Derivations and Formulae

In this chapter we present the formal derivation and mathematical details of the relativistic wave-like solution given by Eq. ( 5.1 ) discussed in the previous chapter. We first derive the solution and then present various details relating to the integrals and formulae for the de Broglie wave energy. As we have previously mentioned, even though the de Broglie wave energy is given simply by E ( x , t ) = f 0 ( x + c λ t ) $$\mathcal {E}(x, t) = f_0 (x + c \lambda t)$$ , the determination of the wave energy as a function of velocity E ( u ) $$\mathcal {E}(u)$$ is far more complicated, and there appears to be no simple single expression such as that for the particle energy, namely, e = e 0[1 − (u∕c)2]−1∕2. In the final section of this chapter, for the sake of completeness, we present an alternative derivation that is based on the hyperbolic representation u = c tanh θ $$u = c\tanh \theta $$ rather than the trigonometric representation u = c sin ϕ $$u = c\sin \phi $$ . The following chapter deals with Lorentz invariances and functional dependence of the assumed underlying linear forces f and g given by ( 5.2 ).