- Email: [email protected]
Determination of the quantized velocity of a single photon confined into a one-dimensional cavity
Optik 125 (2014) 6245–6246
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Determination of the quantized velocity of a single photon confined into a one-dimensional cavity M.A. Grado-Caffaro ∗ , M. Grado-Caffaro Scientific Consultants, C/ Julio Palacios 11, 9-B, 28029 Madrid, Spain1
a r t i c l e
i n f o
Article history: Received 9 November 2013 Accepted 15 June 2014 Keywords: Photon One-dimensional cavity Ultrarelativistic Non-zero rest-mass Wave-particle dualism.
a b s t r a c t Within a standing-wave approach, we determine the quantized velocity of a single photon in a onedimensional cavity by considering the photon as an ultrarelativistic particle of non-zero rest-mass. As a matter of fact, this ultrarelativistic nature is regarded as well as the mathematical expression involving the photon modes in the cavity, and de Broglie relationship for the wave-particle dualism. © 2014 Elsevier GmbH. All rights reserved.
Photons in cavities appear as a very interesting issue which, as other related subjects, has a number of open questions from both the experimental and theoretical points of view. In particular, with respect to theoretical research, solving questions relative to a single photon confined in a cavity with simple geometry can provide useful information which may be employed satisfactorily as a solid basis for achieving further findings. In fact, analytical single-particle approaches with relatively (or even extremely) simple geometries may be successful instead unnecessary computer simulations for many particles and complicated geometries. Certainly, the aforementioned single-particle models are usual in theoretical and mathematical physics. The spirit of the present note is inscribed into the above context given that, besides, photons do not interact mutually. As a matter of fact, we will consider the onephoton dynamics in a one-dimensional cavity. At this point, we emphasize that photons are non-interacting particles. On the other hand, we note that, in reality, photons are non-zero rest-mass particles [1–3]. Although this mass is extremely small, the fact that it is non-zero is certainly realistic [1–3]. Let us consider a single photon in a one-dimensional cavity. It is well-known that the photon oscillation modes corresponding to standing waves in the cavity obey the following quantization law relative to permitted wavelengths: nn = 2l
(1)
∗ Corresponding author. E-mail address: [email protected] (M.A. Grado-Caffaro). 1 www.sapienzastudies.com. http://dx.doi.org/10.1016/j.ijleo.2014.08.005 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
where n denotes (quantized) photon wavelength, l is the length of the cavity, and n = 1, 2, . . . On the other hand, the de Broglie formula relative to the waveparticle dualism reads n = h/(mn vn ), mn being the photon relativistic mass which is given by:
mn (vn (n )) =
m0 1−
v2n (n )
(2)
c2
where m0 is the photon rest-mass (which, in principle, will be assumed as independent of wavelength, a fact which will be confirmed later) and vn (n ) is the magnitude of the (quantized) / 0, it is clear wavelength-dependent photon velocity [1]. Since m0 = that this velocity is smaller than c although it is extremely near c; this fact tells us that the photon is an ultrarelativistic particle. Combining formulas (1), (2) and the de Broglie relationship, it follows:
hn
c 2 − v2n (n ) = 2lm0 c 2
(3)
We have said that the photon is really an ultrarelativistic particle so the quantity under the symbol of square root in Eq. (3) is sufficiently small. It is smaller than the square root so, in order to get an approximate ultrarelativistic expression from Eq. (3), we will replace the aforementioned square root by the quantity under its symbol. Also, since vn () ≈ c, we will use that c 2 − v2n (n ) =
6246
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 125 (2014) 6245–6246
(c + vn ())(c − vn ()) ≈ 2c(c − vn ()). Then, under the above conditions, formula (3) becomes:
vn (n ) ≈ c 1 −
m0 n 2h
(4)
where, of course, as one gets from Eq. (1), we have that n = 2l/n which, inserted into (4), yields the following relation in terms of n:
vn ≈ c 1 −
m0 l hn
(5)
Looking at formula (5), we see that the second factor on the right-hand side of (5) must be extremely close to unity, that is, hn m0 l which implies that n m0 l/h so vn → c as n→ ∞. This means that velocity quantization takes place for increasingly less mutually separated values of n such that n m0 l/h, which is equivalent to say that the velocity quantization is very weak (quasi-quantization). In other words, the velocity spectrum is quasi-continuous from a sufficiently large value of n (n m0 l/h). This is confirmed by the fact that from relation (5) it follows for the velocity gap: vn ≡ vn+1 − vn ≈
m0 l hn(n + 1)
(6)
As expected, from expression (6) it follows that vn → 0 as n→ ∞. In addition, by formula (6), we get that vn ≈ m0 l/hn2 . On the other hand, since from formula (1) we have that n = 2l/n, then it is clear that n is quasi-constant for a quasi-continuous set of n-values sufficiently close to infinity. Therefore, m0 is practically independent of wavelength for the aforementioned values so this fact is confirmed by our initial assumption on the wavelengthindependence of the photon rest-mass. In summary, in fact we have studied the multiphoton dynamics in a one-dimensional cavity by taking into account that photons are non-interacting particles which behave as standing waves in accordance with a well-known law relative to permitted wavelengths. Our formulation is easily extrapolable to treat two-dimensional and three-dimensional cavities. On the other hand, fundamental aspects of the physics of laser cavities can be tackled by using a methodology similar to our preceding approach. References [1] R. Lakes:, Experimental limits on the photon mass and cosmic magnetic vector potential, Phys. Rev. Lett. 80 (1998) 1826–1829. [2] M.A. Grado-Caffaro, M. Grado-Caffaro:, A zero-point energy approach for estimating the photon mass, Optik 117 (2006) 93–94. [3] M.A. Grado-Caffaro, M. Grado-Caffaro:, Photon velocity from the Klein-Gordon equation, Optik 121 (2010) 2094–2095.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Determination of the quantized velocity of a single photon confined into a one-dimensional cavity M.A. Grado-Caffaro ∗ , M. Grado-Caffaro Scientific Consultants, C/ Julio Palacios 11, 9-B, 28029 Madrid, Spain1
a r t i c l e
i n f o
Article history: Received 9 November 2013 Accepted 15 June 2014 Keywords: Photon One-dimensional cavity Ultrarelativistic Non-zero rest-mass Wave-particle dualism.
a b s t r a c t Within a standing-wave approach, we determine the quantized velocity of a single photon in a onedimensional cavity by considering the photon as an ultrarelativistic particle of non-zero rest-mass. As a matter of fact, this ultrarelativistic nature is regarded as well as the mathematical expression involving the photon modes in the cavity, and de Broglie relationship for the wave-particle dualism. © 2014 Elsevier GmbH. All rights reserved.
Photons in cavities appear as a very interesting issue which, as other related subjects, has a number of open questions from both the experimental and theoretical points of view. In particular, with respect to theoretical research, solving questions relative to a single photon confined in a cavity with simple geometry can provide useful information which may be employed satisfactorily as a solid basis for achieving further findings. In fact, analytical single-particle approaches with relatively (or even extremely) simple geometries may be successful instead unnecessary computer simulations for many particles and complicated geometries. Certainly, the aforementioned single-particle models are usual in theoretical and mathematical physics. The spirit of the present note is inscribed into the above context given that, besides, photons do not interact mutually. As a matter of fact, we will consider the onephoton dynamics in a one-dimensional cavity. At this point, we emphasize that photons are non-interacting particles. On the other hand, we note that, in reality, photons are non-zero rest-mass particles [1–3]. Although this mass is extremely small, the fact that it is non-zero is certainly realistic [1–3]. Let us consider a single photon in a one-dimensional cavity. It is well-known that the photon oscillation modes corresponding to standing waves in the cavity obey the following quantization law relative to permitted wavelengths: nn = 2l
(1)
∗ Corresponding author. E-mail address: [email protected] (M.A. Grado-Caffaro). 1 www.sapienzastudies.com. http://dx.doi.org/10.1016/j.ijleo.2014.08.005 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
where n denotes (quantized) photon wavelength, l is the length of the cavity, and n = 1, 2, . . . On the other hand, the de Broglie formula relative to the waveparticle dualism reads n = h/(mn vn ), mn being the photon relativistic mass which is given by:
mn (vn (n )) =
m0 1−
v2n (n )
(2)
c2
where m0 is the photon rest-mass (which, in principle, will be assumed as independent of wavelength, a fact which will be confirmed later) and vn (n ) is the magnitude of the (quantized) / 0, it is clear wavelength-dependent photon velocity [1]. Since m0 = that this velocity is smaller than c although it is extremely near c; this fact tells us that the photon is an ultrarelativistic particle. Combining formulas (1), (2) and the de Broglie relationship, it follows:
hn
c 2 − v2n (n ) = 2lm0 c 2
(3)
We have said that the photon is really an ultrarelativistic particle so the quantity under the symbol of square root in Eq. (3) is sufficiently small. It is smaller than the square root so, in order to get an approximate ultrarelativistic expression from Eq. (3), we will replace the aforementioned square root by the quantity under its symbol. Also, since vn () ≈ c, we will use that c 2 − v2n (n ) =
6246
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 125 (2014) 6245–6246
(c + vn ())(c − vn ()) ≈ 2c(c − vn ()). Then, under the above conditions, formula (3) becomes:
vn (n ) ≈ c 1 −
m0 n 2h
(4)
where, of course, as one gets from Eq. (1), we have that n = 2l/n which, inserted into (4), yields the following relation in terms of n:
vn ≈ c 1 −
m0 l hn
(5)
Looking at formula (5), we see that the second factor on the right-hand side of (5) must be extremely close to unity, that is, hn m0 l which implies that n m0 l/h so vn → c as n→ ∞. This means that velocity quantization takes place for increasingly less mutually separated values of n such that n m0 l/h, which is equivalent to say that the velocity quantization is very weak (quasi-quantization). In other words, the velocity spectrum is quasi-continuous from a sufficiently large value of n (n m0 l/h). This is confirmed by the fact that from relation (5) it follows for the velocity gap: vn ≡ vn+1 − vn ≈
m0 l hn(n + 1)
(6)
As expected, from expression (6) it follows that vn → 0 as n→ ∞. In addition, by formula (6), we get that vn ≈ m0 l/hn2 . On the other hand, since from formula (1) we have that n = 2l/n, then it is clear that n is quasi-constant for a quasi-continuous set of n-values sufficiently close to infinity. Therefore, m0 is practically independent of wavelength for the aforementioned values so this fact is confirmed by our initial assumption on the wavelengthindependence of the photon rest-mass. In summary, in fact we have studied the multiphoton dynamics in a one-dimensional cavity by taking into account that photons are non-interacting particles which behave as standing waves in accordance with a well-known law relative to permitted wavelengths. Our formulation is easily extrapolable to treat two-dimensional and three-dimensional cavities. On the other hand, fundamental aspects of the physics of laser cavities can be tackled by using a methodology similar to our preceding approach. References [1] R. Lakes:, Experimental limits on the photon mass and cosmic magnetic vector potential, Phys. Rev. Lett. 80 (1998) 1826–1829. [2] M.A. Grado-Caffaro, M. Grado-Caffaro:, A zero-point energy approach for estimating the photon mass, Optik 117 (2006) 93–94. [3] M.A. Grado-Caffaro, M. Grado-Caffaro:, Photon velocity from the Klein-Gordon equation, Optik 121 (2010) 2094–2095.