31:T6ef,function *TDSESimulation() { potential() // set up potential texture while (playing) { // regl.poll() ++mutable elapsedTime if (elapsedTime > animationTimeLimit) { mutable elapsedTime = 0; } // step through approximation, rendering each step to a frame buffer setupDefault({time: elapsedTime}, () => { psi_n({wave_position: wave_position, wave_unit_components: wave_unit_components}); k1(); k2(); k3(); k4(); combine_k(); }) camera(function () { setup3DDefaults(() => { if(complex_components_active) { drawProbabilityAmplitudeReal(); drawProbabilityAmplitudeComplex(); } if(complex_phase_active) { drawComplexPhase(); } if(selected_visualization == "probability") { drawProbability(); } if(selected_visualization == "probability_amplitude") { drawProbabilityAmplitude(); } drawPotential(); }) }); yield regl.container } if (!playing) { while (true) { camera(function () { setup3DDefaults(() => { if(complex_components_active) { drawProbabilityAmplitudeReal(); drawProbabilityAmplitudeComplex(); } if(complex_phase_active) { drawComplexPhase(); } if(selected_visualization == "probability") { drawProbability(); } if(selected_visualization == "probability_amplitude") { drawProbabilityAmplitude(); } drawPotential(); }); }); yield regl.container } } }32:T630,drawProbability = regl({ uniforms: { texture: kCombined, }, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; ${probability} float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.yx; return WORLD_HEIGHT*(-1.0 + 2.0 * remap_probability(texture2D(texture, uv).rg, 0.7)); } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform mat4 projection, view; void main() { pos = xzPosition; vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.yx; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }`, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform sampler2D texture; ${probability} ${color_map} void main () { vec2 tex = texture2D(texture, vUv).rg; gl_FragColor = vec4(color_map(intensity(tex)), 1.0); }`, })33:T6a0,drawProbabilityAmplitude = regl({ uniforms: { texture: kCombined, }, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.yx; float h = texture2D(texture, uv).r * 1.5; return WORLD_HEIGHT*(-1.0 + 2.0 * h) / 2. - .5; } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform mat4 projection, view; void main() { pos = xzPosition; vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.yx; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }`, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform sampler2D texture; ${color_map} float nudge_intensity(float x) { return 1. - pow((1. - x), 2.); } void main () { vec2 tex = texture2D(texture, vUv).rg; float h = pow(abs(tex.r), 0.9); // (tex.r + 1.) / 2.; gl_FragColor = vec4(color_map(nudge_intensity(clamp(h, 0., 1.))), 1.); }` })34:T618,drawProbabilityAmplitudeReal = regl({ uniforms: { texture: kCombined, }, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; uniform sampler2D texture; float shading(float x) { return 1. - pow((1. - x), 8.); } void main () { vec2 tex = texture2D(texture, vUv).rg; float h = shading(abs(tex.r) * 0.5); gl_FragColor = vec4(h, 0.0, 0.0, 1.0); }`, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.xy; float h = texture2D(texture, uv).r; return WORLD_HEIGHT*(-1.0 + 2.0 * h) / 2. - .5; } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; uniform mat4 projection, view; void main() { vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.xy; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }` })35:T61b,drawProbabilityAmplitudeComplex = regl({ uniforms: { texture: kCombined, }, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; uniform sampler2D texture; float shading(float x) { return 1. - pow((1. - x), 8.); } void main () { vec2 tex = texture2D(texture, vUv).rg; float h = shading(abs(tex.g) * 0.5); gl_FragColor = vec4(0.0, 0.0, h, 1.0); }`, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.xy; float h = texture2D(texture, uv).g; return WORLD_HEIGHT*(-1.0 + 2.0 * h) / 2. - .5; } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; uniform mat4 projection, view; void main() { vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.xy; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }` })36:T719,drawComplexPhase = regl({ uniforms: { texture: kCombined, }, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.xy; float h = texture2D(texture, uv).r + texture2D(texture, uv).g; return WORLD_HEIGHT*(-1.0 + 2.0 * h) / 2. - .5; } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform mat4 projection, view; void main() { pos = xzPosition; vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.xy; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }`, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; varying vec2 pos; uniform sampler2D texture; #define PI 3.141592653589793 #define hue2rgb(h) clamp(abs(mod(6.*(h)+vec3(0,4,2),6.)-3.)-1.,0.,1.) float shading(float x) { return 1. - pow((1. - x), 8.); } void main () { vec2 tex = texture2D(texture, vUv).rg; float hr = shading(abs(tex.r) * 0.5); float hg = shading(abs(tex.g) * 0.5); gl_FragColor = vec4((hr + hg) * hue2rgb(atan(tex.r,tex.g)/(2.*PI)), 1.); }` })37:T64e,drawPotential = regl({ uniforms: { texture: potentialBuffer, }, vert: ` uniform float WORLD_SIZE; uniform float WORLD_HEIGHT; uniform sampler2D texture; float getHeight(vec2 xz) { vec2 uv = vec2(0.5, 0.5) + xz.yx; return clamp(WORLD_HEIGHT*(-1.0 + 2.0 * texture2D(texture, uv).r), -1.0, 5.0) - 0.01; } vec3 getPosition(vec2 xz) { return vec3(WORLD_SIZE*xz.x, getHeight(xz), WORLD_SIZE*xz.y); } precision mediump float; attribute vec2 xzPosition; varying vec2 pos; varying vec3 vPosition; varying vec2 vUv; varying vec3 vNormal; uniform mat4 projection, view; varying float height; void main() { pos = xzPosition; height = getHeight(xzPosition); vec3 xyzPosition = getPosition(xzPosition); vec2 uv = vec2(0.5, 0.5) + xzPosition.yx; vUv = uv; float eps = 1.0/16.0; // approximate the normal with central differences. vec3 va = vec3(2.0*eps, getHeight(xzPosition + vec2(eps,0.0)) - getHeight(xzPosition - vec2(eps,0.0)) , 0.0 ); vec3 vb = vec3(0.0, getHeight(xzPosition + vec2(0.0, eps)) - getHeight(xzPosition - vec2(0.0, eps)), 2.0*eps ); vNormal = normalize(cross(normalize(vb), normalize(va) )); vPosition = xyzPosition; gl_Position = projection * view * vec4(xyzPosition, 1); }`, frag: ` precision mediump float; varying vec2 vUv; varying vec3 vNormal; varying float height; uniform sampler2D texture; void main () { float tex = texture2D(texture, vUv).r; gl_FragColor = vec4(vec3(1., 1., 1.) , clamp(ceil(height + 0.99), 0.0, 1.0) * 0.5); }` })38:[] 2f:{"title":"3D View of TDSE Simulation","cells":[{"id":0,"value":"md`# 3D View of TDSE Simulation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":75,"value":"TDSESimulation()","pinned":false,"hidden":false,"mode":"ojs"},{"id":493,"value":"md`Select potential:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1098,"value":"viewof potential_selection = html`\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":391,"value":"md`Drag to adjust barrier height (automatically sets potential to barrier):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":380,"value":"viewof barrier_height = {\n const element = html``\n return element\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":89,"value":"md`Step ${elapsedTime} of ${animationTimeLimit}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":870,"value":"md`Simulation speed:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":865,"value":"viewof dt = {\n const element = html``\n return element\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":48,"value":"viewof playToggle = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":142,"value":"viewof restart = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1428,"value":"viewof selected_visualization = html`\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1348,"value":"md`Select Color Map:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1351,"value":"viewof color_map = html`\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":848,"value":"md`Display probability amplitude (complex phase):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":856,"value":"viewof complex_phase_active = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":765,"value":"md`Display probability amplitude (complex components):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":635,"value":"viewof complex_components_active = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1039,"value":"md`## Simulation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1049,"value":"md`Imports from base simulation notebook:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":35,"value":"import\n{\n mutable elapsedTime,\n shading_function,\n kCombined,\n potentialBuffer,\n k1,\n k2,\n k3,\n k4,\n combine_k,\n potential,\n setupDefault,\n wave_unit_components,\n psi_n,\n regl,\n}\nwith\n{\n TDSESimulation as TDSESimulation,\n playing as playing,\n animationTimeLimit as animationTimeLimit,\n potential_selection as potential_function,\n dt as dt,\n wave_position as wave_position\n}\nfrom \"@flimsyhat/tdse-simulation\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":1043,"value":"md`New simulation:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":66,"value":"$31","pinned":true,"hidden":false,"mode":"ojs"},{"id":1013,"value":"md`## Shaders`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1091,"value":"mutable wave_position = [0.5, 0.25]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1019,"value":"md`Functions for drawing the various potentials:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":194,"value":"barrier = `\nfloat potential(vec2 p) {\n return float(${barrier_height}) * float(p.y < (0.5 + dx/2.0) && p.y > (0.5 - dx/2.0));\n}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":453,"value":"two_slit_potential =`\nfloat potential(vec2 p) {\n return 2. * float(p.y < (0.5 + dx) && p.y > (0.5 - dx))\n * float(p.x < 0.4 || p.x > 0.45) * float(p.x < 0.55 || p.x > 0.6);\n}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":774,"value":"harmonic_potential = {\n return `\n float potential(vec2 p) {\n return pow((p.x - 0.5) * 2.0, 2.) + pow((p.y - 0.5) * 2.0, 2.);\n }`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1052,"value":"scatter = {\n return `\nfloat potential(vec2 p) {\n float radius = pow((p.x - 0.5) * 2.0, 2.) + pow((p.y - 0.5) * 2.0, 2.);\n if ( radius < 0.005 )\n return 3.0;\n else \n return 0.0;\n}\n`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1023,"value":"md`Setting up shared defaults:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":881,"value":"setup3DDefaults = regl({\n uniforms: {\n u_resolution: ctx => [ctx.framebufferWidth,ctx.framebufferHeight],\n WORLD_SIZE: 10.0, // the size of the world on the x and z-axes.\n WORLD_HEIGHT: 1.0, // the height of the world.\n },\n attributes: {\n xzPosition: xzPosition\n },\n elements: elements,\n blend: { // enables alpha blending\n enable: true, \n func: {\n srcRGB: 'src alpha',\n srcAlpha: 'src alpha',\n dstRGB: 'one minus src alpha',\n dstAlpha: 'one minus src alpha',\n },\n },\n});","pinned":false,"hidden":false,"mode":"ojs"},{"id":1298,"value":"md`Conversion from complex ${tex`\\psi`} to probability ${tex`|\\psi|^2`}:\n\n${tex`|x + iy|^2 = x^2 + y^2`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1230,"value":"probability = `\n float probability(vec2 complex_number) {\n return pow(complex_number.r, 2.) + pow(complex_number.g, 2.);\n }\n\n float remap_probability(vec2 rg, float factor) {\n return pow(probability(rg), factor);\n }\n\n float clamped_probability(vec2 rg) {\n return clamp(remap_probability(rg, 0.7), 0., 1.);\n }\n\n float nudge_intensity(float x) {\n return 1. - pow((1. - x), 6.);\n }\n \n float intensity(vec2 rg) {\n return nudge_intensity(clamped_probability(rg));\n }\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1357,"value":"probability_amplitude = `\nfloat intensity(vec2 rg) { \n return clamp((tex.r + tex.g + 1.) / 2., 0., 1.);\n}\n\n`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1026,"value":"md`Actual shaders:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1409,"value":"md`Plotting ${tex`|\\psi|^2`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":59,"value":"$32","pinned":true,"hidden":false,"mode":"ojs"},{"id":1403,"value":"md`Plotting the real component of the wavefunction, ${tex`Re(\\psi)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1245,"value":"$33","pinned":true,"hidden":false,"mode":"ojs"},{"id":533,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":620,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":829,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":172,"value":"$37","pinned":true,"hidden":false,"mode":"ojs"},{"id":1275,"value":"import {inferno, viridis, turbo} from \"@flimsyhat/webgl-color-maps\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1191,"value":"md`Experiment with drawing grid lines, not using currently:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1147,"value":"grid = `\nfloat grid(vec2 p, float res) {\n return step(res,cos(p.x * 256.)) + step(res,cos(p.y * 256.));\n}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":996,"value":"md`## Building the mesh:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"N = 128","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"size = 0.5","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"xmin = -size","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"xmax = size","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"ymin = -size","pinned":false,"hidden":false,"mode":"ojs"},{"id":18,"value":"ymax = size","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"xzPosition = {\n let xzPosition = []\n var row\n var col\n for (row = 0; row <= N; ++row) {\n var z = (row / N) * (ymax - ymin) + ymin\n for (col = 0; col <= N; ++col) {\n var x = (col / N) * (xmax - xmin) + xmin\n xzPosition.push([x, z])\n }\n }\n return xzPosition\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"elements = {\n let elements = []\n var row\n var col\n // create plane faces.\n for (row = 0; row <= (N - 1); ++row) {\n for (col = 0; col <= (N - 1); ++col) {\n var i = row * (N + 1) + col\n\n var i0 = i + 0\n var i1 = i + 1\n var i2 = i + (N + 1) + 0\n var i3 = i + (N + 1) + 1\n\n elements.push([i3, i1, i0])\n elements.push([i0, i2, i3])\n }\n }\n return elements\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1033,"value":"md`## 3D turntable camera`","pinned":false,"hidden":false,"mode":"ojs"},{"id":31,"value":"createCamera = require('https://bundle.run/regl-camera@2.1.1')","pinned":false,"hidden":false,"mode":"ojs"},{"id":57,"value":"camera = createCamera(regl, {\n center: [0, -1.5, 0],\n theta: 7. * Math.PI/6.0,\n phi: Math.PI/6.0,\n distance: 20.0,\n damping: 0,\n noScroll: true,\n zoomSpeed: 0, // disables zoom\n renderOnDirty: false,\n rotationSpeed: 0.5,\n element: regl.container\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":998,"value":"md`## Utilities for pausing/playing/restarting the simulation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":42,"value":"mutable playing = false;","pinned":false,"hidden":false,"mode":"ojs"},{"id":44,"value":"{\n playToggle;\n mutable playing = !mutable playing;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":387,"value":"{\n potential_selection;\n barrier_height;\n restart;\n reset();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":423,"value":"reset = () => {\n mutable elapsedTime = -1;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":46,"value":"animationTimeLimit = 4000;","pinned":false,"hidden":false,"mode":"ojs"},{"id":1005,"value":"md`## Other Imports`","pinned":false,"hidden":false,"mode":"ojs"},{"id":501,"value":"import {select} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W38","stdlib":"1","titling":"auto"}