{"id":174,"date":"2015-04-01T00:19:15","date_gmt":"2015-04-01T04:19:15","guid":{"rendered":"http:\/\/tasharp.com\/?p=174"},"modified":"2017-10-09T18:33:17","modified_gmt":"2017-10-09T22:33:17","slug":"this-one-weird-skill-will-make-you-better-at-mathematical-analysis","status":"publish","type":"post","link":"https:\/\/tasharp.com\/index.php\/this-one-weird-skill-will-make-you-better-at-mathematical-analysis\/","title":{"rendered":"Visualizing mathematical analysis"},"content":{"rendered":"

In this post, I’d like to muse about the visualizations that are used when thinking about math, and how they can become quite beautiful as well as useful<\/p>

When adding\u00a0\"7+8\", many people imagine stacking blocks and then\u00a0see<\/em>\u00a0\"5\" spilling over the \"10 mark. This visual thinking made\u00a0the answer seem “obvious” after doing it many times.\u00a0 When doing calculus, simple picture\u00a0tricks\u00a0are\u00a0taught less often. \u00a0If confronted with \"\int_0^1 we might quickly\u00a0follow the memorized rule (from the fundamental theorem of calculus) that you add \"+1 to the exponent and divide by that number. \u00a0\"\int_0^1. \u00a0But one can also\u00a0remember that\u00a0integration finds the area\u00a0under\u00a0a curve. \u00a0With a picture in mind like the one below, we know that the red line cuts the unit square into two parts, each with area \"\frac{1}{2}\". \u00a0 The visual understanding will sometimes be a useful tool.<\/p>

\"integral_of_x\"<\/a><\/p>

This post is supposed to illustrate that\u00a0visual thinking remains useful in more advanced math, especially\u00a0mathematical\u00a0analysis <\/em>(real analysis, complex analysis, functional analysis, differential equations) which deals with objects you can imagine embedded\u00a0in space (that is, with a metric). \u00a0It’s just that the pictures become more and more rarely taught. \u00a0That means it takes a motivated attitude and some\u00a0patience, because often in\u00a0classes you generally have to figure it out for yourself.<\/p>

I’ve found that\u00a0it is worth the effort. \u00a0This post\u00a0will give a couple of examples using\u00a0partial differential equations (PDEs). \u00a0PDEs\u00a0are very useful in engineering disciplines, yet it is not often taught how to picture the solution.<\/p>

For example, let’s consider\u00a0Laplace’s equation.<\/strong><\/p>\n

\"\Delta<\/p>

In classical physics, the electrical potential \"\phi satisfies Laplace’s\u00a0equation in three dimensions everywhere in space that there is no electric charge.<\/p>

Say we want to know\u00a0the potential in a long\u00a0square pipe (a square cylinder), and we are given\u00a0boundary conditions that specify\u00a0the potential of\u00a0each of the four sides. \u00a0We can visualize what this equation is saying. \u00a0The\u00a0equation is local<\/strong> and it says that at every point<\/strong> of the domain, the x-curvature of the potential\u00a0must cancel with the y-curvature of the potential.<\/p>

I picked some boundary conditions (parabolas on the ends and constants at the side) and plotted the\u00a0solution for \"\phi below using Wolfram Alpha.<\/p>

 <\/p>\n

\"00_0\"\"01\"<\/p>

We can visually check that the curvature in x and y are equal and opposite\u00a0at every point. \u00a0Note how the solution\u00a0looks like a stretched membrane. \u00a0That’s because a stretched membrane\u00a0also follows<\/a>\u00a0also\u00a0follows Laplace’s equation. \u00a0The solution\u00a0is essentially the simplest surface you can draw that will connect the boundaries. \u00a0If you can picture gluing an elastic sheet to those boundaries, you can picture the solution to Laplace’s equation.<\/p>

Quantum probability current<\/strong><\/p>

In the example of the Schr\u00f6dinger equation, the rate of change of the\u00a0Hilbert state vector is determined\u00a0by the Hamiltonian operator.<\/p>\n

\"i<\/p>

Operators and state vectors are abstract and\u00a0we are sometimes\u00a0interested in the positions basis specifically, since let’s face it, space seems to be\u00a0a dominant aspect of reality. \u00a0For the 1-dimensional problem of a free particle in the position basis\u00a0we get the\u00a0partial differential equation:<\/p>\n

\"\frac{\partial<\/p>

\"\Psi(x,t) is a complex field, in the sense that it has both real and imaginary components, and we can… really imagine it. \u00a0The equation can be decomposed into two equations, one for the real part and one for the complex part of \"\Psi(x,t), and we can plot it\u00a0using a polar plot at each value of\u00a0x. \u00a0Again, the equation is local and says that the curvature is what matters. \u00a0Where the curvature of the real\u00a0part is positive,\u00a0the imaginary\u00a0part will grow. \u00a0Where the curvature of the imaginary\u00a0part is positive,\u00a0the real\u00a0part will shrink. \u00a0If initially the field looks like a plane wave \"\Psi(x,0) where \"p is a constant, the equation is telling us the corkscrew will turn.<\/p>

\"cscrewlabeled\"<\/a><\/p>

Animation created with Python and ImageMagick
\n<\/a><\/p>

Now for the last part of\u00a0this example, let’s look at the probability current. Usually when this is taught it seems to be a terribly opaque abstract construct. The probability current \"\bold{j} is essentially the quantity that, together with the probability \"P(x,t), satisfies the continuity equation \"\frac{\partial}{\partial when \"\Psi follows the\u00a0Schr\u00f6dinger equation. \u00a0In one dimension it is<\/p>\n

\"j(x,t)<\/p>

It is difficult to picture what this equation is telling us! \u00a0And yet wiki pages and textbooks* everywhere just state it and move on, letting the student grapple with it in the problem set. Let us stubbornly pursue a better picture.\u00a0 Using\u00a0the polar form of the complex quantity \"\Psi and working through a little math we get\u00a0an equation for the flow of that is much\u00a0simpler to picture<\/em>.<\/p>\n

\"j(x)<\/p>

The fraction of probability leaving location x is simply the derivative of the phase. <\/em> \u00a0In three dimensions, \"\frac{d}{dx}\theta which is saying that the probability current is just “flowing uphill” of the phase. That’s what the phase of a wave function has always meant, in case, like me, you weren’t told. This simpler version gives insight that might even be useful to a new wave\u00a0of physics students.<\/p>

Using the example of the corkscrew above, simply stated, the angle \"\theta of the corkscrew increases constantly to the right, so our equation tells us there is uniform probability current to the right.<\/p>

Here are other relations that are useful for visualizing these single-particle dynamics.<\/strong>\u00a0Here I\u00a0use shorthand for the derivatives so \"\frac{d}{dt} and \"\nabla. In the following, ignore the constant factors of \"2m\/\hbar which could be swallowed into the definition of the phase. The phase at a point x will decrease if it is sloped and it will increase due to curvature of A \"\dot. Meanwhile the amplitude A at the point x decreases from phase curvature or from slopes \"\frac{\dot. The probability increases from amplitude curvature (“smoothing out humps”) and decreases wherever the phase is sloped \"\dot . Hope this helps you picture some quantum dynamics.<\/p>

The point of this post is just some examples of looking at\u00a0partial differential equations to get a feeling for\u00a0what the solutions look like. \u00a0Locality of a PDE is a powerful property, something that college students often don’t appreciate in the standard science\/engineering curriculum. \u00a0My intuition dramatically increased once I began\u00a0writing numerical codes to solve PDEs.<\/p>

So I encourage the new mathematical scientists and engineers out there, don’t be afraid to break down your math problem and visualize it! \u00a0It is worth the effort.<\/p>

*Nearly all textbooks I have found stop at the abstract form of the probablility current. \u00a0I eventually found however that Sakurai Modern Quantum Mechanics 1994 contains the same simple equation for \"\bold{j} as above.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this post, I’d like to muse about the visualizations that are used when thinking about math, and how they can become quite beautiful as well as usefulWhen adding\u00a0, many people imagine stacking blocks and then\u00a0see\u00a0 spilling over the mark. This visual thinking made\u00a0the answer seem “obvious” after doing it many times.\u00a0 When doing calculus, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[19,11],"tags":[],"class_list":["post-174","post","type-post","status-publish","format-standard","hentry","category-art","category-physics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p5Yxym-2O","jetpack-related-posts":[],"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/posts\/174","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/comments?post=174"}],"version-history":[{"count":124,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/posts\/174\/revisions"}],"predecessor-version":[{"id":840,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/posts\/174\/revisions\/840"}],"wp:attachment":[{"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/media?parent=174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/categories?post=174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tasharp.com\/index.php\/wp-json\/wp\/v2\/tags?post=174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}