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De Broglie wave and Compton wave
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
DE BROGLIE WAVE AND COMFFON WAVE S.N. DAS Theoretical Physics Centre, Department of Physics, Midnapore College, Midnapore- 721101, India Received 10 June 1983 Revised manuscript received 7 March 1984
In contrast to the three-wave hypothesis (TWH) presented earlier [ l ], it is argued in this letter that a massive particle in motion in a Lorentz frame will actually be associated with only two types of waves: (i) a transformed Compton wave and (ii) a superluminal de Broglie wave (B-wave). The subluminal wave (D-wave or D'-wave [2]) cannot be simultaneously correlated with the particle under consideration.
In the three-wave hypothesis (TWH) presented earlier [ 1], it has been assumed that in a Lorentz frame where the particle is at rest, it is associated with an intrinsic nondispersive Compton wave (C-wave). However, in a Lorentz frame where the particle moves with velocity o , i t is assumed to be associated with [ apart from the existing superluminal de Broglie wave (Bwave)] two more waves, viz. (i) a transformed C-wave and (ii) a subluminal wave whose phase velocity is the particle velocity o (D-wave). It may be remarked that b o t h in refs. [1] and [2], the main emphasis was on the introduction o f the concept of a dual wave (D-wave or D'-wave [2] but no theoretical grounds except the statement have been given to introduce the idea of an intrinsic Compton wave (C-wave) and the transformed C-wave that may be associated with the particle respectively at rest and in motion in a Lorentz frame. In view o f this fact, this letter presents a simple derivation which leads to the result that the above-mentioned wave properties may consistently be associated with the particle. Further it will be argued that in a Lorentz frame, where the particle is moving, it may be associated with only two waves: (i) a transformed Compton wave (C-wave) and (ii) a superluminal de Broglie wave (B-wave). The subluminal D-wave or D'-wave [2] cannot be simultaneously associated with the particle under consideration. It is well know [3] that the fundamental expres338
sions for the wave-particle dualism can be written in the form E=hv,
IPBI =h/X B .
(1,2)
Here E is the energy of the particle and v is its de Broglie frequency, PB is the momentum o f the particle and XB is the wavelength of the de Broglie wave (Bwave). Although the above expressions are true both for matter and radiation, an interesting observation may be noted that E cc v, IPBI ~ I/~.B and E oc 1/XB for photons, but E cc v, IPBI cx 1/XB and E q: 1/X B for massive particles. So in order to restore the symmetry between the relations valid for photons as well as massive particles, we now propose that there might exist a new kind of wave of wavelength Xk which is supposed to be proportional to the energy o f the particle and that Xk ~'B for photons but Xk :~ XB for the massive particles. Then one can have E cc v, IPI31 ~ l/X13 and E ~x 1/X k for photons (with Xk = XB); E cx v, LPBI ~ 1/'hi3 and E o: 1/hk for massive particles (with Xk ~TB)Actually we define here that apart from the relation between the energy and frequency o f radiation or massive particles, a new wavelength h k can also be related with the energy such that =
E = O/X k ,
(3)
where the constant of proportionality D is determined from the consideration that ~'k = ~-B for photons but 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 8
PHYSICS LETTERS
Xk ~ XB for massive particles. It is significantly noted that the e n e r g y - m o m e n t u m relation E 2 = p 2 c 2 + m 2 c 4 provides the proper hints. From this relation one can have
4 June 1984
Eq. (5) fulfils the condition that Xk 4: XB for massive particles and Xk = XB for photons (m 0 = 0) and thus it seems that the value o f D should equal hc. Therefore, one can have from eq. (3)
two types o f matter waves viz. (i) the de Broglie wave (B-wave) that corresponds to the three-momentum PB o f the particle [XB = IPB1-1 ] and (ii) the transformed Compton wave that corresponds to the energy E of the particle [Xk = E - 1 ], the proportionality constants being different and are respectively h and hc. Suppose we now define a space (S') dual to real space (S) such that in the dual space, the energy and momentum of the particle becomes E' = pB c and IP'I = E/c. It is then easy to see that the phase velocity of o f the matter-waves associated with the particle in S'space is v and consequently the group velocity O'g = c2/v. Needless to say, the four-momentum of the particle P u = ( E / c, p ) = (PB, E /c) is tl,en a s p a c e time vector. The wavelength o f the de Broglie wave (X~) and the transformed Compton wave (X~) o f the particle moving in dual space (S') are now given as
E = hc/X k .
x'B = h~ ~'1 = he~L" = Xk,
That is,
and
E 2 / h 2 c 2 = p2B/h2 + E 2 / h 2 c 2 ,
(4)
where E 0 = mOc2 is the energy o f the particle. Now putting eq. (3) in eq. (4), one gets x2/X 2 = 1/~,2B + x2/(',/k) 2 , where x = D / h e and in particular, i f x = 1 i.e. D = he, the above relation turns out to be 1/X~ = 1/XB2 + 1/('fk) 2 .
X k = hc/E = h/(p 2 + m 2 c 2 ) I / 2
(5)
(6a)
and (Xk ) 0 = h / m o c .
(6b)
Needless to say, Xk and (Xk) are connected in the form Xk = (Xk) 0 (1 - t32) 1/2 ,
(7)
where/3 = o/c. This new wave aspect o f matter differs from the de Broglie wave in the sense that the de Broglie wave (Bwave) becomes mathematically undefined for a particle at rest whereas the latter one suggests a finite value even for the rest particle and given (Xk) 0 = h/moc. It is interesting to note that (Xk) 0 is identified with the Compton wavelength ~C = h / m a c o f the particle arid so Xk = (Xk) 0 (1 -/32)1/2 = XC(1 _/32)1/2 for the moving particle may be termed as the transformed Compton wave. We have thus presented a simple derivation that has consistently supported the statement [1 ] that a particle at rest and in motion in a Lorentz frame may respectively be associated with an intrinsic non-dispersive Compton wave [XC = (Xk) 0 = h / m o c ] and the transformed Compton wave [Xk = XC(I - / 3 2 ) 1 / 2 ] . From the above discussion, we then find that any massive particle in motion may actually be associated with
•
!
t
S
--
X'k = hc/E' = h/[PBI = XB,
(8a)
(Sb)
implying that the de Broglie wavelength and the transformed Compton wavelength of matter waves (with of = v, Og = c2/v) in the dual space are respectively the same as the transformed Compton wavelength of matter waves (with of = c2/v, Vg = o)in the real space ,x. This fact thus leads to the conclusion that the concept o f the three-wave hypothesis (TWH) [1] seems irrelevant. In fact, a massive particle in motion will be associated in the absolute sense with only two types o f waves (XB, Xk) whether the phase velocity o f the matter waves is of = c2/v or of = o. The author is grateful to Professor S.R. Maiti o f Midnapore College for helpful discussions. ,1 It is gratifying to note that in ref. [2] the wavelength of the D'-wave (with of = o) is also nothing but the transformed Compton wavelength hk of matter waves with of = c2/v.
[1] [21 [3] [4]
R. Horodecki, Phys. Lett. 87A (1981) 95. R. Horodecki, Phys• Lett. 91A (1982) 269. L. de Broglie, C.R. Acad Sci. 180 (1925) 498. J. Aharoni, The special theory of relativity (Clarendon, Oxford). 339
PHYSICS LETTERS
4 June 1984
DE BROGLIE WAVE AND COMFFON WAVE S.N. DAS Theoretical Physics Centre, Department of Physics, Midnapore College, Midnapore- 721101, India Received 10 June 1983 Revised manuscript received 7 March 1984
In contrast to the three-wave hypothesis (TWH) presented earlier [ l ], it is argued in this letter that a massive particle in motion in a Lorentz frame will actually be associated with only two types of waves: (i) a transformed Compton wave and (ii) a superluminal de Broglie wave (B-wave). The subluminal wave (D-wave or D'-wave [2]) cannot be simultaneously correlated with the particle under consideration.
In the three-wave hypothesis (TWH) presented earlier [ 1], it has been assumed that in a Lorentz frame where the particle is at rest, it is associated with an intrinsic nondispersive Compton wave (C-wave). However, in a Lorentz frame where the particle moves with velocity o , i t is assumed to be associated with [ apart from the existing superluminal de Broglie wave (Bwave)] two more waves, viz. (i) a transformed C-wave and (ii) a subluminal wave whose phase velocity is the particle velocity o (D-wave). It may be remarked that b o t h in refs. [1] and [2], the main emphasis was on the introduction o f the concept of a dual wave (D-wave or D'-wave [2] but no theoretical grounds except the statement have been given to introduce the idea of an intrinsic Compton wave (C-wave) and the transformed C-wave that may be associated with the particle respectively at rest and in motion in a Lorentz frame. In view o f this fact, this letter presents a simple derivation which leads to the result that the above-mentioned wave properties may consistently be associated with the particle. Further it will be argued that in a Lorentz frame, where the particle is moving, it may be associated with only two waves: (i) a transformed Compton wave (C-wave) and (ii) a superluminal de Broglie wave (B-wave). The subluminal D-wave or D'-wave [2] cannot be simultaneously associated with the particle under consideration. It is well know [3] that the fundamental expres338
sions for the wave-particle dualism can be written in the form E=hv,
IPBI =h/X B .
(1,2)
Here E is the energy of the particle and v is its de Broglie frequency, PB is the momentum o f the particle and XB is the wavelength of the de Broglie wave (Bwave). Although the above expressions are true both for matter and radiation, an interesting observation may be noted that E cc v, IPBI ~ I/~.B and E oc 1/XB for photons, but E cc v, IPBI cx 1/XB and E q: 1/X B for massive particles. So in order to restore the symmetry between the relations valid for photons as well as massive particles, we now propose that there might exist a new kind of wave of wavelength Xk which is supposed to be proportional to the energy o f the particle and that Xk ~'B for photons but Xk :~ XB for the massive particles. Then one can have E cc v, IPI31 ~ l/X13 and E ~x 1/X k for photons (with Xk = XB); E cx v, LPBI ~ 1/'hi3 and E o: 1/hk for massive particles (with Xk ~TB)Actually we define here that apart from the relation between the energy and frequency o f radiation or massive particles, a new wavelength h k can also be related with the energy such that =
E = O/X k ,
(3)
where the constant of proportionality D is determined from the consideration that ~'k = ~-B for photons but 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 8
PHYSICS LETTERS
Xk ~ XB for massive particles. It is significantly noted that the e n e r g y - m o m e n t u m relation E 2 = p 2 c 2 + m 2 c 4 provides the proper hints. From this relation one can have
4 June 1984
Eq. (5) fulfils the condition that Xk 4: XB for massive particles and Xk = XB for photons (m 0 = 0) and thus it seems that the value o f D should equal hc. Therefore, one can have from eq. (3)
two types o f matter waves viz. (i) the de Broglie wave (B-wave) that corresponds to the three-momentum PB o f the particle [XB = IPB1-1 ] and (ii) the transformed Compton wave that corresponds to the energy E of the particle [Xk = E - 1 ], the proportionality constants being different and are respectively h and hc. Suppose we now define a space (S') dual to real space (S) such that in the dual space, the energy and momentum of the particle becomes E' = pB c and IP'I = E/c. It is then easy to see that the phase velocity of o f the matter-waves associated with the particle in S'space is v and consequently the group velocity O'g = c2/v. Needless to say, the four-momentum of the particle P u = ( E / c, p ) = (PB, E /c) is tl,en a s p a c e time vector. The wavelength o f the de Broglie wave (X~) and the transformed Compton wave (X~) o f the particle moving in dual space (S') are now given as
E = hc/X k .
x'B = h~ ~'1 = he~L" = Xk,
That is,
and
E 2 / h 2 c 2 = p2B/h2 + E 2 / h 2 c 2 ,
(4)
where E 0 = mOc2 is the energy o f the particle. Now putting eq. (3) in eq. (4), one gets x2/X 2 = 1/~,2B + x2/(',/k) 2 , where x = D / h e and in particular, i f x = 1 i.e. D = he, the above relation turns out to be 1/X~ = 1/XB2 + 1/('fk) 2 .
X k = hc/E = h/(p 2 + m 2 c 2 ) I / 2
(5)
(6a)
and (Xk ) 0 = h / m o c .
(6b)
Needless to say, Xk and (Xk) are connected in the form Xk = (Xk) 0 (1 - t32) 1/2 ,
(7)
where/3 = o/c. This new wave aspect o f matter differs from the de Broglie wave in the sense that the de Broglie wave (Bwave) becomes mathematically undefined for a particle at rest whereas the latter one suggests a finite value even for the rest particle and given (Xk) 0 = h/moc. It is interesting to note that (Xk) 0 is identified with the Compton wavelength ~C = h / m a c o f the particle arid so Xk = (Xk) 0 (1 -/32)1/2 = XC(1 _/32)1/2 for the moving particle may be termed as the transformed Compton wave. We have thus presented a simple derivation that has consistently supported the statement [1 ] that a particle at rest and in motion in a Lorentz frame may respectively be associated with an intrinsic non-dispersive Compton wave [XC = (Xk) 0 = h / m o c ] and the transformed Compton wave [Xk = XC(I - / 3 2 ) 1 / 2 ] . From the above discussion, we then find that any massive particle in motion may actually be associated with
•
!
t
S
--
X'k = hc/E' = h/[PBI = XB,
(8a)
(Sb)
implying that the de Broglie wavelength and the transformed Compton wavelength of matter waves (with of = v, Og = c2/v) in the dual space are respectively the same as the transformed Compton wavelength of matter waves (with of = c2/v, Vg = o)in the real space ,x. This fact thus leads to the conclusion that the concept o f the three-wave hypothesis (TWH) [1] seems irrelevant. In fact, a massive particle in motion will be associated in the absolute sense with only two types o f waves (XB, Xk) whether the phase velocity o f the matter waves is of = c2/v or of = o. The author is grateful to Professor S.R. Maiti o f Midnapore College for helpful discussions. ,1 It is gratifying to note that in ref. [2] the wavelength of the D'-wave (with of = o) is also nothing but the transformed Compton wavelength hk of matter waves with of = c2/v.
[1] [21 [3] [4]
R. Horodecki, Phys. Lett. 87A (1981) 95. R. Horodecki, Phys• Lett. 91A (1982) 269. L. de Broglie, C.R. Acad Sci. 180 (1925) 498. J. Aharoni, The special theory of relativity (Clarendon, Oxford). 339