a r X i v :p h y s i c s /0502054v 1 [p h y s i c s .e d -p h ] 10 F e
b 2005
Snell’s Law from an Elementary Particle Viewpoint
D.Drosdoffand A.Widom
Physics Department,Northeastern University,Boston MA 02115
最亮的星星歌词Snell’s law of light deflection between media with different indices of refraction is usually discusd in terms of the Maxwell electromagnetic wave theory.Snell’s law may also be derived from a photon beam theory of light rays.This latter particle physics view is by far the most simple one for understanding the laws of refraction.
PACS numbers:42.15.-i,42.15.Dp,41.85.-p
I.INTRODUCTION Snell’s law of refraction is usually discusd in elemen-tary physics cours wherein the derivations[1]depend on the electromagnetic wave theory of light.The pur-po of this note is to show how the laws of refraction may be derived from the particle (i.e.photon)view of light rays.In p
articular,we show how the photon Hamil-tonian H (p ,r )must be derived.In this work,p and r denote,respectively,the momentum and position of a photon as it moves along the light ray.
The refraction of a light ray is shown in Fig.1.In terms of the indices of refraction,Snell’s law of refraction asrts[2,3]that文章投稿
n 1sin θ1=n 2sin θ2
(Snell ′
s Law).
(1)
The derivation of Eq.(1)from the energy and momentum conrvation laws associated with photon deflection is ex-hibited Sec.II.The index of refraction is then defined in terms of photon energy and momentum.The relationship between the index of refraction and the photon velocity is discusd in Sec.III and from a more general geometric optics limit in Sec.IV.As an application of the particle viewpoint,we consider in Sec.V the gravitational he bending of light rays in a gravitational
出行英语
field.
FIG.1:A light ray moves from a medium with index of re-fraction n 1into a medium with index of refraction n 2.Rays are considered to be made up of photons with momenta p 1and p 2respectively.
II.CONSER V ATION LA WS
We assume that the two rays in Fig.1are made up of photons with momenta p 1and p 2respectively.We also assume that the photon energies are E 1and E 2respec-tively.Since there is translational invariance in direc-tions parallel to the plane parating the two media,the photon momentum components parallel to the plane are conrved ;i.e.
p 1sin θ1=p 2sin θ2.
(2)
The energies of the photons are also
E 1=E 2.
(3)
In terms of physical photon energy and momentum,the indices of refraction are defined by
n 1=cp 1
E 2.(4)
Eqs.(2)-(4)imply
n 1sin θ1=n 2sin θ2,
(5)
which constitutes a simple yet rigorous derivation of Snell’s law.
Although similar to other treatments which discuss a formal energy and a formal momentum [4,5],we stress that in our treatment we only refer to the physical photon energy and momentum.For example,E has conventional units of Joules and momentum p has conventional units of Joule-c/meter .In other more formal treatments,the so-called “energy”and “momentum”do not have the usual physical dimensions of energy and momentum.
III.
PHOTON VELOCITY
The definitions of the indices of refraction in Eq.(4)are equivalent to the more usual definitions
n 1=
c
u 2
.
(6)if one does not in general identify u with the physical velocity of the photons.The velocity of a photon in a ray is determined by
v =
∂E
2 In general for light rays moving through continuous
media[6]
u=|v|since E
∂p .(8)
In the Maxwell electromagnetic wave theory[7,8,9]of light,u is the pha velocity while v is the group velocity. Let us consider this in more detail from a purely particle physics viewpoint.
青年强则国家强IV.GEOMETRICAL HAMILTONIAN OPTICS In inhomogeneous continuous media,the index of re-fra
ction in general depends on the photon energy as well as position.Eq.(4)implies the particle energy restriction
E=
c|p|
∂p and˙p=−
∂H(p,r)
[n+E(∂n/∂E)]wherein u=
E p
np雨水的诗句
.(12)
The force on the photon f=˙p obeys
f=E grad n
c2
+...,(18)
independently of the energy E of the photon.For ex-
曾巩的诗ample,a spherical astrophysical object will induce in the
neighboring space a potentialΦ=−(GM/r)and thereby
a spherical lens with index of refraction
变色龙契诃夫原文n(r)=1+
2GM
c 2
1+4GM
c
(angular momentum),
R s=
2GM
端午节手抄报模板3 together with Eq.(20)imply a radial photon momentum
p r=±E
1+ 2R s r 2,(23)
and its associated scattering angular deflection Θ=π−2 ∞r min bdr
1+(2R s/r)−(b/r)2
(24) as shown in Fig.(2).The impact parameter thereby obeys
b=−R s cot(Θ/2).(25) For small angles and large impact parameters[11,18],
Θ=−2R s
c2b
(b≫R s),(26)
which was ud by Einstein to predict the bending of light around the sun[19,20].We note in passing,the geometrical optics refraction cross ction
dσ
sinΘdΘ =R2s
