The deBroglie Wavelength

Internal Vibrations

as always with Einstein, we start with E = mc^2

(define (E0 m) (m * ('c ^ 2))) 
    

(organisational note: if you like to see a formula nicely formatted, call its according function)

(E0 'm) 
    

deBroglies first hypothesis was to assume that every particle has a hypothetical internal vibration at frequency nu0 which relates to the rest energy in rest frame of particle (only there this energy-frequency relation holds)

(define (nu_naught E0) (E0 / 'h)) 
    

particle travels at velocity vp

(define (vp beta) (beta * 'c)) 
    

(define (beta v) (v / 'c)) 
    

(define (gamma beta) (1 / (sqrt (1 - (beta ^ 2))))) 
    

time dilation: internal vibration is slower for observer. so the frequency-energy relation does not hold: the frequency indeed decreases instead of increasing with energy. this is the conundrum deBroglie solved. so hang on.

(define (nu_one nu_naught gamma) (nu_naught / gamma)) 
    

sine formula for internal vibration. we do not know what exactly vibrates so we set the amplitude to one

(define ((internal-swing nu_one) t)
   (sin (* '2pi nu_one t))) 
    

((internal-swing 'nu_one) 't) 
    

calculate the phase of the internal swing at particle point x = v * t

(define ((internal-phase nu_one v) x)
  (asin ((internal-swing nu_one) (x / v)))) 
    

(is-equal (* '2pi 'nu_one ('x / 'v))
           ((internal-phase 'nu_one 'v) 'x)) 
    

personal note: to me, this is the sine-part of a standing wave, the standing vibration.

A general Wave

now for something completely different: general definition of a wave

(define ((wave omega k) x t)
  (sin ((omega * t) - (k * x)))) 
    

with the usual definition of omega

(define (omega nu) ('2pi * nu)) 
    

and the simplest possible definition for the wave-vector k: a dispersion free wave traveling at phase-velocity V

(define (k omega V) (omega / V)) 
    

I think here is a good place for a reminder that you can check the JS code by adding a ; up front

;(define (k omega V) (omega / V)) 
    

calculate the phase of the wave

(define ((wave-phase nu V) x t)
  (asin ((wave (omega nu) (k (omega nu) V)) x t))) 
    

(is-equal (* '2pi 'nu ('t - ('x / 'V)))
           ((wave-phase 'nu 'V) 'x 't)) 
    

Phase difference

calculate the phase difference between the vibration and some wave at time t = x / v as a function of the ratio of the frequencies

(define ((phase-difference x v nu V) ratio)
  (((internal-phase (ratio * nu) v) x)
    - ((wave-phase nu V) x (x / v)))) 
    

(is-equal (* '2pi 'nu ((('ratio - 1) * ('x / 'v)) + ('x / 'V)))
          ((phase-difference 'x 'v 'nu 'V) 'ratio)) 
    

state the general ratio of frequencies that keeps the vibration of the particle in phase with some wave of velocity V in terms of the velocity of the particle

(define (phase-ratio v V) (1 - (v / V))) 
    

(solves (phase-ratio 'v 'V)
        (phase-difference 'x 'v 'nu 'V)) 
    

the Energy of the particle for the observer

(define (Ev E0 gamma) (E0 * gamma)) 
    

we assume the deBroglie wave has the frequency: energy divided by Planck's constant. reminder: this relation holds in every frame of reference, especially for the observer who is not in the rest frame.

(define (nu Ev) (Ev / 'h)) 
    

now that nu is set, calculate the physically viable ratio of the frequencies in terms of beta

(define (physical-ratio beta)
  ((nu_one (nu_naught 'E0) (gamma beta))
   / (nu (Ev 'E0 (gamma beta))))) 
    

(is-equal (1 - ('beta ^ 2))
          (physical-ratio 'beta)) 
    

state, in terms of the particle velocity beta, the value of the physical phase-velocity V that keeps the vibration and the deBroglie wave in phase

(define (phase-velocity beta) ('c / beta)) 
    

(solves (phase-velocity 'beta)
  (lambda (V) ((physical-ratio 'beta)
               - (phase-ratio (vp 'beta) V)))) 
    

note: the phase-velocity is always greater than the speed of light. It is independent of the position x and the mass of the particle

the relativistic momentum is defined as

(define (p m v gamma)
  (* m v gamma)) 
    

calculate the deBroglie wavelength (by dividing the phase-velocity by the frequency) and show that it indeed is h divided by the momentum

(define de-broglie-wavelength
  ((phase-velocity (beta 'v))
    / (nu (Ev (E0 'm) 'gamma)))) 
    

(is-equal ('h / (p 'm 'v 'gamma))
           de-broglie-wavelength) 
    

personal note: one can see this upside down. V and nu define not only the deBroglie phase wave but also a standing wave (standing vibration). and the intersection of the two waves gives the trajectory of the particle (and hence its velocity v). Intersection meaning the points where the phase of the wave and the sine-part of the standing vibration-wave have the same value. The mass of the particle is then given by the deBroglie-wavelength and the v. Mass is thus a constant of the motion, the same value in every frame.

Appendix: some formulas to test the Scheme compiler

((1 / 2) / 2) 
    

(3 * (1 / 2)) 
    

(1 + (((2 / 3) - 4) * 5)) 
    

(2 ^ 3) 
    

(1 + (cos 0)) 
    

(+ 1 (cos 0)) 
    

((sin ^ 2) (314 / 100)) 
    

((cos (sin 0)) + (((2 / 3) - 4) * (5 ^ 2))) 
    

(+ (cos (sin 0)) (* (- (/ 2 3) 4) (expt 5 2))) 
    

(3 * (cos 0)) 
    

((3 * cos) 0) 
    

(cos 'x) 
    

((3 * (sin ^ 2)) 'x) 
    

((D (3 * (sin ^ 2))) 'x) 
    

(define value 6) 
    

(1 + value) 
    

(define (funame x) (x + 1)) 
    

(funame 4) 
    

(define ((ffuname x) y) (x + y)) 
    

((ffuname 4) 5) 
    

(define (fulet x) (let ((y 1)) (+ x y))) 
    

(fulet 4) 
    

(1 + 1/2)