2d:null 2e:["$","title",null,{"children":"03 - Enhanced tunnelling of a quantum particle | Matthew Lilley | Observable"}] 33:T45e,md` where we'll be setting ${tex`\hbar=1`} and ${tex`m=1`} to visually simplify things. In the [last tutorial](https://observablehq.com/@mklilley/quantum-particle-tunnelling-through-a-barrier), we saw how a quantum particle could escape a confining "box" (in the form of a [square potential well](https://en.wikipedia.org/wiki/Finite_potential_well)) through a process called [quantum tunnelling](https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_%28OpenStax%29/Map%3A_University_Physics_III_-_Optics_and_Modern_Physics_%28OpenStax%29/07%3A_Quantum_Mechanics/7.07%3A_Quantum_Tunneling_of_Particles_through_Potential_Barriers). The "leakiness" of our confining box could be seen through the exponential decay of the probability of finding the particle inside the box over time. In this tutorial, we will see that this exponential characteristic of quantum tunnelling is not universal and we can in fact speed up the process. We will demonstrate this by using 2 identical square potential wells. As before, we will initially consider the particle to have zero average momentum `34:T60f,md`## Why? The difference between the results above and those from the last tutorial stems from the discrete vs. continuum nature of the energy levels of the systems under investigation. As you may recall from the [introduction notebook](https://observablehq.com/@mklilley/quantum-physics-particle-in-a-box?collection=@mklilley/quantum-physics): > When particles are free, they can have any energy they like. When they are confined however, their energy can only take on discrete values, aka [energy levels](http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qnenergy.html). Because our square well potentials are identical, their discrete energy levels (which we have yet to calculate) are also identical. The oscillations we see above (often referred to as [Rabi oscillations](https://en.wikipedia.org/wiki/Rabi_cycle)) are the result of these identical energy levels interacting with each other through the barrier. A detailed understanding of Rabi oscillations is best developed in the context of a [two-state system](https://github.com/project-ida/two-state-quantum-systems), but we can give a conceptual outline here. Conceptually, each energy level is associated to a specific frequency. An energy level can be thought of as a classical pendulum with that same frequency and the interaction of two levels as the interaction of two coupled pendulums. If you displace only one pendulum, then its maximum amplitude oscillates slowly as it transfers energy to the second pendulum and then back again - as you can see in this video below. ` 35:[] 31:{"id":"ce17f876b3a3d663","slug":"enhanced-tunnelling-of-a-quantum-particle","title":"03 - Enhanced tunnelling of a quantum particle","thumbnail":"a74c57e04cd756b806120e71437c77fb872426a6293bf3c303bf0a65ac58766b","access_level":"public","create_time":"$D2020-12-08T11:02:23.852Z","update_time":"$D2024-11-13T16:48:40.856Z","view_time":null,"first_public_time":"$D2020-12-08T16:59:38.951Z","has_unpublished_edits":null,"fork":{"id":"a609eaf88f13e77e","slug":"quantum-particle-tunnelling-through-a-barrier","title":"02 - Quantum particle tunnelling through a barrier","owner":{"id":"4a28ea9ed826d0a0","login":"mklilley","name":"Matthew Lilley"}},"stars":2,"starred":false,"owner":{"type":"team","id":"4a28ea9ed826d0a0","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":"starter_2024"},"version":1781,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"dcd27f799cb1f0fc","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":null}],"members":[{"type":"individual","id":"dcd27f799cb1f0fc","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":{"id":"61c3b2fecc25e94b","slug":"quantum-physics","title":"Quantum physics - Particle in a box","owner":{"id":"4a28ea9ed826d0a0","login":"mklilley","name":"Matthew Lilley"}},"type":"notebook","body":{"title":"03 - Enhanced tunnelling of a quantum particle","cells":[{"id":0,"value":"md`# 03 - Enhanced tunnelling of a quantum particle`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":802,"value":"md`\n## What?\nIn this third notebook of the series on quantum particles inside \"boxes\", we are once again solving the time-dependent [Schrödinger equation](http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schrcn.html#c1) in 1-D, i.e.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":804,"value":"md`\n${tex.block`i\\hbar\\frac{\\partial\\psi}{\\partial t} = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2\\psi}{\\partial x^2} + V\\psi`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":809,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1224,"value":"md`\n\n**p0** = ${viewof p0.bind(html``)}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"md`Let's see what happens.\n\nRun simulation ${viewof simulate.bind(html``)}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":40,"value":"graph = plot([x,psi0_squared,V_plot],[ \"x\", \"|Psi|^2\",\"V_norm\"]) // Initialise the simulation plot","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1249,"value":"{\n let button\n \n if(simulate){\n button = html``;\n }\n else{\n button = html``;\n }\n\n button.onclick = () => {\n reset()\n }\n\n return button;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":963,"value":"md`Below, we keep track of the probability of finding the particle inside the left well. We can see that it does not follow an exponential decay as in the last turorial but instead it oscillates and it does so on a timescale that's faster than the previous exponential one. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"graph_P_well = plot([nj.array([[0,0]])],[ \"time\", \"P_well\"]) // Initialise the simulation plot","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1531,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1696,"value":"html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1700,"value":"md`\n This classical transfer of energy only works well when the pendulum frequencies are the same - the same is true for the energy of the quantum energy levels.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1713,"value":"md`\nWell defined Rabi oscillations only present themselves in systems with discrete energy levels, i.e. confined systems. In the last tutorial, we simulated the tunnelling of a particle from a confined potential to zero potential, i.e. from trapped to free. When the particle is free it can have a continuum of energies which contribute a continuum of frequencies to the Rabi oscillations. The exponential decay that we saw last time is the [result of these many contributions](http://info.phys.unm.edu/~ideutsch/Classes/Phys522S16/Lectures/Phys522_Lect26.pdf). It is curious that the overall result is a slower decay in the probability as compared to the non-continuum case that we are studying in this tutorial - something that we will return to in a later tutorial.\n\n` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1527,"value":"md`## How?\nTo understand the nuts and bolts of the algorithm being used here, head back over to the [first notebook](https://observablehq.com/@mklilley/quantum-particle-moving-in-a-harmonic-potential). Everything is almost the same as the last tutorial, only the potential and initial conditions are different.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":901,"value":"md`**Simulation parameters**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1084,"value":"md`Be aware that this simulation box is much larger than last time. As you change parameters you might need to wait a few seconds for the simulation grid to catch-up...alas JavaScript has its limits.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"md`Time step:\n\n**dt** = ${viewof dt.bind(html``)}`\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1449,"value":"md`Time steps between each plotting frame:\n\n**n** = ${viewof n.bind(html``)}`\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"md`\nSize of simulation box: \n\n**xmax** = ${viewof xmax.bind(html``)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":54,"value":"md`Simulation grid spacing:\n\n**dx** = ${viewof dx.bind(html``)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":56,"value":"numx = {\n if(dx){\n if(dx !=0){\n return Math.round(2*xmax/dx +1)\n }\n else { // This is in case the input is 0, we don't want the whole notebook to show errors and make the screen jump all over the place\n return 0\n }\n }\n else{ // This is in case the input is empty, we don't want the whole notebook to show errors and make the screen jump all over the place\n return 0\n }\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":866,"value":"md`Absorbing boundary strength:\n\n**${tex`\\lambda`}** = ${viewof lambda.bind(html``)}\n
\n\nAbsorbing boundary width:\n\n**${tex`w/x_{max}`}** = ${viewof wOverXmax.bind(html``)}\n`\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":875,"value":"w = wOverXmax*xmax","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":898,"value":"md`**Build simulation grid**\n\nFirst we need to make a column of ${tex`x`} values`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"x = nj.arange(numx).multiply(dx).subtract(xmax).reshape(numx,1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1080,"value":"md`From these ${tex`x`} values we can calculate the corresponding ${tex`k`} values for Fourier space as we did last time.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"k = fftfreq(x)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":904,"value":"md`**Potential**\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1138,"value":"md`We construct the potential by \"gluing\" together several constant pieces to create a [piecewise function](https://en.wikipedia.org/wiki/Piecewise)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1096,"value":"md`\n**barrier_width** = ${viewof barrier_width.bind(html``)}`\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1115,"value":"md`\n**barrier_height** = ${viewof barrier_height.bind(html``)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1097,"value":"md`\n\n**well_width** = ${viewof well_width.bind(html``)}`\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1046,"value":"V = {\n let numBarrier = Math.round(barrier_width/dx)\n let numWell = Math.round(well_width/dx)\n let numLeft = Math.round((numx - numWell*2 - numBarrier )/2) // This will create a double well centred at zero\n let numRight = numx - numLeft - numWell*2 - numBarrier\n \n let barrier = nj.ones(numBarrier).multiply(barrier_height)\n let well = nj.zeros(numWell)\n let left = nj.ones(numLeft).multiply(barrier_height)\n let right = nj.ones(numRight).multiply(barrier_height)\n\n\n let V = nj.concatenate(left,well,barrier,well,right)\n return V.reshape(numx,1)\n\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1515,"value":"md`Later on, we'll need to integrate the probability density inside the left well, so it will be convenient to know the index values where it begins and ends.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1255,"value":"well_start = Math.round((numx - Math.round(well_width/dx)*2 - Math.round(barrier_width/dx) )/2)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1260,"value":"well_end = well_start + Math.round(well_width/dx)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1213,"value":"md`Let's also create a normalised version of the potential so that we can always plot it alongside the probability density - it will always have a height of 1 on the graph.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1193,"value":"V_plot = V.divide(barrier_height)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":895,"value":"md`**Exponentials**\n\nWe can now create the complex exponentials that allow us to advance the solution as in the \\`\\`psi_dt\\`\\` function - just as we did last time.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"expK = cexp(k.pow(2).multiply(-dt*(2*Math.PI)**2.0))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1134,"value":"expV = cexp(V.multiply(-dt/2))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":419,"value":"expABC = {\n let absorb = nj.tanh(x.subtract(xmax).divide(w)).add(2).subtract(nj.tanh(x.add(xmax).divide(w))).multiply(-lambda*dt)\n \n let expABC = nj.exp(absorb)\n \n expABC = nj.concatenate(expABC, nj.zeros([numx,1]) ) // need to add 2nd column of zeros for the zero imaginary part\n \n return expABC\n\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":922,"value":"md`**Initial conditions**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1147,"value":"md`We can represent a particle moving with average momentum momentum ${tex`p_0`} localised in space around ${tex`x_0`} by the following [wave packet](http://farside.ph.utexas.edu/teaching/qmech/Quantum/node25.html):\n\n${tex.block`\\psi_0 = \\left(\\frac{2}{\\pi \\sigma^2}\\right)^{1/4} e^{-(x-x_0)^2/\\sigma^2 + ip_0x}`}\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1332,"value":"md`We'll place the particle in the centre of the left potential well so that`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1330,"value":"x0 = - barrier_width/2 - well_width/2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1336,"value":"psi0 = {\n let sigma2 = 2\n let norm_constant = Math.pow(2 / (Math.PI * sigma2), 0.25);\n let psi0 = nj.concatenate(nj.exp(x.subtract(x0).pow(2).multiply(-1).divide(sigma2)), nj.zeros([numx,1]) )\n psi0 = cmultiply(psi0,cexp(x.multiply(p0)))\n\n return psi0.multiply(norm_constant);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1380,"value":"md` Now we can create ${tex`|\\psi_0|^2`} for initialising the first plot ` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1369,"value":"psi0_squared = cnorm2(psi0)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1398,"value":"md`and ${tex`P_{\\text{well}} = \\int _{\\text{well}} |\\psi_0|^2 dx`} for the second plot` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1415,"value":"P_well0 = psi0_squared.slice([well_start,well_end]).sum()*dx","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":925,"value":"md`**Simulation**\n\nFirst, let's set up the variables that we'll need.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1344,"value":"mutable psi=psi0 // This sets the initial condition for the simulation of psi\n\n// Note, for more info on \"mutable\" see https://observablehq.com/@mbostock/mutable-values","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1374,"value":"md`We create an additional variable psi_plot so that we have the flexibility visualise the simulation every **n** timesteps.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1349,"value":"mutable psi_plot = psi0","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1436,"value":"md`We need to create two arrays to hold evolving values of P_well and also the time for the second plot`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1407,"value":"mutable P_well = nj.array([P_well0]) // this creates an array to store the evolving values of P_well","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1424,"value":"mutable time = nj.array([0]) // this creates an array to store the evolving time for plotting along with P_well","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1433,"value":"mutable i = 0 // this keeps track of the number of the elapsed timesteps","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1463,"value":"md` Now let's bring all the initialisations together in a function that can be called using the reset button.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1462,"value":"reset = function(){\n mutable psi = psi0;\n mutable psi_plot = psi0;\n mutable time = nj.array([0])\n mutable P_well = nj.array([P_well0])\n mutable i =0\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1473,"value":"md`And now for the main simulation loop.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":80,"value":"{ \n if(simulate===true) { // Only simulate when simulation checkbox has been checked\n mutable psi = psi_dt(psi, expV, expK, expABC)\n \n // Update the timestep\n mutable i+=1\n \n // Only update the psi_plot for plotting every n time-steps\n if(i%n==0) {\n \n mutable psi_plot = psi // Note this line will trigger psi_squared for the plots and also for the next step below\n \n // Add new data to the P_well and time arrays by concatenating old array with new value\n mutable P_well = nj.concatenate(mutable P_well,nj.array([psi_squared.slice([well_start,well_end]).sum()*dx])) \n mutable time = nj.concatenate(mutable time, nj.array([i*dt]))\n \n }\n\n\n }\n return \"Indefinite simulation loop\"\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":85,"value":"psi_squared = cnorm2(psi_plot)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1477,"value":"md`The last step is to update the graphs at the top of the page with new values.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":941,"value":"graph.update([x,psi_squared,V_plot]) // Update the simulation plot when new values of psi_plot are available","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1007,"value":"graph_P_well.update([nj.stack([time, P_well], -1)]) // Update the simulation plot when new values of P_well are available","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1545,"value":"md`### numjs\nImport the numjs library\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"nj = require('https://cdn.jsdelivr.net/gh/nicolaspanel/numjs@0.16/dist/numjs.min.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1562,"value":"md`### Tutorial specific functions\n\nWe'll need to import several functions that we created in the [first tutorial](https://observablehq.com/@mklilley/quantum-particle-moving-in-a-harmonic-potential).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":679,"value":"import {fftfreq, plot, cmultiply, cexp, cnorm2, psi_dt} from \"@mklilley/quantum-particle-moving-in-a-harmonic-potential\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1551,"value":"md`### Observable \nHere is some observable specific code that makes it conveient to update parameters. See the [first tutorial](https://observablehq.com/@mklilley/quantum-particle-moving-in-a-harmonic-potential) for more detail.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"import {View} from \"@mbostock/synchronized-views\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"viewof simulate = new View(false)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"viewof dx = new View(0.1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":60,"value":"viewof xmax = new View(30)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"viewof dt = new View(0.01)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":869,"value":"viewof wOverXmax = new View(0.05)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":872,"value":"viewof lambda = new View(20)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1104,"value":"viewof well_width = new View(2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1106,"value":"viewof barrier_width = new View(1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1119,"value":"viewof barrier_height = new View(2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1238,"value":"viewof p0 = new View(0)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1455,"value":"viewof n = new View(5)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"