d '
P  P  cos   
P  P  cos   
d

 0 sec-1
Isv sin   
Isv sin   
d '
Now multiplying bots equations with '  dHa = d we find
 P  P  cos    
d'2  
 Isv sin   
 P  P  cos    
d'2  
 Isv sin   
2
2
 P  P  cos    
 1.24  10 35 sec-2
 Isv sin   
2
 P  P  cos    
'  
 1.24  10 35 sec-2
 Isv sin   
2
2
' 0  '
 P  P  cos    
A  '  
 Isv sin   
2
2
 P  P  cos    
' 0  
 1.24  10 35 sec-2
 Isv sin   
2
2
 P  P  cos    
' 0  
 1.24  10 35 sec-2
 Isv sin   
2
2
A  1.24 
2
10 35 sec-2
 P  P  cos    
B  
 Isv sin   
2
So, we found two other integrals of motion. In order to find
d
  P  P  cos     
1  
  A  Isv  sin     
A  dt
2
16
d
  P  P  cos     
1  
  A  Isv  sin    
A  dt
2
or
After some algebraic operations we recognize on the left hand site an integrable function
32
dcos   

a  b  cos   c  cos   
dt

2
 0  
' 0  
  1.571
where
sin    1


 P2  P 2  cos  02  2  P  P  cos  0
a ' 0 
2
2
a  ' 0  
2

 
Isv  sin  0

 P  P  cos  02  2  P  P  cos  0

2
2
-2
gm cm sec
2
2
 
Isv  sin  0
' 0  '
2
b
P  P
Isv
2
' 0  '
2
P  P
Isv
2
 2.479 
 
10 35 sec-2
   '  '0  gsv  Bsv  1  cos   
 ' 0  gsv  Bsv   '  ' 0  gsv  Bsv   1  cos      1.518  10


b  2  cos  0   ' 0  ' 0  gsv  Bsv

2
 
2
cos  0   ' 0

2
33
2
2
2

2
sec-2

 ' 02  gm2  cm4  sec 2  P 2  2  P  P  cos  0  P2
c1  
2
2

Isv  sin  0

c1  2.479 
51


10 35 sec-2
33
2
2
2
2
 ' 0  sec  ' 0  gsv  Bsv  ' 0  2  cos     ' 0  ' 0  gsv  Bsv  2.467 sec-2


b  b  gm  cm
The solution reads
  4  a  c1  b
2
17
cos    

2
c
 2  c  cos  0  b 
b



 2c
 sin  c  t  asin 
34
cos    
 2  c  cos  0  b 
b



 2c
 
2
 sin c  t  asin 
c
where
35
4
 

a  c1  b  4   ' 0  ' 0  sin  0   ' 0  ' 0  gsv  Bsv


4
2
  4  a  c1  b
2
2
2
2
 
4
2

 sin  0  ' 0  ' 0  gsv  Bsv
 
2
2
Therefore cos
 

T0
2
  determined by

c
between the two values
cos  1 and cos  2
36
cos  2 
   b
cos  1 
   b
2
2
c1
c1
T0
 2 :  2 T0  1
between the corresponding values  1 and
depending on the initial condition.
Now we are ready to determine  ( t) and ( t) . By integrating the equation (23) we find :
t  Ha
t
 P  P  cos   
 

g sv  Bsv  t  
dt
2


I

sin


sv
0


P  P

1
g sv  Bsv  t  

d   2
2
 Isv  sin    d 

dt

0
37
P  P
 cos   



P

P
A

I

sin



sv

g sv  Bsv  t  
d
2
2
 P  P  cos    
 Isv  sin   

1
0
 A  Isv  sin    



 0  1.571
18
P P
AIsvsin(  )

1
0


 A  Isv  sin    
P  P

2
0  
38 a
0  g sv  Bsv  t 
P  P
   P  P



 cos      asin 
 cos  0   
  asin
   A  Isv  sin  

 A  Isv  sin 0

38 b
P  P
   P  P



 cos      asin 
 cos  0   
  asin
   A  Isv  sin  

 A  Isv  sin 0

In an analogous way we obtain
39 a
P  P
  P  P



 0   asin 
 cos      asin 
 cos  0  
  A  Isv  sin  

 A  Isv  sin 0

39 b
P  P
  P  P



 0   asin 
 cos      asin 
 cos  0  
  A  Isv  sin  

 A  Isv  sin 0

0  g sv  Bsv  t 
Bsv
e equation, which
0
or  0  , ' 0
0
P
, 0
Isv
P0
Isv
' 0  ' 0  g sv  Bsv.
0
Lagrange equation 20-22 are then equvalent to
:
P0  P
P0  P
  ' 0
'  ' 0
P0
P0
Isv
Isv
P  P0
' 0  ' 0  g sv  Bsv  1
41
'  '  g sv  Bsv
 ' 
P0
P0
P
P
Isv
Isv
Isv
Isv
1

0
 0  
The solution of the latter equations are :
19
 P

 I  gsv  Bsv   t  0   0  3.142
 sv

 ( t)   0

2
 1.571
 ( t)  1.571
 P

  
 g sv  Bsv   t  0   0
 Isv

z
0  Isv    Isv  t  P  t  gsv  Bsv  Isv  0  Isv  1.571
Isv
Th
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 t
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  z is a
constant of motion
 P

 z Isv  
 g sv  Bsv  P  g sv  Bsv  Isv  z0
 Isv

41
P  gsv  Bsv  Isv  0 gm cm2 sec-1
 P

Isv  
 g sv  Bsv   0 gm cm2 sec-1
 Isv

 z0
Further, taking into account the relation (24) and (30) and introducing the angle such that :
'
'
A  cos  
sin  
A  sin 
 P  P  cos    
 '  0
 A  Isv  sin   

  asin 

asin 
 P  P  cos    
 '  0
 A  Isv  sin   
we can write  x and  y in the form
x
y
A  Isv  cos    
A  Isv  sin   
Taking into account the solution ( t) given in (37) we obtain a simple dependence of
-
20
t.
 P  P  cos  0 
 ' 0  0
A  Isv  sin  0 


( t)  
0  gsv  Bsv  t  asin 

( t)  
0    gsv  Bsv  t  asin
 P  P  cos  0 
 ' 0  0
 A  Isv  sin 0 



Cosequently, the dependence of  x and  y on tis simple too. The vector  precesses around
the z-axis with the frequency L gsv  Bsv forming fixed angle  s with the x-axis.
  2 2
 x y 
 s atan
 

z


A  Isv
P  g sv  Bsv  Isv
e canonical angular
momentum also precesses around time-independent magnetic field B Bk
sz
z
P
P  2.342 
95
2 -1
10 gm cm sec
s z  2.342 
95
2 -1
10 gm cm sec
sx

 

44
sy

 
 P  P  cos  0  

  1  sign  0
2

 A  Isv  sin   
A  Isv  cos  0  g sv  Bsv  t  asin 


y
sin 
 P  P  cos  0  

  1  sign  0
2

 A  Isv  sin   
A  Isv  cos  0  g sv  Bsv  t  asin 


x

 and
0
' ( t)
0
we have:
41
x
0 y
sx
0 sy
0 z
0 sz
P  gsv  Bsv  Isv
P
P
But , at the same time the body rotates around z with the frequency gsv  Bsv  
I
sv
which is different from Larmor frequency L  gsv  Bsv
As we are going to prove in the following section, this result reflects the fact that the potential
V  gsv    Bsv
in the Lagrangian(8) comes from the torqe N  Mtopsv  Bsv
which governs the motion of   Isv  sv
according to the well known torqe equation. In fact we shall prove that the Lagrange equations
are equivalent to differential equations for
equation, an give two other proofs of the spin precession equation.
21
IV .The Torque equation and its
equivalence with Lagrange equation
We are going to demonstrate the equivalence of the torque equation
Mtopsv  Bsv  4.123 
77
2 -2
10 gm cm sec
gsv   sv  Bsv  4.123 
'  0 sec-1


d
 sv N
dt
77
2 -2
10 gm cm sec
Mtopsv  Bsv


gsv   sv  Bsv
sin  
wits the Lagrange equation(20,21) and (22) by substituing into the torque equation the
expressions (3)
'  0 sec-1
t  Ha
46
 d cos    '  sin   sin    ' g  B   sin   '  cos    sin    '  
 
 
sv sv
 dt

Ha  5.68 
17
10 sec
  0 sec-1
47
d
 sin   sec  '  cos    sec  sin   sec  '
dt 
gsv  Bsv   cos    '  sec  sin   sin  '  sec  '  sec 

48
d 
'  cos     ' 
dt
From eq.(48) we obtains immediately one of the integral of motion
49
 '  cos      '
z
 x0
Isv
Isv
But , this equation is equvalent to the Lagrange equation (20) , the relation between the constants
being :
50
z
P  gsv  Bsv  Isv
summing the resultant expression, we find the Lagrange (22).The third equvalence between the
Lagrange and torque equations may be established after the following operations. First, we
multiply (20) with 51
'  sin   
P
Isv

P
Isv
 cos   
''  
Differentiation of the latter equation gives :
22
52
''  sin     2  cos     '  ' 
P
Isv
 '
respectively and then summing the resultant expression. Hence the equvalence is proved.
V. Hamilton's equations for the magnetic top
From (16) we eassily derive Hamilton's equations for the magnetic top
'
d
H
d P
'
d
H
dP
P
Isv
P  P  cos   
Isv  sin   
53a
'
P'
d
 d



2
P  P  cos   
d
H
d P


 g sv  Bx  cos    By  sin  
Isv  sin   
H
2

Bx  sin  By  cos   cos    B
 gsv 
sin   
 Bx  sin   By  cos   

sin   


 gsv  
P  cos     P  cos     P  P 
2
2
 g sv  By 
z
2
cos   P  cos    P
sin   


sin    P  P  cos     

 1  cos       gsv  Bx  

sin   
2
 

2
 2

sin    P  P  cos      
 P  cos     P2  cos     P  P   1  gm  cm2  cos    2   gsv  Bx  
  
2
sin   


 
cos    P  cos     P



  gsv  By 

2
sin   


P'
d
 H
d
53b
P'
d
 H
d
By taking B along z axis, we obtain the simpler equations
'
P'
'
P
P'
P
Isv
P  P  cos      P  cos     P
Isv  sin   
P  P  cos   
Isv  sin   
const
2
2
 g sv  Isv  Bsv
0
23
P  P  cos   
'
Isv  sin   
const
P
P'
2
0
which were derived form the Lagrange equations (20) and (21). By combining the equations for
P' through
P'
''
Isv
we find the Lagrange equations (25).
Now we shall show that Hamilton's formalisme for magnetic top leads also to the torque
this purpose we shall use the Poisson-bracket formalism.
By applying the general dynamical for any quantity u{q.a,p.a) in phase space (q,a,p,a) for the
 d     d H   d     d     d H    d  d H    
 dq i   d  dp i  dq i   d   dp 1 d  i j 
      j   
j
    i 

For the Poisson brackets of spin components we after some calculation
d
1
dt
 i H
 x  y
 z  gsv  Isv  Bsvz
 z  gsv  Isv  Bsv  2.342 
 x  z
  z  y 
95
2 -1
10 gm cm sec
 y  gsv  Isv  Bsvy
 x  gsv  Isv  Bsv.x
We have also from(16)
56
d
x
H
d x
Isv
d
y
d y
d
d z
H
H
Isv
z
Isv
By supstitution (56) and (57) into (55) we find again the torque equation (45), i.e.
57






d
x
dt
gsv   y  Bx   z  By
d
y
dt
gsv   z  Bx   x  Bx
58
d
z
dt
gsv   x  By   y  Bx
24
It is well known that it follows from (58) that  2 is a constant of motion
59
d 2

dt
0
Before we start to quantiye this system let us note that due to the equalities
d
i
dq


d
si
dq


d
si
dp
d
s i  gsv  Isv  Bi
dq
60
d
i
dp
d
s i  gsv  Isv  Bi
dp
we have the follwing important relations
61
 i  j s i s j
Taking this relation into account we find the Poisson brackest of the components of the canonical
angular momentum or spin vector s.
62
s i s j
( i j k)  s k
as wellas the dynamical equation for s
58 '
d
s
dt
g sv  s  Bsv
VI. Quantum magnetic top
In order to quantze the motion, we shall aply two standard quantization procedures.1) Cannonical
quantization and 2) Schrodinger quantization. The third form of quantization, the path integral
formalism, will be discussed separately.
1) Canonical quantization
It is well known that in the framework of this formalism one passes from the classical to the
quantum case by replacing the classical dynamical variables f(p,q) , g(p,q), etc. by operators F,G,
etc.in some Hilbert space of states, in such a way that the Lie product in the space of classical
functions, defined as a Poisson bracket :
d  d
d  d
( f  g)  f   g   f   g
d
q
d
p
 
 dp  dq
is replaced by the Dirac commutator (quantum Poisson bracket)
 ( F G)0  ( i  h)  1  (F  G  G  F)   ( i  h) 1  (F G)
which now plays the role of the Lie product in the space of operators.The Dirac Lie product
conserves the structure of Lie algebra of classical functions with Poisson bracket as the Lie
product. The equation of motion for a dynamical variable F now reads
d
F
dt
1
i h
 ( F  H)
( F  H)
Q
where H is Hamilton operator associeted with the classical Hamiltonian H(p,q).
The basic quantity of the magnetic top is cannonical angular momentum s. Taking into account
the Poisson bracket (62 )
of the components of s and the requirement that the quantum Poisson bracket (s.i,s.j)^0 have to
25
conserve the structure of the classical Lie algebra we may immediatly write the Dirac bracket of
the components s.i of the operator of cannonical angular momentum s.
s i  s j
 ( i j k)  s k
It follows strainghtforwardly that the commutators of the components of s have to be :
s i  s j s i  s j  s j  s i


i  h   ( i j k)  s k
One further step leads now to Hamilton operator of the quantum magnetic top. Inthe classical
Hamiltonian (16) canonical angular momentum s has to be substituted by the operator s.
s
H
2
s
2
Isv
g sv  Isv  Bsv
2
2
 g sv  s  Bsv 
2
g sv  Isv  Bsv
2
2
Isv
 gsv  s  Bsv 
2
2
2
 2.319 
1077 gm cm2 sec-2
The components of the well known Pauli spin operatpor
0 1

x  
 1 0
1 0

z  
 0 1 
0 1

y  
 1 0 
 sv
0
5.855  1094 

 x  
 gm cm2 sec-1
2
0
 5.855  1094

 sv
0
5.855  1094 

 y  
 gm cm2 sec-1
94
2
0
 5.855  10

 sv
0
 5.855  1094

 gm cm2 sec-1
94
2
0
5.855  10 

satisfy the commutation relations (65) and therefore Pauli operators represent one possible
representation of quantum canonical angular momentum operators. But of cource there are many
other bigher dimensional representationss.
In the two-dimensional spin space spanned by two eigenstates of s.z
1
s  
 0
0
s  
 1
 z  
the cotribution of the term
s
2
Isv
g sv  Isv  Bsv
2
2
 g sv  s  Bsv 
2
2
to the eigenstates is constant
(independent of the state) and we argue that those two terms in the quantum Hamiltonian give a
constant energy shift. In this way we conclude that Pauli Hamiltonian
69
  sv 
    Bsv
HP gsv  s  Bsv gsv  
 2 
  sv 
0
2.062  1077 
  x  Bsv  
gsv  
 gm cm2 sec-2
77
 2 

2.062

10
0


is the dynamical part of the Hamiltonan and one of the quantum representation of the magnetic
26
quantum top
One shorthcoming of this representation is that it does not contain quantum analogues of
quantization (22).
ii^0) Schrodinger quantization
operators of canon
i   sv 
d
P
i   sv 
d
P
i   sv 
d
P
d
70
d
d
find the differential representation of the s.x,s.y,s.z.
sx
71
sy
sz
 cos     d i  h   sin   cos     d i  h  sin   d i  h 




sin    d 
sin    d

 d


 sin    d i  h   cos    cos     d i  h  cos    d i  h 




sin    d
sin    d 

 d


d
d
i  h
It is eqsy to see that commutators of the above differential operators sartisfy the commutation
relations (65) . By squaring the operators (71) and by summing the resultant expression we obtain
the differential representation of the operator s^2.
d 2
 d2 2  d2 2 
cot  
1
2
2
2
2
d2 2
d2
2
s
sx  sy  sz
h  cot     h 

h 
h   2 

h
2
2
2
2
2 
sin    d  d 
d
sin     d
d
 d 
The differential representation of the Hamilton operator (66) reads :
 d2 2
d 2
 d2 2  d2 2 
cot   
1
1
d2
2 
d 2
H

h  cot      h 

h 
h   2 

h    g sv  Bsv  i   h  
2
2
2
2
2
2I


sin

d

sin     d 
d   d  
 d 
sv  d 

 d 


g sv  Isv  Bsv
2

2
2
As in the case of Pauli representation, in the subspace spanned by the eigenstates of s^2
associated with the eigenvalue s*(s+1), the contribution of the first two terms to energy
eigenvalues is independent of the states. Thr ramaininig term is another possible representation of
the Pauli Hamiltonian
Bsv
Bsv  k
27
s

1
HP
g sv  i  h  Bsv 
2


d
d
We want to stress here that s is quantum analogue of the canonical angular momentum s and not
coincides with the canonical angular momentum s. In the works of Bopp and Haag (9) and Dahl
(13) the operators (71) and (72) have been derived starting form the free top and from the angular
les at
point with radius vectors r.1 and r.2 (with constant mutual angle u).
73

Isv  sv
P1  P2  P2  r2
corresondence rule (70). Rosen also uses those differential operators (8).
The subsequent procedure of Bopp and Haag in the presence of the field consists in the
the field into the following relation between angular momentum compone
(denoted in their paper by
Mx
x
My
y
cos    P 
sin 
sin  
 P 
sin   cos   
sin  
 P
 cos    P  cos    cos     P 


sin  
 sin  

sin   P  
74
Mz
z
P
But , as it is seen from (11) this latter relation is valid in the absence of the field. In this way
Bopp and Haag obtained the relation
75
M
'M  M'
Isv  sv  gsv  Isv  Bsv
Isv  sv  gsv  Isv  Bsv  3.513 
95
2 -1
10 gm cm sec
which
In this way they found
76
2
g sv  Isv  Bsv

M 
 H 2  I   gsv  M  Bsv 
2
sv 

In the next step Bopp and Haag claim that the quantum analogue of M is the operator (71)
In the above reasoning the justification of the use of the relation (74) in the presence of the
field is missing. Consequently, the theoretical meaning of the relation (75) (the relation (36) in
Bopp and Haag paper) is missing too. In our reasoning, which strictly follows the standard
procedure for the construction of the Hamiltonian (which has to be considered as a function in
phase space
(11) which takes the place of Bopp and Haag relation (360. But , then we define in (12) a new
quantity s and we look for the quantum analogue of this quantity. In this way we make a clear
2
2
this distinction is theoretically justified in the framework of Hamiltonian formalism. Moreover,
28
the analogousdistinction betweein the kinetic momentum mv and canonical momentum p is
standard in the gauge theory of point particles. On the other hand, theoretical status of Bopp and
Haag quantity M,M' and'M has not been established.
The quantization based on the form (66) of the Hamiltonian has one more advantage. One
discovers this advantage if one tries to quantize on the basis of the Hamiltonian expressed
through phase space variables
2
2
2
 2
P

g sv  Bsv Isv
cos   
1
2
H
 P 
 P  2  P  P 
 gsv  Bsv  P 
2
2

2
2  Isv 
sin   
sin   

  
gsv  Bsv  Isv
2
gsv  Bsv  P 
2
P  P 
cos   
sin   
2
2
2
3
 cm
 1.237 
 1.476i 
271
10
1078 gm cm2 sec-2
2 2 -2
gm cm sec
2
 2
P

2
 P 
 P  2.749  10109 gm cm2 sec-2
2

2  Isv 
sin   

The direct substitution of the phase space variables by operators (70) into the above form of H
leads to the operator which differs from Hamilton operator (66a) by the absence of the terms -
1
d
d
This diff
that the use of canonical angular momentum implicitly alleviates this ambiguity and provides the
correct ordering
VII The states of the quantum magnetic top
s
1
2
With Pauli representation of the spin operators, the associeted quantum states are the spinors
   which are linear combinations of two basic states  1  and  0  , namely the eigenstates
 
 0
 1
  
1
0
  
     0    1 
  
The two eigenstates of the Pauli Hamiltonian are very often written in terms o
 I B
 B B
e
2


     
 2  
  cos 
1
0
   eI B  sin    

2 
0
1
 

78
i B
    sin    1    e i B  cos      0  
 2
 0 
 2   1 
In this way of writting one stresses the fact that the eigenstates of the spin Hamiltonian in a
magnetic field are the eigenstates of the component s.B of the spin operator s.
As is well known, the differential operators (71) and (72) can act on larger spaces of states
than the space of Pauli spinors and these spaces are richer in informations than are Pauli states.
 B B
e
2
  
29
The operator s^2 has the eigenvalues s*(s+1) where s takes all integer and half integer values. In
the corresponding subspaces D^s thetwo-valued representations of the Rotation group are
realized (9).
In the case of s=2/2, which is of interest to us here, the basic states of D^1/2 are usually
chosen to be the eigens
u1
2
      

       cos 
2
2   2 



i e

2 
i


2
79 a
u 1
2
      

       sin 
2
2
2
i  e     
2 
i


2
or
u1
2
      

       cos 
2
2   2 



i  e

2 



        cos 
2
2
2

i  e     
2 


i
2
79 b
u1
2
      
i
2
Therefore ,the use of differential operators (71) instead of Pauli operators (67) implies the
description of spin states by probabillity amplitudes u.n and their linear combinations insread by
1
0
matrices   and  and their linear combinations.
 0  1

Pauli spinor   ?
  
range and abstruse
quantum-mechanical object fitted into the general quantum-mechanical framework. From this
advantage follows the second one. It is telated to the understanding of the law of transformation
of spin states under rotation.

The property of spinors  
  

 B 

cos 

2 




 i x
 i x
 1  B   
2
2
 B 




Rz       e
e
e

 
sin
  i  B 
 2 
e


 B 


cos 

 2 








30

 B 


cos 


 2 




 i x
 i x
 1  B   
B  
2
2





Rz       e
e
e

  i  B  sin  2  

e

 B  


cos 

 2 
has been the subject of studies (both theoretically and experimentally), discussions and
controversies (25-31). The source of controversies lies in the difficulties to physically
1
0
understandthis property. Namely, if one uses for the states   and   the usual physical
 0
 1
picture of the spin vector alongz-axis, one can hardly understand what is the physical reason for
the phase changes by It seems that these difficuilties are removed if one interpretes the spin property as a
Rz   
u1
2
     
1
i e
2
(      )

  
2
cos 
2

2
81
Rz   
u 1
2
1
     
i e
2
(      )

  
2
sin 
2

2
In our study of the classical magnetic top we saw that to the simple precession of spin with
frequency -g.sv*B.sv corresponds a more complicated motion of the magnetic top, in which the
spin is along the z-axis) the body rotates with frequency -g
spin vector changes by ly leads to the initial orientation.
We expect that those differeces in the motion of the angular momentum and of the body, in
the classical case, have their counerparts in the quantized motions. They might explain the
strange transformation properties of spinors under rotation. But, the full understanding requires
more detailed study of the quantized motion of the magnetic top.
gsv  Bsv  3.521 
g sv  Bsv 
P
Isv
P
Isv
 3.521 
 18 -1
10
sec
 7.042 
 18 -1
10
sec
 18 -1
10
sec
VIII. The Asymmetric Magnetic Top
It seems worthwhile to generalize the above study to the case of an asymmetric top for which the
longer valid. Istead the following relation holds
31
82

 Ii  i  ei
i
where e.i are unit vectors along the body fixed frame for which the moment inertia tensor is
diagonal. In addition we shall assume, insread of relation (1), the more general relation between
the kinetic angular momentum and magnetic momentum
M
 gi   i  ei  gi  Ii  i  ei
i
i
83
Consequently , the Lagrangian of the magnetic top in a magnetic field B=Bk reads
i  1  1
84
2
 Isv  sv2
Isv  sv

L
 Mtopsv  Bsv

g sv  Isv  sv  Bsv
2
2

i
i
 i
where B.i are the components of B in body-fixed frame
85
 sin    sin   
B
Bi  ei B   sin   cos   
i
cos   






i
Isv  sv
2
2


i

sv  Bsv1 



and
Bi Ii  gi  Bi

Isv  sv
2
 Mtopsv  Bsv  5.154 
2
77
2 -2
10 gm cm sec
i
Bsv1  Isv  gsv  Bsv

Isv  sv
2

2
i

 gsv  Isv  sv  Bsv 
5.154

77
2 -2
10 gm cm sec
i
Isv  sv
2

2
i
 sv  Bsv1 
3.217  10
89
eV
i
Bohr radius by coeficient

Isv  sv
2
2

i
 sv  Bsv1
i
9.74 
10 gm cm sec-2
 5.292 
10 9 cm
85
As in the case of symmetric top, the three coordinates which determine the orientation of the top
in the laboratory frame are :
q1

86
q2

q3

32
Since the components of angul
of the angles, it is appropriate to write the set of relations (4) in matrix form
87
 C( q)  q'
where
 cos    sin    sin   0 
C( q )  sin    sin     cos    0 


0
cos   
1 

88
 ' 
q'  ' 
 
 ' 
Using this notation we write the Lagrangian as
84 a
2
1 
 C  q'   B
L

Ii  
Cin ( q)  q1n  
 im n  i
2 




in
 i
 n

Consequently ,the canonical momenta are :
89

Pk


 I  Cin(q)  q'n  Cik  Ak
d
L
d k'k
i
90
g 1  I1  sin     cos     g 2  I2  sin     sin     cos   

2
2
2
2
2
A B   g 1  I1  sin     sin     g 2  I2  sin     cos     g 3  I3  cos   


g 3  I3  cos   

We shall define the quantity :
T
.k  Pk  A k
Ii 
Cin( q) q'n  Cin
Cin Ii  Cik  q'n 


i
n
in


 
.k
 Cni
T





 A 


 A 
 A 
  

 Cik  q'n
ni
or in matrix form
T

C  C  q'n
where
Cij Ii  Cij
and
T
C
is the transposed matrix of C
 I1  cos    I1  sin    sin   0 


C  I2  sin    I2  sin     cos    0 


0
I3  cos   
I3


33
It follows from (91) that
1
q' C
where
 T 
 C
1

cos   
sin   


0


I1
I2








sin

cos

1
0
C


I1  sin   
I2  sin  0


 sin    cos    cos     cos    1 

I3 
I1  sin   
I2  sin   


 cos    sin    sin    cos   

  
sin   
 sin  


1
CT  
cos   
cos   
 cos   
 sin   sin   
sin   

0
0
1








-1*(C^T)^-1 wich reads :

cos   
sin   
cos    sin      1
cos     sin     cos     1
1



 



I
I1
I2
sin   
sin   

 I1 I2 
 1

2
2
2
 sin  
cos     sin     1
cos    
cos     sin   
cos  
1
1


 







2
2

I2
I2
sin   
sin     I1
sin   
 I1 I2 

 I1

2
2
2
2
 cos     sin     cos      1  1  cos      sin      cos      cos      sin   2I  cos   


 I 
1
I

2
2

I2
sin   

 1 I2  sin     I1
 2   sin    
2
g
1
C
 T 
 C
1
2
94
with the aid of matrix elements g.ik, the relation (92) read
q'k
 gki  .i  gki  Pi  Ai
i
i
Having expressed velocities q'.k in terms of momenta P.i we are now ready in construct the
Hamiltonian of the asymmetric top starting from the general relation
  I1 
2
H p  q'  L ( p  q'  A  q')   
 [ ( C  q')i] 
 i 2 




Using (92) the first two terms take the form
2
1
1
T 1
1
  C T  1  .j 1    C 1  CT  1  
p  ( q'  A  q')   q'   C   C    .1

.k  Cki  
 2
2 

 i
i
 k
 j
It folows now that
96
1
 Ii

1 
1
T 1
  i2 T 1  ( p  A)  C 1  CT   ( p  A) 1  ( p  A )  G  ( p  A) 
H
      C   C  
2
2
2
2





  

i
H
1
2
 ( p  A)  G  ( p  A)
1
2
 gik  ( q)  ( p  A) .i  ( p  A) . k
Eqs. (92a) and (96) suggest to interpret g.ik(q) as the metric tensor in the space of the kinetic
34
momenta
The more explicit form of H reads ;
2
2

cos    
cos   
2
1
2
2
2
H I    sin     I1 

P

A


P

A

  P  A    









2
2
I2
sin   
sin   




1

 P  A 
2  I3


cos     sin   
I2  I1

p  A   P  A 

I1  I2
sin   
 cos     sin     cos     I2  I1


 p  A
sin   

 I1  I2
 cos     sin    2 cos    2 


   p  A 

2
I1
I2


sin




2

 
  P  A   
 


  P  A  

For the symmetric top (I.1=I.2=I.3) and for g.1=g.2=g.3=g, the above Hamiltonian reduces to the
form given in (18').
Hamilton and Lagrange equations follow directly from the above expressions for Hamiltonian
and Lagrangian.
Canonical angular momentum
Let us now express the canonical angular momentum s through the canonical momenta


I1  1  e1
i

CT 
1

 
CT 
I1 
i
1
Cik  q'k  e1
k
 T
P C

i
1
I1.k 

Cik  C
1
 T 
 C
1
 
k
A
97
 cos    sin   sin     cos    

  P  A  
sin   
sin   

 

 
   P  A  
cos    cos   
 sin   sin   sin    cos      P  A 
 

  
0
0
0


If we compare the latter relation with the relation (12) and (13) we conclude that in the phase
irst term is the same function of phase space
variables as is the function s defined in (12). So,we shall call the quantity
 cos     P  sin     P  sin     cos     P 


 

sin   
sin   


T 1
cos   
cos   
s C   p 
s i  ei
sin     P 
 P 
 cos     P 






sin 
sin 
i


P


the canonical angular momentum, or simply spin of the asymmetric top. The components of s in
the laboratory frame are identic
1*A)*e.i we shall give the name
CT  1  A .i  e
a


i
i
Its components in the body frame are :
a1
g1  I1  sin    sin  
35
a2
a3
g2  I2  sin    cos   
g3  I3  cos   
The relation (97) turns into :

sa
Now it is matter of simple algebra to express the Hamiltonian of the asymmetric magnetic top in
terms of its spin
100
H
1
1
T
C C
2


1
2
1
 Cji   j  Cik
 k
jik
1
2

  j   k   Cjk  Cik 
1
jk
i
1
2

j  k 
jk
jk
Ij
2

j
2
j
A remarkable simplification occurs if we choose the constants g.1 that we introduces in (83) to
satisfy g.1^2 =g^2/I.1. Then the Lagrangian and Hamiltonian become
L
1
2
  B
2
g
1
g
2
101
H
2
2
Components of the spin vector in the body frame satisfy the folowing equations of motion :
 d H    d    d s   d s   d  
d H   s
s'j ( H  s )
 d 
 d   i i
  dp i   dq i  dp i  dq i
k
k
k
k
i
i 








i
k 
i 



 i s i s i  ai s i ai s i  ijn  s n  ai s j
n
ai s j
  ddqks i   ddpks j

k
s'j

s i  ai



 s  a  s

i j
 ij n
i
n
Appendix A: Top with magnetic moment fixed in the body frame
A top is fully characterized and specified by its coupling. In this paper we have defined and
studied magnetic top characterized by a velocity dependent magnetic moment. In order to make
more clear our argumentation that the magnetic top is the more appropriate classical model of
spin we shall present here a theory of the top which carries the magnetic moment M attached to
the body. That implies that the magnetic moment M is independent of the angular velocity (for if
Ii
Consequently, the coupling with magnetic field B is velocity independent.
V
M  B
This potential has the smae form as the potential top (with mass M.sv and center of mass
coordinate R) in the gravitational field g.
V
Msv  R  gsv
M being analogous to M.0*R playing the role of gravitationald field g.
We shall deal here with the axialy symmetric top (I.1=I.2) and shall assume that M is along the
body z-axis, i.e. M=M.z. Then, making B along the z-axis of the laboratory frame (B=Bk), wich
does not reduce thr generality of our results,we write the ineraction potential V in the form
A3
36
Ij

j
s j 
2
V
M  B  cos   
The Lagrangian is differnce of kinetic and potential energy terms
A4
I1  2
I3
I3
 I1
2
2
2
2
2
2
L TV
  1  2 
 3  M  cos   
  '  '  sin    
  '  '  cos      M  B  cos   
2 
2
2
 2
In order to construct the Hamiltonian we folow the usual procedure. Canonocal momenta are the

P
P
P
'
d
d '
d
d'
I1  '
L
I1  sin    '  I3   '  '  cos      cos   
2
L
d
d '

I3   '  cos     '
L
 P 
I 
 .1 
P  P  cos  
'
I1  sin   
'
P
I3
2


 P  P  cos    
cos   
I2  sin   
2
By substituting the latter expression into T and
pq
'  P  '  P  '  P
one obtain
2
2
2
2
 P 2
P
P
P  cos   
cos    

pq 2T 2



 P  P 
2
 2  I1 2  I3 2  I  sin   2 2  I  sin   2
I1  sin    
1
3


and consequently

H      P  P P



'  P  '  P  '  P       P  P P  V
TV
P
2
2

I1
P
2
I3
2

P  cos   
2

2
2
P
2
I1  sin   
I3  sin   
2
I1  sin   
Therefore , in agreement with the general theory, to the Lagrangian with velocity independent
potential there corresponds a Hamiltonian which is a simple sum of kinetic and potential energy
terms. The Hamiltonian (16) of the magnetic top does not have this property, again in agreement
with the general theory, since the interaction term in The Lagrangian (7) is dependent on
Hamilton's equations of motion
The first three Hamilton1s equations are the eqs.(A6) .The remaining three read :
P  cos     P  cos     P  P 
2
P'
d
d
H
I1  sin   
2
 1  cos   

cos   
2
 P  P 
2
2
 M  B  sin   
37
2
d
P'
d
H
A9
d
P'
d
H
Comparing the equations (A6) and (A9) with Hamilton's equations (54) of the magnetic top we
the equations for
magnetic moment is identical to the gravitational top, the corresponding Hamilton's equation are
to be found in literature (24). Here we shall review the well known qualitative analysis (19)
Two immediate first integrals of motion are :
A10a

P I3  '  '  cos   
A10b
I3  3
P0
I1  sin    '  I3  '  I3   '  '  cos    

2
P
P0
Since the system is conservative the total energy is the third integral o f motion.
A10c
E
TV
I1
2

 '  '  sin   
2
2

2
I3
2

P0
2
 M  B  cos   
I3
Only three additional quadratures are needed to solve the problrm. From the above three integrals
A11
'
P0  P0  cos   
I1  sin   
2
A12
'
P0
I3
 cos    
P0  P0  cos   
I1  sin   
2
A13
2
 I1 2 P0  P0  cos     
P 
  ' 
  M  B  cos    E  0  E'
2
 2
2  I3

2I1  sin   



The equation (A13) differs from the corresponding equation (31) of the magnetic top by presence
re going to see, due to the presence of this term the equation
(A13) leads to an eltptic integral (with cubic polynomial under the integral sign) On the other
hand the equation (30) leads to the equation (32) with square polynomial and therefore is
integrable. From (A13) it follows :
A14
1

2
2 2

'  I1  sin   
 sin    2  I1   E'  M  B  cos      P0  P0  cos   
A15


38
COS (  ) ( 1)
T




d

2 I 

 1
COS (  ( 0) )
1
cos    .
 dx
2
2 
 1  cos     E'  M  B  cos     P0  P0  cos    
2


Since the solution of the equation (A15) cannot be writwn in an analytic form, the sme is valid
the equation *A13) are known (19). They are pictured on Fig.3 in which the possible shapes for
the locus of the body axis on the unit sphere are indicated. Recalling that M was assumed to be
along e.x. this figure presents also the motion of the magnetic moment M fixed with the body.
top wich move with respect to the body. So, M performs a complicated motion (precession with
Acknowlegments
One of the authors (M.B) would like to thank Professor Abdus Salam, the International Centre for
Theoretical Physics, Trieste.
39