The Probability Density Operator for the K- G Equation
P.S.C. Bruskiewich
Mathematical-physics, University of British Columbia, Vancouver, BC
Dr. Maurice Pryce
Quantum-Physics, University of British Columbia, Vancouver, BC
This paper was written in 1982 with the kind assistance of Dr. Maurice Pryce
of UBC but never published. The well - aged manuscript remained lost in
the author's papers for thirty years, until recently rediscovered.
1.0 The Non-Relativistic Free Particle Equation
The non-relativistic Schrodinger equation for a single particle with no external forces
acting on it is derived from the Mass - Energy expression in the classical limit,
772 2 2, 24
L = p c +m Q c
1.1
where we can separate out the rest energy and the kinetic energy of the particle, namely
E p =+m c .11 +
f 2 \
P
2 2
K m c j
m c
f
2
f 2 W
P
\rn\c 1 ))
1.2
= E +E
rest energy Kinetic Energy
The classical portion of the mass-energy expression is the kinetic energy term, and so let
us focus on that term
K
Kinetic Energy ,«•» 13
By replacing the c-operators by the q-operators [1]
E -^ih —
dt
p->-iftV 1.4
where V is the gradient operator in the coordinate system the particle motion is being
described (Euclidean, non-Euclidean, cylindrical, spherical, hyperbolic, elliptic, etc), the
use of the q-operators yields the non-relativistic Schrodinger equation for a free particle,
in — y/ = V y/
2.., 1.5
dt ' 2m
where the quantum state function y/ is some function of the position and kinematics of
the particle and is non-relativistic. [2]
The standard interpretation of the state function y/ is an interpretation first proposed by
Max Born and is that the state function is Hermitian, and the probability of the state
function is given by the square of its amplitude, (y/* denotes the complex conjugate of
the state function y/ ) namely
p = y/ y/ i- 6
For a free particle, since the particle is not being affected by external forces, when we
integrate the probability of the state over all space we arrive at a constant value, namely
j pdV = j y/*y/dV = constant
All Space
All Space
1.7
2.0 The Quantum Conservation of Current
Take the partial differential in time of the probability operator,
( \
[ y/*y/dV
^All Space
= ^
dt
= I
All Space
f
d_
dt
¥
\
J
r a \ \
¥ + ¥
d
¥
J
1 I
V in J A11 '
v y All Spa
* v>
2m
\dt j
¥ + ¥*
J
dV
2.1
( tf „ 2 ^
V
2m
V>
dV
r ifi
2m
= v:
J (^(vV)-(vV»^
V-"-/ All Space
V
2m y
All Space
Collecting all the terms within the integral and expressing the integrand as an expectation
value we find
I
All Space
— /?+vq
dt H
V 2m j
( ¥ *(V ¥ )-(V ¥ *) ¥ )
>dV = 22
We define a probability current operator j , such that
J =
( ih\
v
2m
J
{yf\Vy,)-{Vy,*)y,
2.3
The classical conservation of current law carries over to a quantum conservation of
current law, provided we view the expression within the integrand as an expectation
function, which is meaningful only when is integrated over all space, namely
d_
dt
p + VUj =0
2.4
With regards to the probability current, if the state function y/ does not have a complex
dependence on the space variable, then it has no current associated with it.
Consider as examples the state functions e ±,kx and cos(fcc). It is evident that e ±,kx are
traveling wave state functions. Choose the state function
y/ cc e
ilex
J =
ifi
2m
(2ik) =
f fik^
= v
\m )
2.5
which is a current in the positive v direction.
Similarly for the conjugate state function
y/ cc e
-ikx
ih
2m
(-2ik) =
V m )
-v
2.6
It is evident then that cos (Ax) represents a stationary state made up of two equally
weighted e ±,kx state functions traveling in opposite directions, namely
1
cos(kx) = -(e lkx +e lkx )
For such a stationary state function as cos(kx) , since k ■■
n
v h J
2.7
v , the two opposite
traveling waves, which represent currents traveling in opposite directions, sum to zero.
3.0 The Probability Density for the K- G Equation
The simplest relativist wave equation is derived by taking the relativistic Energy-
Momentum equation, but retain both the rest energy and kinetic energy terms, namely
2 , 2 2
= p +m c
once again replacing the c-operators by the q-operators, we find
3.1
2 o2
r d
■2^72
2 2.
3.2
c 2 dr
y/ = -HVy/ + mcy/
Notice that unlike the non-relativistic case which is first order in time, the relativistic
Klein - Gordon equation is second order in time.
Let us retain the form of the probability current operator j , such
(-— 1
v 2m J
J
K-G
(v m (vv)-(vv m )v)
3.4
By dimensional analysis across the equivalent sign, this means then that the density
function p for the K - G equation must be first order in time, since the q-operator for
Energy is to first order in time.
Without loss of generality let us consider a candidate density function Z
f
Z = A
f
¥
V v
d_
dt
f
¥
d A }
dt
¥
¥
yui j
+ B(y/*y/)t + Cy/*y/
3.5
We find then that
d
—Z = A
dt
( *2 \
¥
\
d
dt
¥
( *2 \
yvi J
d
dt
¥
¥
+
B(y/*ys),
since
3.6
— r
dt-
(^V)
= 0^C = 0&B— (y/*y/)
— r
dt-
t =
3.7
where B is some function of the rest mass of the particle. Consider a candidate density
function Z to be made up of two densities p + and p_ such that
Z = P + +P-=P
3.8
By inspection
P + ~-
ifi (
Imc 2 K
* d }
. dt J
imc *
+ y/y/t
n
and
P-~-
ifi
2mc
(d * >
imc 2 *
Mf lift
yj y/i
n
3.9
3.10
When the two terms to Z are added together the probability density becomes
P = P + +P
ifi
2mc
¥
(d \ fd
V
dt
¥
J
V
dt
¥
\
J
¥
3.11
It is interesting to note that the terms in the probability density function linear in time
sum to zero. This makes sense given the quantum conservation of current law is an
expectation function.
4.0 Implications of Probability Density on Positive Definiteness
Since the initial conditions, the state function y/ and — y/ , can be arbitrarily prescribed,
dt
the probability density for the K -G equation outlined in 3.11 may assume negative as
d
well as positive values. Likewise, since the state function yr and — y/ are also
dt
functions of the spatial coordinates it is evident that the probability density for the K -G
equation may assume positive values in some regions and negative values in other
regions.
In essence there is a loss of positive definiteness in the probability density for the K -G
equation. In a one particular model, a negative probability density is inherent to the K -
G equation and does not depend on the choice of the current density function j K _ c .
As a result of this inherent short coming, the Klein - the Gordon equation is not a good
candidate for a single particle relativistic equation. The K -G equation can be used in a
field theory for particles with spin 0.
Acknowledgement
The author would like to thank Dr. Maurice Pryce of the Department of Physics at the
University of British Columbia for his kind and thoughtful comments on this paper.
When Dr. Pryce realized that I had never read the 1926 paper by Paul Dirac titled The
Fundamental Equations of Quantum Mechanics, nor any of the original papers written by
Bohr, Schrodinger, Heisenberg and Dirac, Dr. Pryce gave me a copy of the book
"Sources of Quantum Mechanics" as a personal gift. It was Maurice Pryce's belief that
to truly understand the Foundations and the thought process of the intellectuals who
developed a subject as rich and sublime as quantum mechanics, one needs to read the
original papers.
{see recent author's notes below)
Notes:
[1] The designation c-operator for classical operator and q-operator for quantum operator
was a designation first proposed by Paul Dirac.
[2] The quantum rules were first presented in a consistent fashion by in a 1926 paper by
Paul Dirac. see Paul Dirac, The Fundamental Equations of Quantum Mechanics or his
book on Quantum Mechanics.
References:
Paul Dirac, The Fundamental Equations of Quantum Mechanics, Proc. Roy. Soc. A 109
(1926), 642-653
Author's Recent Notes:
This paper was written in 1982 with the kind assistance of Dr. Maurice Pryce ofUBC.
The well aged manuscript, which was never published, remained lost in the author 's
papers for thirty years, until recently rediscovered in an old box with other manuscripts.
The manuscript was transcribed and posted to archive.org.
As part of his scientific career Dr. Pryce did service for the Royal Navy and Admiralty
Research. The writer first met Dr. Pryce in 1981 when the author was a young man
serving as a Naval Reserve Officer in the Canada 's Navy and an undergraduate physics
student at UBC. Over several years, before his death in 2003, the author periodically
had a chance to talk physics with Dr. Pryce.
Maurice joined the Department of Physics at UBC in 1968. He had a reputation of being
rather precise in his thinking and not suffering fools and foolishness quietly. The author
is proud to say that he never was on the receiving end of any sharp words from Dr.
Pryce. In 1985 the author would arrange for Dr. Pryce to write his "Recollections of
Paul Dime", which was published in the journal of the UBC Physics Society and is now
available at archive.org.
Manuscripts were written and notes kept of the Pryce-Bruskiewich conversations and
they may be posted to archive.org in the near future. This paper, the Probability
Density Operator of the K - the G Equation, is the second time any of these notes or
manuscripts have appeared in print
In the measure of connections to Paul Dirac, Maurice Pryce is an Dirac-Alpha-Measure
and the author is a Dirac-Beta-Measure, having written a paper with the kind and
thoughtful comments of a physicist, who himself was a pupil of and worked with Paul
Dirac. Maurice Pryce was one of Dr. Paul Dirac 's finest student.
© 2012 Patrick Bruskiewich