Waves, Particles, and Facts

Abstract

The logic of notions like see, observe, witness, notice, data, evidence, and facts has, for my purposes at least, been thoroughly explored. For the next few chapters I should like to see how the morals drawn from our work so far can be applied in cases of actual scientific perplexity. The present chapter, therefore, as well as the two to follow, will be addressed, not to logical exploration (as the previous chapters were), but to the application of the fruits of our study to typical cases of experimental research.

Appendix

For calculations of the general case see, for example, De Broglie (1930, 146–148) (non-relativistic) and Compton (1926, 265–268 and Appendix 6) (relativistic).

1.1 The Compton Long Wave-Length Shift Calculated for a Scattering Angle (ø) of 90°

• X-component momentum obeys the equation $m{v}_x=\frac{hv}{c}$

• and for the Y-component momentum $m{v}_y=-\frac{h{v}^{\prime }}{c}$

• where v′ is the frequency of the scattered quantum.

Energy conservation is given by the formula (neglecting relativistic effects)

$$ hv=h{v}^{\prime }+\frac{1}{2}m\left({v}_x^2+{v}_y^2\right). $$

Now we have from the momentum formula that

$$ {v}_x^2+{v}_y^2=\frac{h^2}{m^2{c}^2}\left({v}^2+{v^{\prime}}^2\right). $$

Substituting this into the energy formula we obtain

$$ \frac{v-{v}^{\prime }}{v^2+{v^{\prime}}^2}=\frac{1}{2}\frac{h}{m{c}^2} $$

and putting the result in terms of the x-ray wave lengths $\lambda =\frac{c}{v}$ and ${\lambda}^{\prime }=\frac{c}{v^{\prime }}$ we get

$$ \frac{\lambda {\lambda}^{\prime}\left({\lambda}^{\prime }-\lambda \right)}{\lambda^2-{\lambda^{\prime}}^2}=\frac{1}{2}\frac{h}{mc}. $$

Since the shift in wave length (λ ^′ − λ) is quite small in comparison with the wave length itself, λ ² + λ ^′2 = λλ ^′ approximately and

$$ {\lambda}^{\prime }-\lambda =\frac{h}{mc}. $$

This is exactly the wave-length shift as found by experiment for 90° scattering. At smaller scattering angles the shift is less, becoming zero at ø = 0°; and at ø = 180° it attains the value of 2 h/mc.