Volume 105A, number 9 PHYSICS LETTERS 12 November 1984

COMMENTS ON "DE BROGLIE WAVE AND COMPTON WAVE"

P. MUKHOPADHYAY Physics Department, Jadavpur University, Calcutta 700032, India

Received 25 July 1984

This note points out that the "new wavelength" of Das is really not new since such a wavelength was introduced by us in the literature in 1978. Further, it has been shown that the so-called "new wavelength" has been introduced by a procedure which is physically as well as mathematically unjustified.

The aim of this note is to emphasize that the "new wavelength" Xk of ref. [1] is far from being new. This is because such a wavelength was suggested by the present author [2] in 1978. In this context it is worth mentioning that the wavelength, suggested by us [1 ], was called the de Broglie wavelength of sec- ond kind and it was denoted by ?'s. It is highly in- teresting to note that ~ is given by the expression )'k = hc/E which is identical to the expression for ?'S which is also given by )'S = hc[E. In the following we demonstrate that the method, followed in ref. [ 1 ], for obtaining the expression given above for )~k is physically meaningless and mathematically absurd.

For our purpose indicated above, we proceed by noting that the so-called new wavelength ~ has been expressed as [1]

E = D/~. k (1)

by requiring that h k = }'B for the photon but ?'k =/= XB for massive particles; )~B being the de Broglie wave- length. At this point, it is strongly emphasized that the condition ~k = }'B for the photon is physically de. void of any meaning and mathematically under'reed. This point becomes immediately transparent if we recall the well-known fact that the de Broglie waves are actually matter waves. This in turn means that the de Broglie wavelength refers to a massive particle. Clearly, then, the photon (being a massless particle) cannot be associated with the de Broglie wavelength which is, in fact, undefined for it. This is evident from the following well-known relation,

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XB = h/P = h/my = h(1 - [32)1/2 /mOv ,

where/t = v/c. Obviously, )'B is undefined for the pho- ton for which m 0 = 0, v = c and hence/3 = 1. This in turn implies that the constant D, occurring in eq. (1), cannot be determined. To see this point, we may pro- ceed to determine the constant D by the method fol- lowed in ref. [ 1 ]. The well-known relation

E 2 = p2c2 + m2c 4

may be recast as

E2/h2c 2 = p2/h2 + E2/h2c 2 , (2)

where E 0 = mo c2. Now, from eq. (1) it follows that

e 2 (3a)

and

E 2 = D 21(;~k) 2 . (3b)

With the help of eqs. (3a) and (3b), eq. (2) can be given the following form [ 1 ]

(n21h2 c2Xl lX2) = l/X2 B +(D21h2 c2)[l/(hk)2] , (4a)

which takes the form

x 2/X2k = 1];k 2 + x 2/(;kk)20 , (4b)

if we write x =D/hc as in ref. [1]. Now, to determine the constant D, eq. (4b) has been considered in ref. [1] for the photon. Obviously, eq. (4b) reduces to

= 1/xg (4c)

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Volume 105A, number 9 PHYSICS LETTERS 12 November 1984

for the photon. Then, it has been argued [1] that x = 1, i.e. D = hc as h k = ;k B for the photon. However, we have already emphasized that X B is undefmed for the photon. This means that the quantity x and as such the constant D cannot be evaluated from eq. (4c). As a consequence, eq. (1) cannot be given the form

x k = hc/F (5)

by setting D = hc. That the constant D really remains undetermined becomes evident from the fact that eq. (5) is undefined for the photon. This point can be easily checked by rewriting eq. (5) as

X k = hc/E = h / m c ,

E = mc 2 = m0c2(l - /32) -1/2 = (h[mocX1 - / 3 2 ) 1/2 ,

(Sa)

which is obviously undefined for the photon. We can conclude, therefore, the quantity ~k given by eq. (1) remains undefined as the constant D occurring in the same equation has not been determined. Stated differ- ently, the author of ref. [1] has not been able to in- troduce any new wavelength.

References

[1] S.N. Das, Phys. Lett. 102A (1984) 338. [2] P. Mukhopadhyay, Ind. J. Phys. 52A (1978) 176.

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