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A two-wave hypothesis of massive particles
Volume
A TWO-WAVE
15 September 1986
PHYSICS LETTERS A
117, number 9
HVPOTHESIS
OF MASSIVE
PARTICLES
S.N. DAS Theoretical Physics Centre, Department
of Physics, Midnapore
Received 10 April 1986; accepted for publication
College, Midnapore
721 IOI.
India
29 July 1986
In contrast to the three-wave model published earlier, this letter shows that the waves associated with massive particles can be reduced to two: the de Broglie wave and the transformed Compton wave, which are dual to each other.
In the recently proposed three-wave model (TWM) [1,2], a massive particle in motion is associated with three waves, viz. (i) an internal nondispersive standing C-wave and two external ones, (ii) a superluminal, i.e. the de Broglie B-wave and (iii) a subluminal D-wave whose phase velocity is the particle velocity, the frequencies and wavelengths of the above waves are defined as [l] E=hvc=hv,=hv,, A, = h/me,
04 A, = h/mu,
A, = h/mu, (lb,c,d)
where m is the mass of the particle with velocity u, c is the velocity of light, u is the phase velocity of the B-wave and h is the universal Planck constant. A similar idea was also put forward by Horodecki in some earlier papers [3,4]. We like to point out here that the wavelength X, given for the dual wave (D-wave) is not physically meaningful. In particular, the dispersion relation for the D-wave expressed in an implicit form, w’D = c2k:, - c2k,ZP-2,
(2)
is not covariant under Lorentz transformations, where k, = (m,c)h-‘, /3 = v/c and m. is the rest mass of the particle. This difficulty was first observed by Horodecki [4] and he then introduced the D’-wave [4] instead of the D-wave [3] to resolve this anomaly. Thus both Kostro [1,2] and Horodecki [3,4] considered three waves for a 436
satisfactory explanation of the wave properties of a massive particle. The main purpose of this letter is to show that one can fully explain (keeping consistency with relativity theory) the wave properties of a massive particle by associating only two waves: the de Broglie wave and the transformed Compton wave (TCW). The three-wave model thus seems to be irrelevant. The possible significance of the TCW is also discussed. The wave property involves four parameters v, respectively the freA, up and vt representing quency, wavelength, the group and phase velocity of the waves. The wave-particle dualism of matter [5] given by E=hv,
IPI =h/X
implies that the frequency corresponds to the energy and the wavelength corresponds to the momentum of the particle. Further, the group velocity us of the matter wave is known to be equal to the particle velocity u. Thus v + E, A + v,? For the de I P 0 up + v. Then what about Broglie B-wave, vr = c2/v > c is, of course, not directly measurable. However, since the phase velocity is an indispensable property of a wave, it is apparent from the wave-particle dualism that a massive particle in motion with velocity v might also be endowed with a non-observable velocity (say, pseudovelocity) u’ such that one can then complete the set of correspondences: v + E, X --, ) p I, up + u and of + u’. Such a proposition also seems transparent from the concept that “the
0375-9601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 117. number 9
PHYSICS LE-M-ERS A
particle picture and the wave picture are merely two different aspects of one and the same physical reality” [6], thereby indicating that the same number of parameters would be involved both in the particle and wave picture. In fact, it will be seen that the concept of a pseudovelocity (u’) is inevitable from the standpoint of the physical nature of the dual or transformed Compton wave. The concept of velocity was initially introduced in physics when space and time were considered absolute. But the theory of relativity asserts that space and time are not an absolute, but rather a relative concept and are placed on the same footing, xP=(ct=xO, x). It is then desirable that one should not consider only the variation of spacedistance x with respect to time t, or stated differently, the variation of distance x with respect to time-distance x0 = ct to define the particle velocity u = dx/dt = c dx/dx,, but one should also simultaneously consider the variation of the timedistance x0 with respect to space-distance x, which leads one to define another velocity (say, pseudovelocity) u’ in the form U’ = c dx,/dx. Actually, it is emphasized here that a massive particle in motion (say, along the x-axis) is simultaneously associated with two velocities, (i) the actual velocity IJ = c dx/dx, 0: rate of change of x with respect to x0, and (ii) an unobservable velocity u’ = c dx,/dx a rate of change of x0 with respect to x, and one always gets u’u = c*, a relation similar to the relation urus = c*, found from the wave aspect of matter. A comment should be made here. The concept of pseudovelocity u’ ‘should not be considered trivial because it is already present in an implicit form in relativity as is evident from the time part of the Lorentz transformation, t’ = y(t - ux/c*) = y (t - x/u’), y = (1 B*)- “* . Further, the ratio of the energy E and momentum p of the particle moving with velocity u is found to be equal to its pseudovelocity u’: E/p = u’. The de Broglie wave and Compton wave play an fundamental role both in relativistic and nonrelativistic quantum mechanics. In a previous paper [7], the present author has shown that a massive particle in motion is associated with only two waves, viz. (i) the existing de Broglie wave (B-wave) which corresponds to the momentum of ,
15 September 1986
the particle [X, ap-‘1 and (ii) the transformed Compton wave (TCW) which corresponds to the energy E of the particle [X, a E-‘1. However, in ref. [7] the characteristics, in particular the frequency Yk, the phase velocity u, and the group velocity ugk of the TCW have not been stated clearly. Now it will be seen from the concept of pseudovelocity u’ that the said parameters are respectively uk = muc/h, uR = u and I+ = c*/u = u’, thereby indicating that the TCW is actually dual to the de Broglie B-wave. From the well known relativistic relation E * = p*c* + m&z4 as well as the relation E = pu’ mentioned above, one obtains p*u’* =p*c* + mic4, which, by taking p = u/c and /3’ = u’/c, can alternatively be written as mo( /3’* - 1)-l.‘* =p/c
= m/l.
(3)
Now considering m’ = mo(/3’* - 1)-l/* = rnp > 0 as the variation of mass m, of the particle having the pseudovelocity u’, the corresponding energy E’ and momentum p’ may be defined as E’=
m’c* =pc,
p’=
m’u’= E/c.
(40)
We then get two sets (E, p) and (E’, p’) for the energy and momentum corresponding to two velocities u and u ’ of the particle. In the de Broglie matter wave, (ue, X a) are correlated -with (E, p). In an analogous manner, a second kind of matter wave may now be introduced, whose frequency ( vk ) and wavelength (X k ) are correlated with E’ and p’ in the form E’ = hzy,
p’ = h/X,,
which, by using expressed as vk = muc/h,
eqs. (4a) and
A, = h/me.
Eq. (5b) can be defined A, = (&J,(l (&),,
= h/m,c.
(4b) can now be
-P*)“*,
alternatively
(5a,b) as (6a) (6b)
Since (A,),, is identified with the Compton wavelength Xc = h/m,c of the particle, this second 437
Volume 117, number 9
wave may be termed the transformed Compton wave. The phase velocity u, of the TCW is u, = vkXk = u, the particle velocity. It may be recalled that the phase velocity of the D-wave (dual wave) proposed in the TWM [1,2] is also supposed to be the particle velocity u. The dispersion relation (2) for the D-wave is, however, not consistent with relativity. On the other hand, the dispersion relation for the TCW, now expressed as w; = k;c* - k,2c2,
(7)
is found to be covariant under Lorentz transformations and therefore, consistently supports the view that this TCW is actually dual to the de Broglie B-wave in the sense that its phase velocity is the particle velocity. The wave function #k for the TCW may now be given in the form $k = exp[i(k,x
- car)],
(8)
representing a plane wave of (i) frequency yk = muc/h, (ii) wavelength A, = h/me and (iii) phase velocity uk equal to the particle velocity u. We find a similar result in a paper by Elbaz [8], in which an amplitude wavefunction in the form u = exp[ i( kx - ii%)]
438
= 0,
where 0 = c-*a2/a t* - v *. The relations between the two wavelengths and frequencies are found as Ai2 = Xi* + Xi2, V’B =
2 yk+Q,
A, = &a,
2 vk
=&I,
v,=ch,‘.
A, = cv;‘,
(lOa) (lob)
In the rest frame: X,=X,, A,= cc; vk =O, vg= v0 and at velocity c: A, = A,, vk = va. This is the case of photons. From the above discussion we conclude that a massive particle in motion is actually associated with two waves: the TCW (A, a E-‘) and the existing B-wave (A, axp-'),which are dual to each other. Thus it seems sufficient to consider a two-wave hypothesis instead of three waves in order avoid inconsistency with relativity theory. The author desires to express his gratitude and thanks to an anonymous referee for some helpful suggestions. He also wishes to express his cordial thanks to Professor P.P. Roy of Viswavarati University and Professor SC. Samanta of Midnapore College for useful discussions.
References
with k = h/me, Q = muc/h, has been defined by considering that not only the time part of the Lorentz transformation but all the relativistic transformation properties of the coordinates of a massive particle should be taken into account. Eq. (8). together with the dispersion relation (7) yields immediately the wave equation [9] appropriate for the k-wave (TCW) as (CI - m$z*/h*)#,
15 September 1986
PHYSICS LETTERS A
(9)
[l] L. Kostro, Phys. Lett. A 107 (1985) 429. [2] L. Kostro, Phys. Lett. A 112 (1985) 283. [3] R. Horodecki, Phys. Lett. A 87 (1981) 95. [4] R. Horodecki, Phys. Lett. A 91 (1982) 269. (51 L. De Broglie, CR. Acad. Sci. 180 (1925) 498. [6] .M. Jammer, The philosophy of quantum mechanics (Wiley-Interscience, New York, 1974) p. 68. [7] S.N. Das, Phys. Lett. A 102 (1984) 338. [8] C. Elbaz, Phys. Lett. A 109 (1985) 7. [9] C. Elbaz, Phys. Lett. A 114 (1986) 445.
A TWO-WAVE
15 September 1986
PHYSICS LETTERS A
117, number 9
HVPOTHESIS
OF MASSIVE
PARTICLES
S.N. DAS Theoretical Physics Centre, Department
of Physics, Midnapore
Received 10 April 1986; accepted for publication
College, Midnapore
721 IOI.
India
29 July 1986
In contrast to the three-wave model published earlier, this letter shows that the waves associated with massive particles can be reduced to two: the de Broglie wave and the transformed Compton wave, which are dual to each other.
In the recently proposed three-wave model (TWM) [1,2], a massive particle in motion is associated with three waves, viz. (i) an internal nondispersive standing C-wave and two external ones, (ii) a superluminal, i.e. the de Broglie B-wave and (iii) a subluminal D-wave whose phase velocity is the particle velocity, the frequencies and wavelengths of the above waves are defined as [l] E=hvc=hv,=hv,, A, = h/me,
04 A, = h/mu,
A, = h/mu, (lb,c,d)
where m is the mass of the particle with velocity u, c is the velocity of light, u is the phase velocity of the B-wave and h is the universal Planck constant. A similar idea was also put forward by Horodecki in some earlier papers [3,4]. We like to point out here that the wavelength X, given for the dual wave (D-wave) is not physically meaningful. In particular, the dispersion relation for the D-wave expressed in an implicit form, w’D = c2k:, - c2k,ZP-2,
(2)
is not covariant under Lorentz transformations, where k, = (m,c)h-‘, /3 = v/c and m. is the rest mass of the particle. This difficulty was first observed by Horodecki [4] and he then introduced the D’-wave [4] instead of the D-wave [3] to resolve this anomaly. Thus both Kostro [1,2] and Horodecki [3,4] considered three waves for a 436
satisfactory explanation of the wave properties of a massive particle. The main purpose of this letter is to show that one can fully explain (keeping consistency with relativity theory) the wave properties of a massive particle by associating only two waves: the de Broglie wave and the transformed Compton wave (TCW). The three-wave model thus seems to be irrelevant. The possible significance of the TCW is also discussed. The wave property involves four parameters v, respectively the freA, up and vt representing quency, wavelength, the group and phase velocity of the waves. The wave-particle dualism of matter [5] given by E=hv,
IPI =h/X
implies that the frequency corresponds to the energy and the wavelength corresponds to the momentum of the particle. Further, the group velocity us of the matter wave is known to be equal to the particle velocity u. Thus v + E, A + v,? For the de I P 0 up + v. Then what about Broglie B-wave, vr = c2/v > c is, of course, not directly measurable. However, since the phase velocity is an indispensable property of a wave, it is apparent from the wave-particle dualism that a massive particle in motion with velocity v might also be endowed with a non-observable velocity (say, pseudovelocity) u’ such that one can then complete the set of correspondences: v + E, X --, ) p I, up + u and of + u’. Such a proposition also seems transparent from the concept that “the
0375-9601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 117. number 9
PHYSICS LE-M-ERS A
particle picture and the wave picture are merely two different aspects of one and the same physical reality” [6], thereby indicating that the same number of parameters would be involved both in the particle and wave picture. In fact, it will be seen that the concept of a pseudovelocity (u’) is inevitable from the standpoint of the physical nature of the dual or transformed Compton wave. The concept of velocity was initially introduced in physics when space and time were considered absolute. But the theory of relativity asserts that space and time are not an absolute, but rather a relative concept and are placed on the same footing, xP=(ct=xO, x). It is then desirable that one should not consider only the variation of spacedistance x with respect to time t, or stated differently, the variation of distance x with respect to time-distance x0 = ct to define the particle velocity u = dx/dt = c dx/dx,, but one should also simultaneously consider the variation of the timedistance x0 with respect to space-distance x, which leads one to define another velocity (say, pseudovelocity) u’ in the form U’ = c dx,/dx. Actually, it is emphasized here that a massive particle in motion (say, along the x-axis) is simultaneously associated with two velocities, (i) the actual velocity IJ = c dx/dx, 0: rate of change of x with respect to x0, and (ii) an unobservable velocity u’ = c dx,/dx a rate of change of x0 with respect to x, and one always gets u’u = c*, a relation similar to the relation urus = c*, found from the wave aspect of matter. A comment should be made here. The concept of pseudovelocity u’ ‘should not be considered trivial because it is already present in an implicit form in relativity as is evident from the time part of the Lorentz transformation, t’ = y(t - ux/c*) = y (t - x/u’), y = (1 B*)- “* . Further, the ratio of the energy E and momentum p of the particle moving with velocity u is found to be equal to its pseudovelocity u’: E/p = u’. The de Broglie wave and Compton wave play an fundamental role both in relativistic and nonrelativistic quantum mechanics. In a previous paper [7], the present author has shown that a massive particle in motion is associated with only two waves, viz. (i) the existing de Broglie wave (B-wave) which corresponds to the momentum of ,
15 September 1986
the particle [X, ap-‘1 and (ii) the transformed Compton wave (TCW) which corresponds to the energy E of the particle [X, a E-‘1. However, in ref. [7] the characteristics, in particular the frequency Yk, the phase velocity u, and the group velocity ugk of the TCW have not been stated clearly. Now it will be seen from the concept of pseudovelocity u’ that the said parameters are respectively uk = muc/h, uR = u and I+ = c*/u = u’, thereby indicating that the TCW is actually dual to the de Broglie B-wave. From the well known relativistic relation E * = p*c* + m&z4 as well as the relation E = pu’ mentioned above, one obtains p*u’* =p*c* + mic4, which, by taking p = u/c and /3’ = u’/c, can alternatively be written as mo( /3’* - 1)-l.‘* =p/c
= m/l.
(3)
Now considering m’ = mo(/3’* - 1)-l/* = rnp > 0 as the variation of mass m, of the particle having the pseudovelocity u’, the corresponding energy E’ and momentum p’ may be defined as E’=
m’c* =pc,
p’=
m’u’= E/c.
(40)
We then get two sets (E, p) and (E’, p’) for the energy and momentum corresponding to two velocities u and u ’ of the particle. In the de Broglie matter wave, (ue, X a) are correlated -with (E, p). In an analogous manner, a second kind of matter wave may now be introduced, whose frequency ( vk ) and wavelength (X k ) are correlated with E’ and p’ in the form E’ = hzy,
p’ = h/X,,
which, by using expressed as vk = muc/h,
eqs. (4a) and
A, = h/me.
Eq. (5b) can be defined A, = (&J,(l (&),,
= h/m,c.
(4b) can now be
-P*)“*,
alternatively
(5a,b) as (6a) (6b)
Since (A,),, is identified with the Compton wavelength Xc = h/m,c of the particle, this second 437
Volume 117, number 9
wave may be termed the transformed Compton wave. The phase velocity u, of the TCW is u, = vkXk = u, the particle velocity. It may be recalled that the phase velocity of the D-wave (dual wave) proposed in the TWM [1,2] is also supposed to be the particle velocity u. The dispersion relation (2) for the D-wave is, however, not consistent with relativity. On the other hand, the dispersion relation for the TCW, now expressed as w; = k;c* - k,2c2,
(7)
is found to be covariant under Lorentz transformations and therefore, consistently supports the view that this TCW is actually dual to the de Broglie B-wave in the sense that its phase velocity is the particle velocity. The wave function #k for the TCW may now be given in the form $k = exp[i(k,x
- car)],
(8)
representing a plane wave of (i) frequency yk = muc/h, (ii) wavelength A, = h/me and (iii) phase velocity uk equal to the particle velocity u. We find a similar result in a paper by Elbaz [8], in which an amplitude wavefunction in the form u = exp[ i( kx - ii%)]
438
= 0,
where 0 = c-*a2/a t* - v *. The relations between the two wavelengths and frequencies are found as Ai2 = Xi* + Xi2, V’B =
2 yk+Q,
A, = &a,
2 vk
=&I,
v,=ch,‘.
A, = cv;‘,
(lOa) (lob)
In the rest frame: X,=X,, A,= cc; vk =O, vg= v0 and at velocity c: A, = A,, vk = va. This is the case of photons. From the above discussion we conclude that a massive particle in motion is actually associated with two waves: the TCW (A, a E-‘) and the existing B-wave (A, axp-'),which are dual to each other. Thus it seems sufficient to consider a two-wave hypothesis instead of three waves in order avoid inconsistency with relativity theory. The author desires to express his gratitude and thanks to an anonymous referee for some helpful suggestions. He also wishes to express his cordial thanks to Professor P.P. Roy of Viswavarati University and Professor SC. Samanta of Midnapore College for useful discussions.
References
with k = h/me, Q = muc/h, has been defined by considering that not only the time part of the Lorentz transformation but all the relativistic transformation properties of the coordinates of a massive particle should be taken into account. Eq. (8). together with the dispersion relation (7) yields immediately the wave equation [9] appropriate for the k-wave (TCW) as (CI - m$z*/h*)#,
15 September 1986
PHYSICS LETTERS A
(9)
[l] L. Kostro, Phys. Lett. A 107 (1985) 429. [2] L. Kostro, Phys. Lett. A 112 (1985) 283. [3] R. Horodecki, Phys. Lett. A 87 (1981) 95. [4] R. Horodecki, Phys. Lett. A 91 (1982) 269. (51 L. De Broglie, CR. Acad. Sci. 180 (1925) 498. [6] .M. Jammer, The philosophy of quantum mechanics (Wiley-Interscience, New York, 1974) p. 68. [7] S.N. Das, Phys. Lett. A 102 (1984) 338. [8] C. Elbaz, Phys. Lett. A 109 (1985) 7. [9] C. Elbaz, Phys. Lett. A 114 (1986) 445.