Quantum Harmonic Oscillator: Power series method in Maple


In the previous blog post What is Computational Physics (Science)?, I ended the post with the following figure

Graph of the probability distribution of the 100th state of the quantum
harmonic oscillator (generated using the power series method).

and stated that I might write a post on how to solve the Quantum harmonic oscillator numerically using the power series method (the other method being the ladder operator method [1]) and generate that figure. This post is just about that.

Ok. First I need to clear the cache with the restart command, import the PDEtools (to solve the pde SE) and Maplets[Elements] (necessary if you want to generate a maplet with a slider) packages.

with(PDEtools): #we need to use the dchange command later in the solution

What is Computational Physics (Science)?


As a senior physics undergraduate I have come to believe that scientific computation must be part of the physics curriculum. It is true that physics students are required to study and master many topics, languages, techniques, and skills like mathematics, linguistics, & science communication, still I think that computational physics should be a major part of the curriculum. It is not logical to be in the age of supercomputers and the physics curriculum remain bound to pen and paper as it used to be before the advent of computers! I am not suggesting that physics should all be done on computers; absolutely not. The student must acquire the necessary theoretical and mathematical concepts and skills, besides the physics thinking, before delving in computational physics! What use would a computer have if its user doesn’t know what he wants to use it for? In other words, how would a physics student who hasn’t studied classical mechanics be able to solve a classical mechanics problem on a computer? He will surely not be able to do so, since he will not be able to appropriately instruct the computer due to his lack of conceptual physics and paper & pen problem solving skills. In short, “a computer is as dumb as its user is dump, and a computer is as smart as a smart user; the smarter and knowledgeable the user, the more productive and efficient the computer is”!

The computer is a little over 70 years old. The first computer, many articles & resources claim, is the “Electronic Numerical Integrator And Computer“, or ENIAC for short, which is not technically correct. Many other computers preceded ENIAC most of which were developed for military purposes (e.g; calculation of artillery, cryptoanalysis, etc…) and were analogue (or electro-mechanical) & programmed by punched cards. ENIAC was a room-sized computer that required several people to operate by turning on/off switches that made use of vacuum tubes the ancestor of the modern transistor.

One particularly interesting electromechanical machine (could be called a computer) was the “bombe” [1] which was [designed] by the mathematician Alan Turing to be used to crack the Enigma, the code used by the Nazi to encrypt messages.

Working rebuilt bombe at Bletchley Park [2].
Interior of the rebuilt bombe at Bletchley Park.
The bombe was in part successful in breaking the Enigma. Moreover, Alan Turing has impacted the modern day internet as well; everyone of us using the internet have definitely faced the “CAPTCHA” which are used to counter-bots & make sure the user is an actual human being & not a bot (from robot). CAPTCHA is an abbreviation for “Completely Automated Public Turing test to tell Computers and Humans Apart”. And yes, Turing in CAPTCHA is the same as Turing the mathematician of the 1940’s, though the original Turing test was a human against a machine test not the other way round!

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