Note on a Simple Derivation of Planck's Formula from Special Relativity B. Dencel Washington University, St. Louis, MO 63130 Joseph
Textbooks of physical chemistry2 and of general physics3 still present the events of the period of the old quantum theory (1900-1925) in an historical manner. For example, the Planck formula, eqn. (I), E = hu
(1)
for the quantization of electromagnetic radiation is usually presented as a hold and ground-breaking proposal, which it was in 1905. Now, 75 years later, the Planck formula hardly seems more daring than, say, Newton's law of gravitation. Everyone has heard of quanta by now. While it is important not to lose sight of the large role played by the investigations of the "classical" experiments of blackbody radiation, the photoelectric effect, the Bamer series in the atomic s ~ e c t r u mof hydrogen, etc., it . is no longer necessary in a modern presentation of quantum theorv to trot out all of these ex~erimentsin chronological order. The following discussion presents a simple alternative way of introducing the important Planck formula. While the treatment here does not pretend to he an original idea, it is to he found in very few textbooks and, therefore, may not he too well known. The discussion that follows shows how eqn. (1) may he deduced from ideas of special relativity and the hypothesis of wave-particle duality. Effect of Relativity Consider two frames of reference, S and Sf. For simplicity, the frames are arranged so that the x and x'axes coincide, and the other pairs of corresponding axes are parallel. Frame S' is moving at a constant velocity of Vrelative to frame S in the +x direction, and time is counted beginning from the instant a t which the origins 0 and 0' are coincident. The
At' = y A t [ l
- (VZlc2)]
Since y > 1, time intervals in the frame S', as measured by an observer in frame S (i.e., At), appear longer or "dilated" than when measured by an observer in S' be., At'). I t also follows that a particle with velocity:-component u, in the frame S has a velocity x -component u , in the frame S' of , dx' =-
"
I
dt'
Finally, the Lorentz transformation, eqn. (21, can he applied to the study of conservation of momentum and of energy. This leads to the result that the relativistic energy of an object moving a t a velocity of u relative to an observer is given by E = mc2, where m = moll - ( " / c ) ~ ] - ' / ~
(5)
and mo is the mass of the object when it is at rest. Frequency Ratio Now consider an experiment in which a laser positioned at 0 emits a pulse of light in the +x direction at time t = 0 (i.e., when 0 and 0' are coincident). An observer at 0' records the time, At', for exactly one cycle of oscillation of the wave to reach him. The frequency of the light is then In frame S the spatial separation of the pair of events (release of the light from O', arrival of the light a t 0') is the distance the frame Sf has moved, Ax = VAt, and so to an ohserver at 0, the distance traversed by the wave is one wavelength (A) plus Ax. Since this distance is covered by the pulse a t a speed of c, the elapsed time for the observer at 0 is
Then, according to special relativity, the followingrelations, known collectively as the Lorentz transformation, exist hetween coordinates and times in the two frames.4
*' = ",(x - V t )
after making use of A.u = c. Because the two events take place a t the same position (at x' = 0)as far as the observer a t 0' is concerned, Ax' = 0 and eqn. (3h) applies. I t follows from eqn. (7) that
y'=y,z'=z
t' =",It - ( V x / c Z ) ] [ l - (V/c)2]-'JZ
", =
(2)
These equations may he used to investigate pairs of events. Thus, we have a*' = y ( A r - V A t ) At' = ?[At
- (VAxle2)l
(34
and if two events occur separated in time hut at the same place in frame S', then Ax' = 0 and Ax = VAt, whence
'
Correspondence should be addressed to the authw at Department of Chemistry, University of Missouri. St. Louis, MO 63121. For example, see Bromberg, J. P., "Physical Chemistry." Allyn & Bacon, Inc., Boston, 1980, p. 486. For example, see Halliday, 0.. and Resnick, R.. "Fundamentals of Physics," 2nd ed., John Wiley & Sons. Inc., New York, 1981, p. 777. A derivation may be found in Richtmyer. F. K.. Kennard, E. H.. and Cooper, J. N., "Introduction to Modern Physics," 6th ed.,McGraw-Hill Book Co., New York, 1969, pp. 57-58. Volume 60
Number 8 August 1983
645
The limit of this expression as u, is allowed to approach a value of c (the case of a photon) is
and combination of equations (6)and (8) yields finally 11"' = Iru[l - (Vlc)]l-'
.
or
= lim
(E'IE)
us-
