Download Relativistic and non-relativistic differential equations for the quantum

Relativistic and non-relativistic differential equations for the
quantum particles
Yusuf Z. Umul
Cankaya University, Engineering Faculty, Electronic and Communication Dept., Balgat-Ankara,
Turkiye
[email protected]
Abstract: An important contradiction in the actual usage of the non-relativistic energy and
momentum relations of a quantum particle is analyzed. This analysis led us to obtain new
forms of differential equations for the mechanics of a non-relativistic quantum particle.
Also a new equation is derived for the relativistic case which yields the same differential
equation for the non-relativistic approximation. These equations are extended for a
particle, moving in a potential.
Key words: Quantum mechanics, Schrödinger equation, Quantum optics
1. Introduction
The quantitative representation of the behavior of a quantum particle was first expressed by
Schrödinger who obtained a wave equation by taking into account the non-relativistic energy
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relation [1]. In a series of following papers, he applied his equation to the solution of various
eigen-value problems [2-5]. The solution of this differential equation gives a wave function,
interpretation of which yielded different schools of quantum mechanics [6, 7]. A second
quantitative approach was the matrix formulation of the quantum states, which was developed by
Heisenberg, Born and Jordan [8, 9]. After the intensive criticisms of the Copenhagen school,
Schrödinger managed to show that his differential equation approach was equivalent to the
matrix formalism [10].
The Schrödinger equation is based on the Planck-Einstein equations, which connect the wave
and particle behavior of the quantum particles into each other. The differential equation is
obtained by the jointly usage of the Planck-Einstein equations with the non-relativistic energy
relation of classical dynamics. This approach can be found in many text books [11]. The author
of this paper has recently pointed out a contradiction which can be realized by the consideration
of the special theory of relativity [12].
In this study, our aim is to show a contradiction in the actual usage of the non-relativistic
energy and momentum equations. These relations are important since they are the basis of the
Schrödinger equation. The contradiction forces one to obtain a new differential equation for the
mechanics of a non-relativistic quantum particle. First of all, the contradiction and its reason will
be summarized for the sake of completeness. Then a differential equation, which leads to no
contradiction, will be obtained. We will also derive a relativistic equation for the quantum
mechanics and show that this equation directly yields the one, obtained for the non-relativistic
case.
2. Theory
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First of all we will derive an important relation between the group and phase velocities of a
quantum particle. The energy of a relativistic particle can be given by
E = mc 2
(1)
where m is the mass of the particle which has the expression of
m=
m0
1−
v g2
.
(2)
c2
c is the velocity of light. v g is the group velocity. m0 is the rest mass. Equation (1) can also be
written as
E=p
c2
vg
(3)
for p is the momentum of the particle which is equal to mv g . Planck showed that the energy of a
quantum particle has the relation of
E = hw
(4)
where h and w are the angular Planck’s constant and angular frequency, respectively [13]. The
equation of
vg v p = c 2
(5)
can be found by using Eq. (4) in Eq. (1) and taking into account the relation of w = kv p . k is the
wave-number which is equal to 2π / λ where λ is the wave-length. v p is the phase velocity. A
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more detailed derivation is given in the Appendix. Equation (5) is well known in literature and it
will be our indicator since it is also valid for the non-relativistic case. It is important to note that
we are taking into account the comments of de Broglie for the definitions of the phase and group
velocities [14, 15].
Now we will consider the energy relation of
E=
1
m0 v g2
2
(6)
for a non-relativistic particle. Equation (6) is valid for v g << c . In literature, the relation of
p = m0 v g is used for the momentum. De Broglie also showed that momentum is equal to hk . If
these relationships and Eq.(4) is used in Eq. (6), one obtains
v g = 2v p
(7)
which leads to the result of v g = 2c . It is apparent that this value of the group velocity
contradicts with the condition of v g << c . We can conclude that the straightforward usage of
p = m0 v g with Eq. (6) leads to erroneous equations. This outcome yields us to two operations.
First of all we must determine the correct value of the non-relativistic momentum and then a new
differential equation must be derived. The equation of Schrödinger is not absolutely correct since
it is developed from the relation of p = m0 v g and Eq. (6).
The momentum of a relativistic particle can be directly evaluated from Eq. (6) by using the
relations of p = hk and w = kv p in Eq. (4). The expression of
4
1
m0 v g α
2
p=
(8)
can be obtained where α is equal to v g / v p . Since we have the representation of the momentum
of a non-relativistic particle, now we can derive a differential equation for the quantum theory.
Equation (6) can be rewritten as
hw =
2 p2
m0 α 2
(9)
hw =
2h 2 k 2
.
m0α 2
(10)
which also yields the equation of
We will take into account the same procedure which is used in most of the text books in order to
obtain the Schrödinger equation [7]. The relations of
∂2
k =− 2
∂x
2
, w=−j
∂
∂t
(11)
for a plane wave which has the expression of exp[ j (wt − kx )] . As a result the differential
equation of
m0α 2 ∂ψ
∂ 2ψ
−j
=0
2h ∂t
∂x 2
(12)
which can be generalized to a three dimensional form as
∇ 2ψ − j
m0α 2 ∂ψ
= 0.
2h ∂t
5
(13)
ψ represents the wave function. Equation (13) is a differential equation for the mechanics of a
free and non-relativistic particle. It is apparent that the equation’s form is the same with the
Schrödinger equation, but the coefficients are different.
As a second step, we will obtain a differential equation for a relativistic particle. Equation (1)
can be considered and rewritten as
hw = mv g2
c2
v g2
(14)
for this case. Equation (14) yields
h 2k 2
hw =
mα
(15)
when Eq. (5) is used with the relations of p = mv g and p = hk . As a result, we can obtain
∇ 2ψ − j
mα ∂ψ
=0
h ∂t
(16)
for a relativistic free particle by considering the transition between Eq. (10) to Eq. (13). m
reduces to
 1 
m ≈ m0 1 + α 
 2 
(17)
for v g << c . When Eq. (17) is used in Eq.(16), The differential equation of
∇ 2ψ − j
m0α  1  ∂ψ
=0
1 + α 
h  2  ∂t
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(18)
can be obtained. Since the non-relativistic relation of energy is considered as in Eq. (6) instead of
E=
1
m0 v g2 + m0 c 2
2
(19)
in the literature, the term of
j
m0α ∂ψ
h ∂t
(20)
is missing in Eq. (13). Now we will develop a relativistic differential equation for the mechanics
of a quantum particle which is moving in a potential of U. We can represent the potential by the
equation of
U = hw0 .
(21)
The value of w0 can always be determined, since we know U and h . Equation (1) takes the form
of
E − U = mc 2
(22)
which can be rewritten as
h(w − w0 ) =
h2k 2
mα
(23)
according to Eq. (15). Before transforming to the differential equation, we will arrange Eq. (23)
as
 w0  h 2 k 2
hw1 −
.
=
w  mα

7
(24)
It is apparent that the term of w0 / w is dimensionless. As a result one obtains the differential
equation of
∇ 2ψ − j
mα  w0  ∂ψ
=0
1 −

w  ∂t
h 
(25)
for a relativistic quantum particle, moving in a potential. The equation for a non-relativistic case
can be developed by considering the approximation of Eq. (17) for v g << c . This leads to the
expression of
∇ 2ψ − j
m0α  1  w0  ∂ψ
= 0.

1 + α 1 −
h  2 
w  ∂t
It is apparent that Eq. (26) reduces to Eq. (18) for U = 0
(w0 = 0) .
(26)
This representation is
reasonable. The wave equation of
∇ 2ψ −
1 ∂ 2ψ
=0
v 2p ∂t 2
(27)
can be written directly when the relation of w = kv p is taken into account. For a particle at rest,
the value of the phase velocity approaches to infinity according to Eq. (5). This limit leads to the
equation of
∇ 2ψ = 0
(28)
for the group velocity is equal to zero. Equation (26) yields the same result for this case since α
is equal to zero for a particle at rest.
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3. Conclusion
In this study we showed that the expression of the momentum, defined by p = m0 v g , can not
be used with Eq. (6) since such an operation creates a contradiction. We derived an expression,
given by Eq. (8), for the momentum of a non-relativistic particle and developed differential
equations for the relativistic and non-relativistic cases, including a free particle and a particle,
moving in the potential. The relativistic equations have the same form with the Schrödinger
equation and can be directly reduced to the non-relativistic equations by using the
approximation, given in Eq. (17). These equations do not have the problems that the KleinGordon equation presents, like negative energy solutions.
Appendix
In this section the derivation of Eq. (5) will be performed. First of all we will consider two sets
of equations that are related with the wave and matter properties of a quantum particle. The
matter based energy and momentum relations of a particle can be given by
E = mc 2
(A.1)
p = mv g .
(A.2)
E = hw
(A.3)
and
The same quantities can be written as
and
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p = hk
(A.4)
according to the wave properties of the particle. It is apparent that these equations are relativistic
because of the term of m which is equal to
m=
m0
1−
(A.5)
v g2
c2
and gives
 1 v g2
m ≈ m 0 1 +
 2 c2





(A.6)
for v g << c . At this point, we will investigate the ratio of the energy and momentum which
satisfies
E c2
=
.
p vg
(A.7)
It is important to note that Eq. (A.7) is independent from the relativistic effects since the term of
m is vanished because of the ratio. This means that Eq. (A.7) is also valid for v g << c . The same
ratio yields
E w
= = vp
p k
according to the wave properties of a particle. The equation of
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(A.8)
vg v p = c 2
(A.9)
can be obtained by equating Eq. (A.8) to Eq. (A.7). Equation (A.5) can be rewritten as
m=
m0
1−
(A.10)
vg
vp
which also leads to
 1 vg
m ≈ m 0 1 +
 2v
p





(A.11)
for v g << v p . This condition shows the consistency of Eq. (A.9) which is also valid for the nonrelativistic case because the ratio of the energy and momentum is always independent from the
relativistic effects.
References
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[3]. E. Schrödinger, Ann. Phys. 80 437 (1926).
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[6]. N. R. Hanson, Am. J. Phys. 27 1 (1959).
[7]. L. E. Ballentine, Am. J. Phys. 40 1763 (1972).
[8]. W. Heisenberg, Z. Phys. 33 879 (1925).
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[9]. M. Born, P. Jordan, Z. Phys. 34 858 (1925).
[10].
E. Schrödinger, Ann. Phys. 79 734 (1926).
[11].
Phillips A. C.; Introduction to Quantum Mechanics; Wiley: West Sussex, 2003.
[12].
Y. Z. Umul, arXiv:0712.0967v1.
[13].
M. Planck, Ann. Phys. 309 553 (1901).
[14].
L. De Broglie, Phil. Mag. Lett. 86 411 (2006).
[15].
P. Weinberger, Phil. Mag. Lett. 86 405 (2006).
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