Emmy, the Algebra System: deBroglie wavelength

A particle riding its wave whose length depends on the former’s momentum
Author
Affiliation

Markus Agwin Kloimwieder

Scicloj

Published

January 27, 2026

In the following I follow deBroglies arguments. There is also a version with infix notation.

Internal Vibrations

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

(define (E0 m) (* m (square 'c)))
(show-tex
  (E0 'm))

\[m\,{c}^{2}\]

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 (square beta)))))

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 (* 2 'pi nu_one t)))
(show-tex
  ((internal-swing 'nu_one) 't))

\[\sin\left(2\,\pi\,{\nu}_{one}\,t\right)\]

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 (* 2 'pi 'nu_one (/ 'x 'v))
          ((internal-phase 'nu_one 'v) 'x))
NoteOUT 
2026-01-29T15:03:21.167Z runnervmkj6or WARN [emmy.util.logic:22] - Assuming (= (asin (sin (/ (* 2 nu_one pi x) v))) (/ (* 2 nu_one pi x) v)) in asin-sin

\[2\,\pi\,{\nu}_{one}\,\left(\frac{x}{v}\right)\]

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) (* 2 'pi 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))

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 (* 2 'pi 'nu (- 't (/ 'x 'V)))
          ((wave-phase 'nu 'V) 'x 't))
NoteOUT 
2026-01-29T15:03:21.200Z runnervmkj6or WARN [emmy.util.logic:22] - Assuming (= (asin (sin (/ (+ (* 2 V nu pi t) (* -2 nu pi x)) V))) (/ (+ (* 2 V nu pi t) (* -2 nu pi x)) V)) in asin-sin

\[2\,\pi\,\nu\,\left(t - \left(\frac{x}{V}\right)\right)\]

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 (* 2 'pi 'nu (+ (* (- 'ratio 1) (/ 'x 'v)) (/ 'x 'V)))
          ((phase-difference 'x 'v 'nu 'V) 'ratio))
NoteOUT 
2026-01-29T15:03:21.219Z runnervmkj6or WARN [emmy.util.logic:22] - Assuming (= (asin (sin (/ (* 2 nu pi ratio x) v))) (/ (* 2 nu pi ratio x) v)) in asin-sin
2026-01-29T15:03:21.221Z runnervmkj6or WARN [emmy.util.logic:22] - Assuming (= (asin (sin (/ (+ (* 2 V nu pi x) (* -2 nu pi v x)) (* V v)))) (/ (+ (* 2 V nu pi x) (* -2 nu pi v x)) (* V v))) in asin-sin

\[2\,\pi\,\nu\,\left(\left(\mathsf{ratio} - 1\right)\,\left(\frac{x}{v}\right) + \left(\frac{x}{V}\right)\right)\]

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))

\[\mathsf{root:}\,\left(1 - \left(\frac{v}{V}\right)\right)\]

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 (square 'beta))
          (physical-ratio 'beta))

\[1 - {\beta}^{2}\]

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))))

\[\mathsf{root:}\,\left(\frac{c}{\beta}\right)\]

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)

\[\frac{h}{m\,v\,\gamma}\]

(repl/scittle-sidebar)
source: src/mentat_collective/emmy/debroglie.clj