I recently wrote a Python script for plotting wave functions for the Quantum Harmonic Oscillator. This is one of the few problems in quantum mechanics that has a “nice” solution which can be obtained exactly. In simple terms the quantum harmonic oscillator refers to a quantum particle that is confined by a potential whose strength is proportional to the square of the distance away from the equilibrium position (i.e. the position where there is no force from the potential acting on the particle). A concrete physical situation where this applies might be an electron that is in a uniform magnetic field. There are many other physical situations that can either be informed by or very well approximated by the harmonic oscillator model.

Below are some plots of the wavefunction for a particle in a harmonic oscillator potential. Here *n* denotes the “energy level” where *n = 0* corresponds to the ground state of the system. The energy level takes on discrete (or “quantized” hence the name “quantum” mechanics) values. The energy of a particle at the *n*-th energy level is

where *ħ* stands for Planck’s constant and *ω* stands for the natural angular frequency of the oscillator (this is related to the proportionality constant for the potential described in the first paragraph). Interestingly the lowest possible energy for a particle confined to this potential (set *n* = 0 in this equation) is not zero! This means it possesses so-called “zero point” energy which is something pretty weird that differs from potentials in the macroscopic world.

A wavefunction is a special mathematical object that encodes information about the dynamics of a quantum system. In general wavefunctions are not directly measurable, and many view them as a mathematical tool we use to represent quantum systems. The wavefunction can be used to compute the probability that, for example, the position of the particle will be within a particular range of values. So without any further adieu, here are the plots:

This is the “ground state” wave function for the quantum harmonic oscillator. The shape of this plot is called a Gaussian. Note that these plots are done in so-called “natural units” (where *c = ħ = 1*) that simplify the calculations a bit.

This is the first excited state wave function, note that it has two “bumps.”

Just for fun, here is the 134th excited state. It’s quite pretty. Notice the larger bumps on either end. As *n* gets large, we expect the oscillations in the middle to sort of “wash out.” These two large bumps on the ends correspond to the classical turning points for the harmonic oscillator. From Wikipedia:

As the energy increases, the probability density becomes concentrated at the classical “turning points”, where the state’s energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.

Here is the Python source code to generate these plots. You will need Python 2.7, NumPy, and matplotlib for the script to run.

I have been exploring for a little bit for any

high-quality articles or blog posts in this kind of area .

Exploring in Yahoo I eventually stumbled upon this site.

Studying this information So i am happy to express that I have an incredibly just right uncanny feeling I

found out just what I needed. I such a lot surely will make sure to do not disregard this web site and provides it a

glance on a constant basis.

Cats are wonderful!

Firstly, TSO support replies inside seconds!

This text shows you a step-by-step guide to how one can eliminate black mould in a bathroom or carpet by using natural, non-poisonous merchandise.

each time i used to read smaller articles which

also clear their motive, and that is also happening

with this piece of writing which I am reading at this time.

My search didn’t give an answer right away. Had to do some hunting and naturally the answer was good and easy.

I don’t care if you contemplate yourself a ‘veteran’ or not.

The company added that in busy journey intervals, wait occasions might double at the bigger airports

for these already ready 30 to 40 minutes.

Specified remote connectivity test(s) to Microsoft Lync server

failed.

In fact, look around on varied forums and you’ll probably realise quite quickly

that they have a really poor popularity for

his or her assist service.

Tsohost have been offering prime quality, feature packed hosting companies to the UK, and overseas, since

2003.

95. If for almost any reason, you are not satisfied, and you are inside the 14 days of date of purchase,

simply call UK Toll Free 0-808-101-3444 (24 hours per day 7

days for each week) and request a return authorization amount and instructions to return the product.

All I can say is this is the corporate that makes me

trust customer service can be a pleasant and helpful expertise.

This stuff is amazing!

Just lately got an e-mail explaining why that they had a couple of minutes downtime on one in all their datacentre servers which was nice as standard.

La importancia de Glutatión en el programa de Dr .

OZ – Immunocal.

Funnel plot of the mean distinction in body weight reduction trials associated with HCA, plotted against sample dimension.

In writing, the facts look very appealing, nevertheless it’s put to the test, the results are usually

less than impressive.

I have dropped twenty-eight lbs since June 11 with no exercising.

It’s important to study all the instructions on the label cautiously.

The dosage should be divided straight into three portions,

which need to be used half an before every

dinner.

Chuah LO, Yeap SK, Ho WY, et al. In vitro and vivo toxicity of Garcinia or even hydroxycitric acid:

a review.

Our intensive study model starts with extensive security studies.

I see you don’t monetize your site, i’v got idea how to earn some

extra money using one simple method, just search in google for;

money making ideas by Loocijano

I am in fact delighted to read this web site posts which

includes lots of helpful data, thanks for providing these information.

Pretty part of content. I just stumbled upon your web site

and in accession capital to assert that I get in fact enjoyed account your blog posts.

Anyway I will be subscribing to your augment or even I success you get admission to

constantly fast.

Hello there! I could have sworn I’ve visited this

website before but after looking at some of the posts

I realized it’s new to me. Anyways, I’m certainly pleased I stumbled upon it and I’ll be book-marking it and checking back frequently!

臨海斎場でお葬式をしたいとお考えのあなたへ。

安心葬儀社ををご紹介しています。

急な遺体搬送・臨海斎場・お葬式・葬儀式場は電話1本で即見積り、最短30分でお迎えに！

まずしなければならない事はご遺体を搬送させることです。

いったん安置すれば検討できる時間が出来ます。

臨海斎場で家族葬をする時のポイントは？先ずお見積りをご説明します。

ご家族の気持ちや故人の想いもきちんとヒアリングを行います。

私たちは聞き取りを大事にしています。

ヒヤリングの中で、ご家族が気付かない気持ちなどを感じ取れるよう、日々鍛錬しています。

その中で各ご家庭に最適なプランを説明させていただきます。

臨海斎場で葬儀を行えば葬儀費用を抑えることが出来ます。

火葬場が併設しているので、交通費も節約できます。

お見積り内容に納得してはじめて葬儀の準備に入ります。

都心部では特に葬儀までに火葬場の日程調整で数日待つことがほとんどです。

故人を病院から搬送・安置させ、気持ちをいったん落ち着ける事が大事です。

お葬式を終えたとき、ありがとうと言ってもらえるように。

大事なのは葬儀料金の安さではなく、大切な人の気持ちを形にすることです。

引越しを業者にお願いする場合、衣紋掛けにかかっているコートなどの衣類を事前に外して畳んだりしておく必要はありません。大家さんもそれを感じており、何度となく苦言をていしたようですが、「飼養していない」の一筋だそうです。

引っ越しの際、業者選びは大事です。「どこでもいい」という訳には決していきません。そして、「選ぶ基準がわからない」「どこがいいのか判断できない」ということでしたら世間によく知られている大手業者を選ぶのが安心できるのではないでしょうか。

あえて業者名を挙げてみますと、テレビコマーシャルで見かけることも多い、アリさんマーク引越社のような会社でしょうか。名のある大手ということで、利用人数が桁違いです。

This is the perfect web site for anybody who would like to find out about this

topic. You know so much its almost hard to argue

with you (not that I actually would want to…HaHa). You certainly put a brand new spin on a subject that has been written about for

decades. Excellent stuff, just wonderful!