Physics

If you want to understand the working of any complex system then you would first try to break everything down to its fundamentals. Then you would understand the fundamentals very clearly and then use that understanding of the fundamentals to completely understand the system. This is the very basis of Physics, which is one of the most successful endeavours of humankind. We have been able to take this approach whereby we break down systems to fundamentals and then use its knowledge to have the most detailed and clear understanding of a lot of things. ...

This is a figure I used for my homework at drexel . I drew it with very little knowledge of tikz at the time. %%A 2021-01-12 \usetikzlibrary{calc,patterns,angles,quotes} \begin{tikzpicture}[ thick, >=stealth, dot/.style = { draw, fill = white, circle, inner sep = 0pt, minimum size = 4pt } ] \coordinate (O) at (0,0); \coordinate (X) at (5,0); \coordinate (Y) at (0,4) ; \coordinate (A) at (-4,0); \coordinate (L) at (0,-1); \coordinate (U) at (0,4); \coordinate[label=above:P] (P) at (2,3); \coordinate (B) at (4,0) {}; \draw [dotted] (O) to node [below] {$d$} (A); \draw [dotted] (O) to node [below] {$d$} (B); \draw [-] (O) to node [below, sloped] {$r $} (P); \draw [dashed] (A) to node [sloped, above] {$r_2$} (P); \draw [dashed] (B) to node [sloped, above] {$r_1$} (P); % mirror \draw [-] (L) -- (U); \fill[black] (A) circle (0.13) node [circle,above,left, label=$-q$] {}; \fill [black] (B) circle(0.13) node [circle, above,right, label=$q$] {}; %\fill [gray] (O) circle(0.19) node [circle, above,left, label=$2q$] {}; \pic [draw, -, "$\theta$", angle eccentricity=1.5] {angle = B--O--P}; \end{tikzpicture}

गणितीय सुत्रहरू अमूर्त कला र कलाकारिता बुझ्नका लागि हामीले त्यस क्षेत्र सम्बन्धित केही ज्ञान हासिल गरेको हुनुपर्छ। नत्र अशिक्षित मानिसलाई कालो अक्षर भैँसी बराबर भनेझैँ, विशेष गरी अमूर्त कलालाई बुझ्न कठीनाइ मात्र होइन त्यसलाई ग्रहण गर्न पनि गाह्रो हुन जान्छ। यो मानेमा भन्दाखेरी कलामा एक किसिमको गहिराई हुन्छ वा भनैँ अर्कै एउटा (वा धेरै, कम्तिमा एउटा) आयाम हुन्छ। सामान्यतः हामी हाम्रा पाँच इन्द्रीयहरूद्वारा के ग्रहण गर्न सक्छौँ वास्तविकताको व्यख्या तिनै अनुग्रहित तरङ्गद्वारा मात्र गर्न सक्छौँ। कलामा हुने यस्तो गहिराईलाई बुझ्ने हाम्रा कुनै पनि भौतिक इन्द्रियहरू छैनन्। यो गहिराई या आयाम भनेको विभिन्न क्षेत्रमा हुन्छ। अर्थोपार्जन गर्ने लगानिकर्ता वा ब्यापारीको क्षेत्रमा उसले देख्ने गहिराई सबैले देख्न सक्दैन्न। सङ्गीतकार वा गायकको आयाम श्रोताले सजिलै बुझ्न सक्दैनन्। यहाँ एक किसिमको प्रष्ट विभाजन गर्न सकिन्छ। सङ्गीत सर्जकले जुन आयाम ताकेर कृती सिर्जना गर्छ श्रोताले त्यही आयामको गहिराई नबुझे पनि त्यो सिर्जनाबाट छुट‍्टै आानन्द लिन सक्छ। श्रोता र सर्जकको बुझाईको अन्तरले सिर्जनालाई कमजोर बनाउँदैन तर सतही सर्जकले बनाएको सङ्गीतले श्रोतालाई त्यही किसिमले बाँध्न सक्दैन। श्रोताको दृष्टिकोणबाट हेर्दा यस्तो सिर्जनामा केही छुटेको महसुष पक्कै हुन्छ तर राम्रो सङ्गीतमा त्यो के विशेशता थियो भनेर पनि खुट्याउन सकिँदैन। यो एउटा स्पष्ट भेद हामी धेरै विषयहरूमा देख्छौँ। ...

Klein Gordan Equation We have seen in this post that Klein-Gordan equation describes a spinless particle with mass $m$. The Klein-Gordan equation formally is $$ \begin{align*} \left( \square + m^{2} \right) \psi = 0 \end{align*} $$Unlike Schrödinger’s equation, this equation doesn’t treat time differently from space derivatives. It can be shown that this in indeed Lorentz invariant. However, the problem with Klein-Gordan equation is that it violates the conservation of probability, which is a serious problem. We need a first order differential equation which would indeed conserve the probability. ...

Schrödinger’s Equation Two of the greatest triumphs in Physics in the first three decades of 20th century are, i) Einstein’s Theory of Relativity", both special and general, and ii) Quantum Mechanics. While both of them are very successful, there is still some friction between them. Special Theory of Relativity taught us that space and time (which we thought were independent), in fact, were just components of more fundamental 4-dimensional spacetime. There are various ways we can formulate quantum mechanics, in any case, Schrödinger’s equation can be thought of as of central importance to the theory. The Schrödinger’s equation is ...

Figure depicting the mass hierarchy of neutrinos with two possible mass ordering, normal mass ordering, and inverted mass ordering. \tikzset{ dimen/.style={<->,>=latex,thin,every rectangle node/.style={fill=white,midway}}, symmetry/.style={dashed,thin}, munu/.style={draw,fill,rectangle,color=red!80,text=black,minimum height=1mm,font=\scriptsize}, taunu/.style={draw,fill,rectangle,color=green,text=black,minimum height=1mm,font=\scriptsize}, enu/.style={draw,fill,rectangle,color=blue!60,text=black,minimum height=1mm,font=\scriptsize}, muone/.pic = { \node[opacity=0] (bummy) {}; \node[munu,minimum width=7mm, right=1mm of bummy] (twomu) {}; \node[taunu,minimum width=6mm,right=0mm of twomu] (twotau) {}; \node[enu,minimum width=7mm,right=0mm of twotau] (twoe) {}; \node{ \tikzpictext}; }, mutwo/.pic = { \node[opacity=0] (eummy) {}; \node[munu,minimum width=12mm,right=1mm of eummy] (threemu) {}; \node[taunu,minimum width=4mm,right=0mm of threemu] (threetau) {}; \node[enu,minimum width=4mm,right=0mm of threetau] (threee) {}; \node{ \tikzpictext}; }, muthree/.pic = { \node[opacity=0] (dummy) {}; \node[munu,minimum height=2mm,minimum width=2mm,right=1mm of dummy] (onemu) {}; \node[taunu,minimum width=8mm,right=0mm of onemu] (onetau) {}; \node[enu,minimum width=10mm,right=0mm of onetau] (onee) {}; \node{\tikzpictext}; }, } \begin{tikzpicture} \draw[->,thick] (-1,0) -- node [sloped, above] {$m^2$} (-1,3) ; \pic ["$\nu_3$"] (normuone) at (0,3) {muone}; \pic ["$\nu_1$"] (normutwo) at (0,1) {mutwo}; \pic ["$\nu_2$"] (normuthree) at (0,0) {muthree}; \pic ["$\nu_2$"] (invmutwo) at (5,3) {mutwo}; \pic ["$\nu_1$"] (invmuone) at (5,2) {muone}; \pic ["$\nu_3$"] (invmuthree) at (5,0) {muthree}; \draw[dimen] (1,.2) --(1,0.8) node [right=2mm] {$\Delta m^2_{\text{sol}}$}; \draw[dimen] (1,1.2) --(1,2.8) node [right=2mm]{$\Delta m^2_{\text{atm}}$}; \draw[dimen] (6,.2) --(6,1.8) node [right=2mm] {$\Delta m^2_{\text{atm}}$}; \draw[dimen] (6,2.2) --(6,2.8) node [right=2mm]{$\Delta m^2_{\text{sol}}$}; \node[munu] (mulab) at (3.6,1) [label=below:$\nu_\mu$] {}; %}{$\nu_\mu$}; \node[taunu] (taulab) at (3.6,2) [label=below:$\nu_\tau$] {}; \node[enu] (elab) at (3.6,3) [label=below:$\nu_e$] {}; \node (norhie) at (1.2,4) {Normal Hierarchy}; \node (invhie) at (6.5,4) {Inverted Hierarchy}; \end{tikzpicture}

Setup first lets setup up some imports import numpy as np import sympy as smp import matplotlib.pyplot as plt plt.rcParams['figure.figsize'] = (16.0, 6.0) smp.init_printing() from sympy.physics import mechanics as mcx #mcx.init_vprinting() mcx.mechanics_printing() #smp.init_printing() Goldstein 1.19 Solve spherical pendulum by lagrangian. t = smp.Symbol('t') g = smp.symbols('g',constant=True); #accleration due to gravity m = smp.symbols('m',real=True,positive=True,constant=True) theta,phi= mcx.dynamicsymbols('theta,phi'); r = smp.symbols('r',constant=True) rdt = smp.diff(r); thd = smp.diff(theta); phd = smp.diff(phi) x = r*smp.sin(theta)*smp.cos(phi); xdt = smp.diff(x,t) y = r*smp.sin(theta)*smp.sin(phi); ydt = smp.diff(y,t) z = r*smp.cos(theta); zdt = smp.diff(z,t); ydt $$r \operatorname{sin}\left(\phi\right) \operatorname{cos}\left(\theta\right) \dot{\theta} + r \operatorname{sin}\left(\theta\right) \operatorname{cos}\left(\phi\right) \dot{\phi}$$V = m*g*r*smp.cos(phi) T = smp.Rational(1,2)*m*(xdt**2+ydt**2+zdt**2); T; smp.simplify(T) $$\frac{m r^{2}}{2} \left(\operatorname{sin}^{2}\left(\theta\right) \dot{\phi}^{2} + \dot{\theta}^{2}\right)$$So the total kinetic energy of the spherical pendulum is ...

We are going to calculate the average kinetic energy of particles whose velocity follow Maxwell’s distribution. We know that the Maxwell’s velocity distribution function is given by the following relation. $$ \begin{align*} f(v) = Av^2e^{-mv^2/2kT} \end{align*} $$Here, $A$ is a normalization constant, $m$ is the mass of each particle, $k$ is Boltzmann’s constant, and $T$ is the temperature of the system. The plot of this distribution looks like The normalization condition gives ...

सबैकुरा पृथ्वीतिर तानिने कुरालाई ब्याख्या गर्न जुन समिकरणको प्रयोग गरिन्छ त्यही समिकरण प्रयोग गर्दा भीमकाय ब्रह्माण्डीय पीण्डहरूको ब्यहोरा लाई पनि सजिलै वर्णन गर्न सकिन्छ। चन्द्रमाले पृथ्वीलाई किन घुम्छ, ग्रहण किन र कसरी लाग्छ भन्ने कुरा त्यही सामान्य सिद्धान्तका आधारमा ब्याख्या गर्न सकिन्छ। कुनै कुराले हामीतर्फ प्रकाश दिँदा (वा परावर्तन गर्दा) हामीले तिनलाई देख्न सक्छौँ। त्यसै सिद्धान्तका आधारमा हामीले आकाशमा ताराहरूलाई देख्न सक्छौँ किनकी ताराहरूले प्रकाश उत्सर्जन गर्छन् । परावर्तन गर्ने प्रकाशका आधारमा हामीलाई हाम्रो पृथ्वीको एउटा चन्द्रमा छ भन्ने थाह छ। उत्सर्जन गर्ने आधारमा सूर्य कहाँ र कतिवाट छ भन्ने थाहा छ।कति बलले कुनै वस्तुलाई तान्छ भन्ने आधारमा हामीले वस्तुको पीण्ड कति छ भन्ने कुरा पत्ता लगाउँछौ। यिनले अरू पीण्डलाई आकर्षण गरेको आधारमा हामीले तिनको पिण्ड र सङ्ख्या भन्न सक्छौँ। हेरेर (यसले उत्सर्ग गर्ने वा परावर्तन गर्ने प्रकाशका आधारमा) हामीले ती कति सङ्ख्यामा वा कहाँ कहाँ छन् भनेर पत्ता लगाउन सक्छौँ। र यी दुवै तरिकाबाट हामीले पत्ता लगाएका ग्रह वा तराको सङ्ख्याका बीच मेल खाएको छ।कुरा यत्ति हो। ...

Introduction Mutually perpendicular electric and magnetic field is generally referred to as Cross field. In this document we work out the trajectory of a charged particle in a cross field. The charged particle is stationary in the beginning. Problem Let a charged particle with charge $q$ be located at the origin with the magnetic field $\vec{B} = B\vec{k}$ and electric field $\vec{E} = E\vec{j}$. We are interested in working out the trajectory of the charged particle. ...