LaTeX for beginning of SICM-Ch01 (#311)

Author: kloimhardt

site/_quarto.yml

@@ -38,9 +38,13 @@ website:
             - href: mentat_collective/emmy/fdg_ch01.qmd
               text: "FDG Chapter 1"
             - href: mentat_collective/emmy/fdg_ch01_ys.qmd
-              text: "FDG Chapter 1 Infix"
-            - href: mentat_collective/emmy/sicm_ch01.qmd
+              text: "FDG Ch01 Infix"
+            - href: mentat_collective/emmy/sicm_original_ch01.qmd
               text: "SICM Chapter 1"
+            - href: mentat_collective/emmy/sicm_ch01.qmd
+              text: "SICM Ch01 Graphics"
+            - href: mentat_collective/emmy/debroglie.qmd
+              text: "deBroglie Waves"
             - href: mentat_collective/emmy/silcm_ch01.qmd
               text: "Lorentz Covariant"
 

src/mentat_collective/emmy/debroglie.clj

@@ -0,0 +1,233 @@
+^{:kindly/hide-code true
+  :clay             {:title  "Emmy, the Algebra System: deBroglie wavelength"
+                     :quarto {:author   :kloimhardt
+                              :type     :draft
+                              :description "A particle riding its wave whose length depends on the former's momentum"
+                              :sidebar  "emmy-fdg"
+                              :date     "2026-01-27"
+                              :category :libs
+                              :tags     [:emmy :physics]}}}
+
+(ns mentat-collective.emmy.debroglie
+  (:refer-clojure :exclude [+ - * / zero? compare divide numerator denominator
+                            time infinite? abs ref partial =])
+  (:require [scicloj.kindly.v4.api :as kindly]
+            [scicloj.kindly.v4.kind :as kind]
+            [mentat-collective.emmy.scheme :refer [define-1 let-scheme lambda]]
+            [civitas.repl :as repl]))
+
+^:kindly/hide-code
+(def prod true) #_"used to check Emmy in Scittle kitchen"
+
+^:kindly/hide-code
+(kind/hiccup
+  [:div
+   [:script {:src "https://cdn.jsdelivr.net/npm/scittle-kitchen@0.7.28-59/dist/scittle.js"}]
+   [:script {:src "https://cdn.jsdelivr.net/npm/scittle-kitchen@0.7.28-59/dist/scittle.emmy.js"}]
+   [:script {:src "https://cdn.jsdelivr.net/npm/scittle-kitchen@0.7.28-59/dist/scittle.cljs-ajax.js"}]
+   [:script {:src "https://cdn.jsdelivr.net/npm/react@18/umd/react.production.min.js", :crossorigin ""}]
+   [:script {:src "https://cdn.jsdelivr.net/npm/react-dom@18/umd/react-dom.production.min.js", :crossorigin ""}]
+   [:script {:src "https://cdn.jsdelivr.net/npm/scittle-kitchen@0.7.28-59/dist/scittle.reagent.js"}]
+   [:script {:type "application/x-scittle" :src "scheme.cljc"}]])
+
+^:kindly/hide-code
+(defmacro define [& b]
+  (list 'do
+        (cons 'mentat-collective.emmy.scheme/define b)
+        (list 'kind/scittle (list 'quote (cons 'define b)))))
+
+^:kindly/hide-code
+(define emmy-env
+  '[emmy.env :refer :all :exclude [Lagrange-equations r->p]])
+
+^{:kindly/hide-code true :kindly/kind kind/hidden}
+(do
+  (require emmy-env))
+
+^:kindly/hide-code
+(kind/scittle
+  '(do
+     (require emmy-env)))
+
+^:kindly/hide-code
+(define string-exp (comp str simplify))
+
+^:kindly/hide-code
+(defn reag-comp [b]
+  (let [server-erg (string-exp (eval b))]
+    (list 'kind/reagent
+          [:div (list 'quote
+                      (list 'let ['a (list 'string-exp b)]
+                            [:div
+                             (when (not prod)
+                               [:div
+                                [:tt 'a]
+                                [:p (list 'str (list '= server-erg 'a))]])
+                             [:tt server-erg]]))])))
+
+^:kindly/hide-code
+(def show-tex-fn (comp kind/tex emmy.expression.render/->TeX))
+
+^:kindly/hide-code
+(defmacro show-tex [e]
+  (if prod
+    (list 'show-tex-fn e)
+    (reag-comp e)))
+
+^:kindly/hide-code
+(kind/scittle
+  '(defn show-tex [e]
+     (->infix e)))
+
+^:kindly/hide-code
+(defmacro is-equal [a b]
+  (list 'if (list '= (list 'simplify (list '- a b)) 0)
+        (list 'show-tex a)
+        "not equal"))
+
+^:kindly/hide-code
+(kind/scittle
+  '(defn is-equal [a b]
+     (if (= (simplify (- a b)) 0)
+       (show-tex a)
+       "not equal")))
+
+^:kindly/hide-code
+(defmacro solves [a f]
+  (list 'if (list '= (list 'simplify (list f a)) 0)
+    (list 'show-tex (list '* (list 'symbol "root:") a))
+    "does not solve"))
+
+^:kindly/hide-code
+(kind/scittle
+  '(defn solves [a f]
+     (if (= (simplify (f a)) 0)
+       (show-tex (* (symbol "root:") a))
+       "does not solve")))
+
+;; In the following I follow deBroglies arguments. There is also a [version with infix notation](https://kloimhardt.github.io/blog/hamiltonmechanics/2024/09/13/debroglie.html).
+
+;; ### Internal Vibrations
+;;as always with Einstein, we start with E = mc^2
+
+(define (E0 m) (* m (square 'c)))
+
+(show-tex
+  (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 (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))
+
+;; 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))
+
+;; 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))
+
+;; ### 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))
+
+;; 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 (square 'beta))
+          (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)
+
+(repl/scittle-sidebar)

src/mentat_collective/emmy/sicm_ch01.clj

@@ -295,6 +295,10 @@
               (make-path-tz (lambda (t) (- (cos t) (harmonic-path t)))
                             0 pi-half 35))))
 
+;;   *Figure 1.1* The difference between the polynomial
+;;   approximation with minimum action and the actual trajectory taken by the
+;;   harmonic oscillator. The abscissa is the time and the ordinate is the error.
+
 (md "Calculate the Lagrange equation of the harmonic oszillator.")
 
 (show-tex-expression