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Optik 127 (2016) 10844–10849
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Optik
journal homepage: www.elsevier.de/ijleo
The relation between wave vector and momentum in
quantum mechanics
Yusuf Z. Umul
Electronic and Communication Dept., Cankaya University, Eskisehir yolu 29. km, Yukariyurtcu Mah., Mimar Sinan Cad., No: 4, Etimesgut,
Ankara, 06530, Turkey
a r t i c l e
i n f o
Article history:
Received 8 July 2016
Accepted 2 September 2016
Keywords:
Quantum mechanics
Wave-number
Diffraction
a b s t r a c t
The fundamental relation of quantum mechanics, which correlates the momentum of a
quantum particle to the wave vector of the matter wave, is re-interpreted. The wave vector,
in the relation, is formulized as the integration of it along the angular coordinates for two
and three dimensional cases. Various evaluations of the wave vector are performed for
different types of waves and the results are discussed. Also the edge diffracted fields are
considered according to the new interpretation.
© 2016 Elsevier GmbH. All rights reserved.
1. Introduction
In 1901, Planck introduced the notion of quanta by explaining the black body radiation problem [1]. He proposed that
the electromagnetic energy was quantized with a proportion to its frequency. Later Einstein used this proposal in order to
put forth a quantitative explanation of the photoelectric effect [2]. These studies showed that the relation
(1)
exists between the energy of the quanta and the angular frequency of the matter wave. is the angular Planck’s constant.
This relation was named as the quantum postulate by Bohr [3]. In 1925, de Broglie obtained the equation
(2)
by a relativistic analysis of a moving quantum particle and a wave that was accompanying it [4]. Eq. (2) shows a relation
between the momentum of a quantum particle and the wave vector of the quantum wave. An alternative form of Eq. (2) is
given by
p=
h
(3)
in a scalar form. h is the Planck’s constant, which is equal to 2 and is the wavelength. De Broglie suggested that the
motion of the quantum particle was determined by a wave and the phase of the wave was giving the trajectory of the
particle [5–8]. However the probabilistic interpretation of Born for the quantum wave dominated the literature [9]. In 1952,
David Bohm decomposed the Schrödinger equation into two parts in terms of the phase and amplitude functions of a
wave [10,11]. Although Madelung [12] proposed this decomposition as a hydrodynamical approach to quantum physics in
1927, Bohm interpreted the two equations in terms of a hidden variable based theory. In the same years, Takabayasi also
E-mail address: [email protected]
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Y.Z. Umul / Optik 127 (2016) 10844–10849
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introduced the same differential equations [13,14]. de Broglie-Bohm interpretation of quantum mechanics was unifying the
wave and particle aspects of the quantum phenomenon, but the main problem was the absence of a physical picture of this
wave-particle association in the macrocosmic scale. At least till 2005.
A group of scientists, working on fluid mechanics, realized a series of experiments which put forth the exact association
of a particle and a wave in 2005 [15–17]. These interesting experiments were related with oil droplets, bouncing on a bath
of same liquid. Couder et al. observed that a droplet does not merge a vertically oscillating liquid surface if the oscillation
frequency is greater than a threshold value. An airbag occurs between the surface and the drop, does not permitting it to
mix with the liquid. However the droplet begins to move on the surface when the oscillating frequency of the bath exceeds
a second threshold value. In this case an attenuating Faraday wave accompanies the droplet and the motion of the drop is
determined by mutual interaction of it and its wave. The interesting point is that this wave-particle formation shows the
effects like diffraction, interference, tunneling and orbiting [18–21]. Another group of scientists began to formulize a pilot
wave theory by repeating the experiments of Couder et al. [22–26]. These latest developments on the area of fluid mechanics
provide a basis for the reconsideration of de Broglie and Bohm’s ideas.
Since the above experiments and theoretical studies on fluid mechanics gave important insights into the wave-particle
nature of quantum mechanics, it is worth to propose a new interpretation on this subject. Because of its vector nature, Eq.
(2) carries an important clue about the particle-wave association. For this reason, I will analyze this equation in terms of the
outputs of the experiments on the motion of droplets. First of all some problems about the direct interpretation of Eq. (2)
will be mentioned. Then I will propose a new interpretation and formulation of this equation as a solution of the problems.
The new approach will be applied to three kinds of waves and the edge diffraction process.
2. Some difficulties about direct evaluation of the momentum-wave vector relation
There are two important points in the experiments of Couder et al. that interests quantum mechanics. 1) The droplet
bounces periodically in a vertical direction and excites a circular wave that propagates in the horizontal direction when
the oscillation frequency of the bath is smaller than a threshold value [27]. 2) The droplet begins to move in a horizontal
direction and the accompanying wave propagates in all directions when the oscillation frequency of the bath exceeds the
threshold level [22,28]. Therefore the pilot wave, proposed by de Broglie and Bohm may also satisfies these two conditions.
If a quantum particle has zero momentum, it will have the energy
E0 = m0 c 2
(4)
according to the relativity theory of Einstein. m0 is the rest mass of the particle. c is the speed of light. A wave with an angular
frequency of
(5)
accompanies the particle according to de Broglie [5]. This case is the counterpart of item 1, given above. The wave is circular
or spherical for the two or three dimensional problems. Thus the wave vector of the wave has the expressions
k = ke
(6)
k = ker
(7)
or
for two or three dimensional cases respectively. The polar and spherical coordinates are defined by (, ϕ) and (r, , ϕ).
However Eq. (2) can not be satisfied for this case, since the left-hand side of this equation is equal to zero when the particle
is stationary. We get something like
(8)
Some people may claim that there is no problem, because the particle does not excite a wave when it is stationary or the
pilot wave does not exist in the physical realm. However there is an alternative solution for the mathematical statement of
Eq. (8).
Now the second item, given above, is considered. The quantum particle is in motion and its relation with the wave vector
of the pilot wave is expressed by Eq. (2). Unless the pilot wave is a plane wave, its wave vector is directed to all of the
positions on the propagation path. However the experiments of Couder et al. show that the accompanying wave is similar
to a circular wave, not a plane wave. Thus the wave factor is also similar to the ones, given by Eqs. (6) and (7), in this case. If
a particle, moving along the x axis is taken into account, the following expressions can be obtained
(9)
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Y.Z. Umul / Optik 127 (2016) 10844–10849
or
(10)
from Eq. (2). So we have a problem in the definition of this equation. Eqs. (9) and (10) point out an uncertainty between
the momentum and wave vector. However this contradicts with Eq. (2), since this deterministic relation directly connects
the momentum of the quantum particle to the wave vector of the wave. The quantum mechanics defines the uncertainty
expression of
(11)
between the momentum and location (x) of a quantum particle. However there can not be any indeterminacy between the
momentum and the wave vector because of Eq. (2). As a result we have three drawbacks in the interpretation of Eq. (2)
according to the experiments of Couder et al. For a stationary particle we met with Eq. (8), which states that the wave vector
of a quantum wave is equal to zero when the particle is stationary. However because of its periodic motion with the angular
frequency ω0 , it excites a quantum wave. For a particle in motion, we obtained Eqs. (9) and (10). These equations state that
although the particle moves in a direction, determined with its momentum, the wave vector shows that the particle moves
in all directions.
3. A new interpretation and formulation of the momentum-wave vector relation
In order to eliminate the two problems, met for the stationary and moving quantum particles in the previous section, I
propose an alternative interpretation for Eq. (2), which can be formulized as
(12)
and
(13)
for two and three dimensional cases respectively. The integrals, at the right-hand sides of Eqs. (12) and (13), show the
vector summation of the wave vector in all directions of propagation. Thus the result will give a wave vector, which lays
along a unique direction. This direction also gives the direction of the particle’s momentum. As a result the representation
of Eq. (2) as in Eqs. (12) and (13) directly resolves the problem for the moving particle.
Equation (12) can be written as
(14)
when Eq. (6) is taken into account for a two dimensional case. Equation (14) reads
(15)
It is apparent that the integrals, in Eq. (15), are equal to zero. This means physically that the particle does not move. If the
wave vector, in Eq. (7) is considered, the integral, in Eq. (13), will be written as
(16)
for the three dimensional case. Eq. (16) becomes
(17)
where Ix , Iy and Iz are equal to
2 cos sin2 dd,
Ix =
=0=0
(18)
Y.Z. Umul / Optik 127 (2016) 10844–10849
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2 sin sin2 dd
Iy =
(19)
=0=0
and
Iz = 2
sin cos d
(20)
0
respectively. All of these integrals are equal to zero. This means that the momentum of the particle is also zero, thus the
particle is stationary. A third type of wave is a plane wave which is propagating along the x axis. Its wave vector can be given
by
k = kex .
(21)
The momentum of the particle can be expressed by
(22)
for the two dimensional case. Thus the relation becomes
(23)
for a plane wave. Eq. (23) means physically that the quantum particle is moving along the positive x direction with a
. These evaluations show that the new interpretation and formulation directly resolves the two
momentum p, equal to
problems, mentioned in the previous section.
4. The edge diffraction phenomenon
As a final example, I will take into account the edge diffracted fields by a perfectly conducting half-plane. The half-screen
is located at x > 0 and y = 0. The edge contour lays on the z axis. The high-frequency asymptotic expression of the diffracted
field can be given by
ud = ui (QE ) D (, 0 )
e−jk
(24)
k
where u represents the z components of the electric or magnetic field intensity [29]. ui (QE ) is the edge point value of the
incident electromagnetic field. D shows the diffraction coefficient. ϕ and ϕ0 are the angles of observation and incidence
respectively. The wave vector of the diffracted wave is equal to Eq. (6). Note the term “k” shows the ray path between
the edge and observation points. The quantum particle (photon in this case) comes to the edge point of the half-plane by
following the ray path of the incident wave. However it will be stationary at this point according to Eqs. (14) and (15). This
means that the particle does not interact with the edge point in the scattering process. Only the pilot wave will be scattered
from the edge of the half-plane and its direction of the wave vector will determine the path of the quantum particle. Although
this statement can be thought of as incorrect, the experimental results of Couder and Fort [30] support my interpretation.
Figure 2 of [30] shows the transmission of two droplets from a slit one by one. It can be seen from the figure that although the
particles pass through the middle part of the aperture, their trajectories deviate on a large scale as if they are being diffracted
by the edge points. Figure 5 of [30] gives the trajectories of 100 droplets. It is clear from the figure that the interaction
probability of the particles with the edges is nearly equal to zero.
5. The relation between wave vectors
de Broglie defined the pilot wave for a moving particle by using the special theory of relativity [4,5]. I propose that a
non-moving particle causes a wave with the angular frequency ω0 , given by Eq. (5). It is a circular or spherical wave for two
or three dimensional cases with a phase velocity of vf , which is smaller than the speed of light. The wave number of this
wave is equal to k0 , which can be determined by the equation
k0 =
ω0
vf
.
(25)
When the particle begins to move with a velocity vg , the frequency of the accompanying wave will be equal to
(26)
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Y.Z. Umul / Optik 127 (2016) 10844–10849
where ␤ can be defined by
ˇ=
1−
v2g
(27)
c2
according to the theory of special relativity. Note that ω is the frequency of the wave at the direction of particle’s motion.
Eq. (25) also leads to
ω=
ω0
ˇ
(28)
according to Eq. (5). Since ␤ is smaller than one, the frequency increases as the particle’s velocity increases like in the Doppler
effect. The wave number, at the direction of particle’s motion, can be given by the equation
kd =
k0
.
ˇ
(29)
Now the relativistic momentum equation of
(30)
is considered. kT is defined by
2
1
kT =
2
k () d
(31)
0
or
1
kT =
4
2 k , sin dd
(32)
=0=0
shows the norm of the vector A.
The equation
for two or three dimensional problems respectively. A
kT =
vg vf
c2
kd
(33)
can be obtained. This relation gives a clue about the structure of the accompanying wave, since it relates the wave number
on the direction of the particle’s motion (kd ) to the norm of the summation of the wave vector in the same direction. For the
plane wave, k will be equal to kT and the equation of
vg vf = c 2
(34)
will hold for the velocities. In this case vf is greater than c.
6. Conclusions
In this letter, I proposed a different interpretation for Eq. (2). Based on some problems, which occur for stationary and
moving quantum particles, I suggested that the wave vector, in Eq. (2), is the summation of the accompanying wave’s wave
vector over the whole contour or surface of propagation. This proposal solved the problems, mentioned for the stationary
and moving particles. The application of the new interpretation also showed that the particles do not interact with the
edge point in the diffraction problem by a half-plane. Instead the accompanying wave interacts with the edge and the
resultant summation wave vectors determine the propagation trajectories of the particles. The experiments of Couder and
Fort also support this behavior [23]. As a final result, I obtained a relation between kd and kT , which will be a guide for the
determination of the accompanying wave’s structure in further studies. My interpretation and formulation do not change
anything with respect to the particle’s aspect. However a new theory will be constructed when the accompanying wave is
considered.
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