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On de Broglie waves and Compton waves of massive particles
Volume 109A, number 1,2
PHYSICS LETTERS
6 May 1985
ON DE BROGLIE WAVES AND COMPTON WAVES OF MASSIVE PARTICLES Claude E L B A Z lnstitut d'Optique, Centre Universitaire d'Orsay, B.P. 43, 91406 Orsay, France
Received 4 December 1984; accepted in revised form 7 March 1985
The Compton wave of a massive particle is associated with an amplitude function u, like its de Broglie wave with the wavefunction ~k of quantum mechanics. The frequencies of the electron amplitude function in the Bohr atom are the subharmonics of the base frequency N1= amc2/h where a = 1/137 is the fine structure constant.
Several authors [ 1 - 3 ] have recently searched for a relation between de Broglie and Compton wavelengths o f massive particles. A controversy has also been established abot~t the possibility of a de Broglie wavelength of the second kind of particles and its undefined character for photons. As these wavelengths play a fundamental role in non-relativistic and in relativistic quantum mechanics, we have thought it interesting to present a contribution to the problem. In his thesis, in 1924 [4], Louis de Broglie introduced the wavelength X -- h / m u by considering a "periodical phenomenon" of frequency E o / h = v 0 and phase S = 2rm0t 0 ,
(1)
associated to a massive particle of energy E 0 = mOc2 at rest. When it is in uniform translation with speed v, we find, by application of relativity: (i) a moving frequency v = v0/(1 - / j 2 ) 1 / 2 ,
(2)
such as
(3)
(ii) a time t related to t o by the Lorentz transformation t = (t o -- OX/C2)/(1 -- ~2)1/2 .
S = 21rvot 0 = 2 ~ v ( t - v x / c 2) = 2 ~ ( v t - X / • B ) ,
where X B = c2/vv = hc2/hvu = hc2/mc2u = h/mv.
(4)
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(5)
We find these calculus in de Broglie's thesis [4]. We can remark that, although the Lorentz transformation acts on space and time coordinates, only the time transformation has been used. All the relativistic transformation properties of the coordinates of a massive particle have not been taken into account and we can search how the relation x 0 = (x - o t ) / ( 1 - / 3 2 ) 1/2
(6)
could be introduced. De Broglie's hypothesis is by itself incomplete. By considering a "periodical phenomenon" with frequency v 0 = E o / h associated to a massive particle, de Broglie gave a physical meaning to one of the relations (3) and defined a phase wave = e x p ( 2 1 r i v o t o ) = exp [2rri(vt - x/XB) ] ,
E = EO/(1 - / 3 2 ) 1/2 = hv = h v o / ( 1 - / 3 2 ) 1 / 2 = m c 2 = m O c 2 / ( 1 - / 3 2 ) 1/2 ,
The phase S, which is a relativistic invariant, can be written, with (2) and (4), as
(7)
expressed in the rest system and in the moving system. In an earlier work [3], we defined an amplitude wave u --- exp(2rrix0/h0) by giving a physical meaning to the relation E 0 = mOc2 .
Volume 109A, number 1,2
PHYSICS LETTERS
6 May 1985
We associated a standing wave e(x, t) = u ( x ) ~ ( t ) , a solution o f Vie = 0, with a massive particle such that
(ii) a phase velocity equal to its moving speed v, and (iii) a frequency N =/3u = m v c / h .
)t o = c/v 0 = hc/hv 0 = hc/moc2 = h / m o c ,
Application. If we use the electron amplitude function u in the hydrogen Bohr atom, where its speed v is
(8)
where Xo is the Compton wavelength of the particle at rest. When it is moving with speed v =/3c, by application of relativity we find: (i) a moving Compton wavelength X = Xo(1 -/32)1/2
(9)
from (3) and (8), and (ii) a space coordinate x related to x 0 by (6). By setting
o = e2i~ = c¢c, where a = 1/137.02 is the fine structure constant, taken as a fundamental constant, the quantization condition in non-relativistic quantum mechanics can be found if we write that, for the electron stable orbits, the frequencies are divisors of the base frequency, N 1 = amOC2/h ,
k = 2ir/X = k0/(1 - / 3 2 ) 1/2 = 27rl)t0(1 -/32)1/2 N n = n-lN1. = 2zwlc,
(1 O)
the amplitude function becomes u = exp(ik0x0) = exp [i(kx - ~ t ) ]
(11)
in a system at rest and in a moving system. This defines a frequency N such that N = ~212n =/3~ol2~ =/3v = m v c / h .
In other words these frequencies are the subharmonics of the base frequency N 1 . Under the same conditions, for atoms with electron numberZ, the frequenciesN Z are, in first approximation, integral multiples of the base frequency N 1 NZn
=
( Z I n ) ~ mocZ lh .
(12)
With (8), (6), (9) and (11) the properties o f the amplitude function are symmetrical to the wavefunction properties o f a particle (2), (4), (5) and (7). The amplitude function of a moving massive particle of mass m and speed o is characterized by (i) a wavelength equal to its Compton wavelength )t = h / m e ,
References [1] [2] [3] [4]
P. Mukhopadhyay, Phys. Lett. 105A (1984) 487. S.N. Das, Phys. Lett. 102A (1984) 338. C. Elbaz, C.R. Acad. Sci. 297 lI (1983) 455. L. de BrogUe, Th~se 1924 (Masson, Paris, 1963).
PHYSICS LETTERS
6 May 1985
ON DE BROGLIE WAVES AND COMPTON WAVES OF MASSIVE PARTICLES Claude E L B A Z lnstitut d'Optique, Centre Universitaire d'Orsay, B.P. 43, 91406 Orsay, France
Received 4 December 1984; accepted in revised form 7 March 1985
The Compton wave of a massive particle is associated with an amplitude function u, like its de Broglie wave with the wavefunction ~k of quantum mechanics. The frequencies of the electron amplitude function in the Bohr atom are the subharmonics of the base frequency N1= amc2/h where a = 1/137 is the fine structure constant.
Several authors [ 1 - 3 ] have recently searched for a relation between de Broglie and Compton wavelengths o f massive particles. A controversy has also been established abot~t the possibility of a de Broglie wavelength of the second kind of particles and its undefined character for photons. As these wavelengths play a fundamental role in non-relativistic and in relativistic quantum mechanics, we have thought it interesting to present a contribution to the problem. In his thesis, in 1924 [4], Louis de Broglie introduced the wavelength X -- h / m u by considering a "periodical phenomenon" of frequency E o / h = v 0 and phase S = 2rm0t 0 ,
(1)
associated to a massive particle of energy E 0 = mOc2 at rest. When it is in uniform translation with speed v, we find, by application of relativity: (i) a moving frequency v = v0/(1 - / j 2 ) 1 / 2 ,
(2)
such as
(3)
(ii) a time t related to t o by the Lorentz transformation t = (t o -- OX/C2)/(1 -- ~2)1/2 .
S = 21rvot 0 = 2 ~ v ( t - v x / c 2) = 2 ~ ( v t - X / • B ) ,
where X B = c2/vv = hc2/hvu = hc2/mc2u = h/mv.
(4)
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(5)
We find these calculus in de Broglie's thesis [4]. We can remark that, although the Lorentz transformation acts on space and time coordinates, only the time transformation has been used. All the relativistic transformation properties of the coordinates of a massive particle have not been taken into account and we can search how the relation x 0 = (x - o t ) / ( 1 - / 3 2 ) 1/2
(6)
could be introduced. De Broglie's hypothesis is by itself incomplete. By considering a "periodical phenomenon" with frequency v 0 = E o / h associated to a massive particle, de Broglie gave a physical meaning to one of the relations (3) and defined a phase wave = e x p ( 2 1 r i v o t o ) = exp [2rri(vt - x/XB) ] ,
E = EO/(1 - / 3 2 ) 1/2 = hv = h v o / ( 1 - / 3 2 ) 1 / 2 = m c 2 = m O c 2 / ( 1 - / 3 2 ) 1/2 ,
The phase S, which is a relativistic invariant, can be written, with (2) and (4), as
(7)
expressed in the rest system and in the moving system. In an earlier work [3], we defined an amplitude wave u --- exp(2rrix0/h0) by giving a physical meaning to the relation E 0 = mOc2 .
Volume 109A, number 1,2
PHYSICS LETTERS
6 May 1985
We associated a standing wave e(x, t) = u ( x ) ~ ( t ) , a solution o f Vie = 0, with a massive particle such that
(ii) a phase velocity equal to its moving speed v, and (iii) a frequency N =/3u = m v c / h .
)t o = c/v 0 = hc/hv 0 = hc/moc2 = h / m o c ,
Application. If we use the electron amplitude function u in the hydrogen Bohr atom, where its speed v is
(8)
where Xo is the Compton wavelength of the particle at rest. When it is moving with speed v =/3c, by application of relativity we find: (i) a moving Compton wavelength X = Xo(1 -/32)1/2
(9)
from (3) and (8), and (ii) a space coordinate x related to x 0 by (6). By setting
o = e2i~ = c¢c, where a = 1/137.02 is the fine structure constant, taken as a fundamental constant, the quantization condition in non-relativistic quantum mechanics can be found if we write that, for the electron stable orbits, the frequencies are divisors of the base frequency, N 1 = amOC2/h ,
k = 2ir/X = k0/(1 - / 3 2 ) 1/2 = 27rl)t0(1 -/32)1/2 N n = n-lN1. = 2zwlc,
(1 O)
the amplitude function becomes u = exp(ik0x0) = exp [i(kx - ~ t ) ]
(11)
in a system at rest and in a moving system. This defines a frequency N such that N = ~212n =/3~ol2~ =/3v = m v c / h .
In other words these frequencies are the subharmonics of the base frequency N 1 . Under the same conditions, for atoms with electron numberZ, the frequenciesN Z are, in first approximation, integral multiples of the base frequency N 1 NZn
=
( Z I n ) ~ mocZ lh .
(12)
With (8), (6), (9) and (11) the properties o f the amplitude function are symmetrical to the wavefunction properties o f a particle (2), (4), (5) and (7). The amplitude function of a moving massive particle of mass m and speed o is characterized by (i) a wavelength equal to its Compton wavelength )t = h / m e ,
References [1] [2] [3] [4]
P. Mukhopadhyay, Phys. Lett. 105A (1984) 487. S.N. Das, Phys. Lett. 102A (1984) 338. C. Elbaz, C.R. Acad. Sci. 297 lI (1983) 455. L. de BrogUe, Th~se 1924 (Masson, Paris, 1963).