Download Variational principle in the conservation operators deduction

A DIFFERENT APPROACH TO DEDUCTION OF THE SHAPE OF
CONSERVATION OPERATORS IN QUANTUM MECHANICS
Dementev N. N.
Chemistry Department, Voronezh State University, Universitetskaya Sq.1,
Voronezh 394006, Russian Federation,
E-mail: [email protected] or [email protected]
Quantum Mechanics acts in terms of operators. And the shape of any certain operator is
usually postulated [1-4]. But, in general, there is no absolute identiy between the shape of
operator and its physical value. Thus no one can absolutely rely on the operators obtained by
such an arbitrary way. This note appeals to improve the situation. As will be seen below, the
shapes of the basic conservation operators can be deduced efficiently with the help of
variational principle applied to some initial relations of quantum mechanics in the case of
monochromic objects. The invariability of the light's velocity along with inevitability of
quanta character of laws are the essential results of the deduction.
Key-words: conservation operators, variational principle, quantum mechanics
Methods
Variational Principle is successfully used in Quantum Mechanics to find out the wave function  of a system. The functions generated in this method are slightly differ from the real
wave function. The Principle stated that the function with lowest energy (among the functions
in the series) is of the best conformity with the real function. In order words, the variation of
total energy of system over the shape of psi-function is equal to zero:

(1)
 E ()    [ ( x, y, z ) H ( x, y, z )] dxdydz  0
where E ( ) means the total energy functional depended on the psi-function  ;

H is the operator of total energy of system, i.e.Hamiltonian ;
sign “ * “ indicates the complex conjugation
It should be stressed that eq.1 is applicable in the method if only Hamiltonian is already exactly known.
Let now assume the wave function is already exactly known. Is it possible that the given wave
function can be related with several Hamiltonians differed in their shapes?
The situation is unreal because different Hamiltonians are gave, in general, a different values
of total energy which correspondent to a different wave functions.
Thus one can made a conclusion that variation of total energy over the shape of Hamiltonian
(for the given psi-function) is equal to zero:


(2)
 E ( H )    ( x, y, z )  ( H ) ( x, y, z ) dxdydz  0

It should be stressed that  (H ) means not the simple increment, but the variation over the
shape of Hamiltonian. One can use this relation for any conservation operator because any

conservation operator (and only conservation operator) Lcons. has to commute with Hamiltonian. Thus we obtain:

   ( x, y, z )  ( Lcons. )  ( x, y, z ) dxdydz  0
(3)
1
As will be seen below, one can successfully use this equation (slightly modified, see below)
as a selection rule in order to find the shape of the conservation operators. Let now detailed

the general presentation for  and Lcons. .
Neither coordinate nor momentum (i.e. impulse) presented  is suitable in our case. (Indeed,
the coordinate presented psi-function must give us the coordinate presented conservation operators. But any conservation operator does not commute with coordinate operator. It means
that any conservation value and coordinate are cannot be determined exactly and simultaneously. Thus the coordinate presentation is not quite correct for conservation operators. On the
other hand, momentum presentation implied that the connection between psi-function and
momentum is already known. And our deduction becomes rather illogical procedure.)
Let further use psi-function presented as follows:
(4)
 ( )   e i
where  is the certain generalized variable;
 is the amplitude factor;
i is the imaginary unit ( i 2  1 );
This presentation is free from doubt speculations of coordinate and momentum presentations.
“  -presentation” reflects only general (and essential) properties of psi-function, namely: psifunction is the complex function and it can be presented as the function of exponential type.
 -presented psi-function must be normalized:
(5)
  ( )( ) d   2  ei ei d  1
  1
(6)
Eq 3 must be rewritten for the  -presented wave function as follows:

   ( )  ( Lcons. )  ( ) d  0
(7)

The very general shape for any operator L is given in [2]. Let further use this shape in  presentation:


2
3
4
L  A0 ( )  A1 ( )
 A2 ( ) 2  A3 ( ) 3  A4 ( ) 4  ...
(8)




where A0 ( ), A1 ( ), A2 ( ), A3 ( ), A4 ( )... are the certain analytic functions
It should be stressed that the number of differential terms in eq 8 must be finite for any certain
operator.
Thus the mathematical task of ours is to find out all possible solutions (presented as eq 8) of
eq 7 with the help from eqs 4,6.
Calculation Procedures
On substituting eq 8 in eq 7, we obtain:

2
3
4
   ( )  [ A0 ( )  A1 ( )
 A2 ( ) 2  A3 ( ) 3  A4 ( ) 4  ...] ( )d  0




(9)
Let export the variation operation  beyond  ( ) for some temporal comfort (it can be
done cause of the invariable character of psi-function due to our assumption that psi-function
is already exactly known):

2
3
4

   ( )[ A0 ( )  A1 ( )
 A2 ( ) 2  A3 ( ) 3  A4 ( ) 4  ...]( )d  0
(10)




The series within square brackets can be folded due to the periodical character of psi-function
(psi-function is the complex function). Indeed, following relations are valid for the psifunction (see eqs 4,6):
2
 (n)
 ( )  i ( n )  ( )
(n)

(11)

3
5
7
 ( )   3  ( ) 

(

)


( )  ...


 5
 7
(12)
2
4
6
8

(

)



(

)


(

)


( )  ...
 2
 4
 6
 8
Eq 10 can be rewritten by using eqs 12, 13:

    ( ){ A0 ( )  [ A1 ( )  A3 ( )  A5 ( )  ...]

 [ A2 ( )  A4 ( )  A6 ( )  ...]
(13)
2
} ( ) d  0
 2
(14)
And then:

2
 B2 ( ) 2 } ( ) d  0


where B1() and B2() are the new certain analytic functions:
B1 ( )  A1 ( )  A3 ( )  A5 ( )  A7 ( )  ...
   ( ){ A0 ( )  B1 ( )
(15)
(16)
(17)
B2 ( )  A2 ( )  A4 ( )  A6 ( )  A8 ( )  ...
Thus the general shape of conservation operator at the stage is:


2
(18)
Lcons.  A0 ( )  B1 ( )
 B2 ( ) 2


.
Next step is to expose the functions A0(), B1() and B2().
Let back the variation operation to its original position:

2
(19)
   ( )  { A0 ( )  B1 ( )
 B2 ( ) 2 } ( ) d  0


and conduct the variation:
 2 
  

2
   B2 ( ) 2  B2 ( )  2 }( ) d  0 (20)
  ( ){ A0 ( )   B1 ( )
 B1 ( ) 


  
  
   ( ){ A0 ( )   B1 ( )

2
2
3
 B1 ( ) 2   B2 ( ) 2  B2 ( ) 3 } ( ) d  0




Use of eq 11 leads to:
   ( ){ A0 ( )  i B1 ( )  B1 ( )   B2 ( )  iB2 ( ) } ( ) d  0
This equation can be rewritten:
   ( ){[ A0 ( )  i B1 ( )   B2 ( )]  [B1 ( )  iB2 ( )]} ( ) d  0
(21)
(22)
(23)
  ( ){ [A0 ( )  iB1 ( ) B2 ( )]  [B1 ( )  iB2 ( )]} ( ) d  0
(24)
The equation must be faithful for any value of the increment “  ”. In extreme case when the
increment tended zero, one can obtain:
(25)
  ( )[B1 ( )  iB2 ( )] ( ) d  0
This equation can be rewritten:
(26)
 [B1 ( )  iB2 ( )]  ( )( ) d  0

Term  ( )( ) is always above the zero because it means the density of probability [1-4].
Consequently, the following decision of eq 26 is valid:
3
B1 ( )  iB2 ( )  0
B1 ( )  iB2 ( )
(27)
(28)
Use of eq 27 in eq 23 leads to:
   ( )[ A0 ( )  i B1 ( )   B2 ( )] ( ) d  0
(29)
[ A0 ( )  i B1 ( )   B2 ( )]  ( ) ( ) d  0
(30)
 A0 ( )  i B1 ( )   B2 ( )  0
(31)
Use of eq 28 in eq 31 leads to:
 A0 ( )  i [iB2 ( )]   B2 ( )  0
 A0 ( )   B2 ( )   B2 ( )  0
 A0 ( )  0
(32)
(33)
(34)
This equation means that A0() is independent from . In order words, it is a constant:
(35)
A0 ( )  A  const
Use of eqs 28, 35 in eq 18 leads to:


2
(36)
Lcons.  A  iB2 ( )
 B2 ( ) 2


The shape of generalized conservation operator is described by this equation.

The mean value of any physical value (i.e. the observable) L related with operator L is given
in [1-4] as:

L  L 
(37)
Thus the observable of our generalized conservation operator will be:

2

(38)
Lcons.   ( ) [ A  iB2 ( )
 B2 ( ) 2 ]  ( ) d


There are six special cases here:


1) Lcons.    A
(39)



2) Lcons.    iB2 ( )
(40)

Lcons will be expressed in the case 2) as:

]  ( ) d
(41)

On the other hand, general eq. 38 must also be faithful. This situation is possible if only the
following equation is valid:
2

(42)
  ( ) [ A  B2 ( ) 2 ]  ( ) d  0

Use of eq 11 in this equation leads to:
(43)
  ( )[ A  B2 ( )] ( ) d  0
(44)
B2 ( )  A
Eq. 40 can be rewritten by use of eq 44 as:



(45)
Lcons.    iA


2

3) Lcons.    B2 ( ) 2
(46)

L cons.    ( ) [iB2 ( )
Lcons will be expressed in the case 3) as:
4
2
(47)
]  ( ) d
 2
On the other hand, general eq. 38 must also be faithful. This situation is possible if only the
following equation is valid:

   ( ) [ A  iB2 ( ) ]  ( ) d  0
(48)

Use of eq 11 in this equation leads to:
(49)
  ( )[ A  B2 ( )] ( ) d  0
(50)
B2 ( )   A
Eq. 46 can be rewritten by use of eq 50 as:

2

(51)
Lcons.     A 2



4) Lcons.  A  iB2 ( )
(52)


2
5) Lcons.  A  B2 ( ) 2
(53)



2
6) Lcons.  iB2 ( )
(54)
 B2 ( ) 2


One can demonstrate that the approach described above being applied to cases 4) and 5) will

lead to the case 1) and that Lcons. is always equal to zero in the case 6). It is important to note
that A is equal to zero in the last case:
A0
(55)
Finally, only the cases 1), 2) and 3) are lead to a different operators:


2


(56)
 A
  iA
  A 2


Conclusions
If we want to estimate our results we have to examine the operators (56). These operators
must be the conservation operators if only the basic premises and the deduction were really
correct. There are three basic conservation laws in Physics, namely: of momentum, of angular
momentum (action), and of total energy. The shapes for coordinate presented operators of an


gular momentum (action) M , momentum P , and total energy H , for the free object (i.e. the
object that does not interact with its surroundings), are given in [1-4] as:




2 2
(57)
M 
P  i
H 
x
2m  x 2
where m is the mass of studied object.
(Here and then the only x -components are taken in order to simplify the situation.)
1) The striking similarity is distinct in the character of construction between the operators
  
  
 ,  ,  and M , P , H , correspondingly. But the feature would rather the fact of hope
than the decisive evidence of the correctness of deduction since no one knows an explicit
dependence   f (x) yet.
2) Operators (56) as well as (57) are the quanta operators. Indeed, operators (56) are all contain the certain constant A as well as operators (57) are all contain the Planck’s constant
 . This resemblance is more ponderable than the 1) one because the quanta character of
operators must be independent on every certain presentation.
L cons.    ( ) [ B2 ( )
5
In connection with the latter it is interesting to consider another one feature of the deduction: the obtained operators (56) are the quanta operators but, however, there is no one
explicit quantum term in the starting equations 4,7,8. Quantum term A is emerged on the
step (35) as a result of boundary condition (7) that is the result of the existence of conservation laws. Thus the following conclusion one can made:
Quanta existence is inspired by the presence of conservation laws.
Vice versa: There are no conservation laws but the quanta ones.
Indeed, the general shape of conservation operator in non-quantum situation (i.e. A  0 ) is
given by eq 54, and one can noticed that the observable is always equal to zero in this
case.
3) With the help from eqs 56,37,38,11,5 one can found the following equality for the observables  ,  ,  :
     A
(58)
There must be the mean value of total energy E and the mean value of total momentum
P among the observables  ,  ,  if the deduction is correct. Thus we obtain:
E
1
(59)
P
At first glance this result looks rather embarrass. But there is nothing strange in eq 59
since the values in (58) are the values in natural unit’s scale. Indeed, the operators (56)
have been obtained as different special shapes of the single generalized operator (36).
It means that eq 58 can be converted into any other unit system (e.g. the International Unit
System) as follows:
E 
(60)
P
where  is the certain constant which value dependent on our choice of this or that unit
system only. One can notice that  has the velocity dimension. It is easy to show (see the
Appendix) that  means the velocity of propagation of the studied object in the case of
inertial movement. Thus we have concluded the invariant character for the velocity of our
objects. It should be recalled here that our deduction is based on the exact knowledge of
the shape of psi-function. This condition take place only for the objects described by the
absolutely monochromic wave functions. It is well established by now that such objects
are the photons. Thus we have finally concluded that the velocity of photons is the invariant. This conclusion is the main postulate of the special relativity theory which numerous
sequences are successfully approved by practice. Thus the reasons of the item are the most
ponderable evidence of the correctness of our deduction.
Appendix
Let show that the following relation is valid in the case of the inertial movement:
E 
(1)A
P
where  is the velocity of propagation of the studied object;
E and P are the mean values of total energy and momentum of the studied object (relative to
the certain observer), correspondingly.
Imagine the observer who want to describe the certain isolate system. Conservation laws
have to be valid in the case since the isolation. And the mean value of changes of the psi
function has to be zero (cause of the relation L   L  ):
d ( x, t )  0
6
(2)A
(where x and t are the spatial and time coordinates, correspondingly)
Let assume that the object movement is inertial. It means that there is no scattering in the values of momentum and total energy and one can exclude the meaning in (2)A:
d ( x , t )  0
(3)A
This equation can be rewritten:


dx 
dt  0
(4)A
x
t


dt  
dx
t
x
Let multiply both parts of eq (5)A by i   , and after the regrouping we obtain:
  dx
 

   i        i  
x  dt
 t 



There are the following identities for H [3], P [3], and  :


 
dx
i  H
 ih
P

t
x
dt
After the integration of (6)A and replacement (7)A we obtain:

E
(5)A

P

(6)A
(7)A
Q.e.d.
References
1. Feynman, R. P., Leighton, R. B., Sands, M. The Feynman lectures on Physics (AddisonWesley publishing company, Inc., Reading, Massachusetts, Palo Alto, London, 1963).
2. Zulicke, von L. Quantenchemie (VEB Deutscher Verlag der Wissenschaften, Berlin,
1973).
3. Fermi, E. Notes on Quantum Mechanics (The University of Chicago Press, 1960).
4. Atkins, P. W. Quanta (Clarendon Press, Oxford, 1974).
7