What Hath Mathematica Wrought ?

        Docs going into infectious disease in the 60s and 70s looked forward to a rather gentlemanly professional career.  While there would be a few calls in the middle of the night, most could be handled by telling the calling doc what antibiotic to use and how much. Then AIDS hit in the 80s and they began living in the hospital combating all sorts of bizarre infections day and night, in a rather different clientele than they were used to. 

        Well, it’s now two weeks into the Quantum Mechanics course, and things are very different than they were 48 years ago, thanks to Mathematica.  Back then we could prove that the integral of sin (n Pi x) times sin (m Pi x) dx was zero if m didn’t equal n.   Seeing that this is true is another matter. If you have access to Mathematica, try this one on for size

          Plot [ Sin [ 3 Pi x] * Sin [ 7 Pi x], {x, 0, 1}]

         You’ll see a bunch of squiggles, the area of which above the line exactly equals that below the lie.  It’s not a proof, of course, but in a way it is MORE convincing than a proof (to me at least).  Back then, we didn’t have the time or the energy to do anything like this.

        But it’s not just the students, it’s the pros teaching them.  The prof in the course never integrates things by himself.  He just tells Mathematica to do it.  Things like

       Integrate [ x Exp [ – ( q – x ) ^ 2], {x, -infinity, infinity}]

       Does anyone know how to do hard core integrations any more?  It reminds me of a science fiction story I read back in the day when people were frightened of what computers would be able to do.  As I recall, it took place in the Defense department.  Computers were at the stage when they were designing newer and better versions of themselves.  The story begins as some inky wretch was brought up before the commanding generals by his superior.  The man had found a way to multiple two 3 digit numbers by hand — the art had been lost.  The brass couldn’t believe it, but they checked it on their hand-held computers and it was so.  There followed an anti-military tirade, ending with the commanding general saying that he could see humans being able to produce a manned missile in the future.

        I was fortunate enough to audit a course on “Ideals, Varieties and Algorithms” a few years ago, given by one of the authors (David A. Cox).  He told me that algebraic geometry became completely transformed by computer algebra systems shortly after he got his PhD.  You just have to see what Mathematica or Maple spits out in the course of eliminating variables from a set of polynomial equations using Groebner bases to do so.  Pages and pages of polynomials.  He was called on by the Mathematica people as a consultant on this topic (presumably to make their algorithms faster) but had to sign a nondisclosure agreement.

        I am now the proud (if poorer) owner of “Physics for Scientists & Engineers” by Giancoli.  Why that book rather than Halliday?  Both of them look like comic books with pretty colored diagrams on every page (photos too).  Well, it’s what Harvard tells their Freshmen to use.  Presumably they know something about physics books.  Hopefully I’ll get through some of it before the course ends.  Hamlitonians are all well and good, but I’ve forgotten Newton’s laws of motion.  Time to crack it open.

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  • kerri  On September 23, 2009 at 12:55 am

    There is a P. Chem book by Dr. Ball. It is nice where the book goes through derivations…. The Eratta of the book is frustrating, but hopefully if the new version comes out any time soon, it will be better…

  • Curious  On September 24, 2009 at 2:05 pm

    Speaking of basic physics, here’s an interesting problem. You are on a boat which floats in a large swimming pool. There’s a 50 pound sack of rocks on the boat. You pick it up and throw it into the surrounding water. The level of the water:
    a. Goes up
    b. Goes down
    c. Stays the same

  • luysii  On September 24, 2009 at 2:49 pm

    Well my guess is that the water level goes down after throwing the rocks overboard. When in the boat the rocks push down on the boat displacing a volume of water equal to the mass of the rocks. At the bottom of the pool it displaces an amount of water equal to its volume — since the density of the rock is greater than that of water this is less displacement.

  • Curious  On September 25, 2009 at 9:39 am

    Bravo! Everyone I know (including myself and excluding my father) hastily gave the first answer. I got this problem out of an old classic, a physics textbook by George Gamow and Charles Critchfield. The book says that Robert Oppenheimer, Felix Bloch and Leo Szilard were all asked this question and they all gave the wrong answer! It’s one of those problems where the first, quick answer is almost always wrong.

  • luysii  On September 27, 2009 at 12:14 pm

    The one who should get the accolades is my wife. I sprung the problem on her, and she insisted on knowing what kind of rock it was. I thought she was just being difficult, as wives sometimes are. She informed me that pumice can be lighter than water. Frightening — she saw through the problem immediately. Being married to a smart woman can be stressful, but it’s never dull.

  • Curious Wavefunction  On October 10, 2009 at 2:16 pm

    You are a lucky man!

  • luysii  On October 29, 2009 at 11:57 am

    It’ s intriguing what Mathematica has done to what we’ ll accept as a proof (assumng you accept what follows below as a proof). From tomorrow’s QM homework problems:

    47. Show that the spherical harmonic Y10 is orthogonal to Y11.That is, that
    = 0


    SphericalHarmonicY[l,m,\[Theta],\[Phi]] gives the spherical harmonic \!\(SubsuperscriptBox[\(Y\), \(l\), \(m\)](\(\[Theta], \[Phi]\))\).  >>

    Y[l_, m_] := SphericalHarmonicY[l, m, \[Theta], \[Phi]]

    Y[2, 0]

    1/4 Sqrt[5/\[Pi]] (-1 + 3 Cos[\[Theta]]^2)

    Integrate[Conjugate[Y[1, 0]] Y[1, 1], {\[Theta], 0, Pi}, {\[Phi], 0, 2 Pi}]


    Does anyone think the last two lines constitute a QED ? (Not quantum electrodynamics but Quod Erat Demonstrandum)

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