Quantum Mechanics Using Back-of-the-envelope Calculations

Quantum Mechanics Using Back-of-the-envelope Calculations YIP Chung On INTRODUCTION Calculations in quantum mechanics are very often lengthy and mathematically involved, and some problems are impossible to get an analytical solution. Our goal, rather than obtaining an exact solution, we try to analyze a problem in quantum mechanics using dimensional analysis and provide a back-of-the-envelope estimate. We choose the ground state problem of a harmonic-quartic oscillator to perform an analytical estimate, as it is a common and useful quantum mechanics problem. Then we use a computer software, Mathematica to solve differential equations numerically, and compare the solutions with the back-of-the-envelope estimate. Above is the Schrà ¶dinger equation for a one-dimensional particle moving in a combination of a harmonic potential of frequency à °Ã‚ Ã…“† and a quartic potential of strength à °Ã‚ Ã¢â‚¬ ºÃ‚ ½. The study of ground state problem of a harmonic-quartic problem is important, as it is a typical system in reality. There are two special cases for a harmonic-quartic oscillator; one is when the strength of the quartic is very small, it becomes a harmonic oscillator, another one is when the strength of the harmonic potential is very small, it becomes a quartic oscillator. Harmonic oscillator is one of the most important model systems in quantum mechanics, one of the examples are simple diatomic molecules such as hydrogen and nitrogen. It is one of the few quantum-mechanical systems which we are able to get an exact, analytical solution. Also, many potentials can be approximated as a harmonic potential when the energy is very low, this provides a great help when studying some very complicated systems. While in reality, it is unlikely that a system is purely harmonic, as most of the time there would be more than one potential acting in a system. So it is important to study a system with multi-potentials, and a harmonic-quartic oscillator, which includes a harmonic potential and a quartic potential, is a good example of that. Our goal, in this project, is to estimate the ground state energy of a harmonic-quartic oscillator making use of back-of-the-envelope calculations, which means that we only involve very few mathematical calculations in our estimate. To specify, we perform dimensional analysis on the equations of the problem we concern, then we compare the results of our estimate with the numerical solution we get from Mathematica, a computer software, to see how close can our estimate get. METHOD We attempt to use dimensional analysis to estimate the ground state energy of the harmonic-quartic problem, and here would be the procedures we would take to perform a dimensional analysis for finding the ground state energy. First we identify the principal units of measurement for the problem, which means the minimal set of units enough to describe all the input parameters of the problem. For this problem, we choose the units of length, [], and energy, [, these two are often chosen in stationary problems in quantum mechanics. Then we identify the input parameters and their units in terms of the chosen principal units. For each of the principal units, we choose a scale which is a combination of the input parameters measured using their units. We may need to determine the maximal set of independent dimensionless parameters: the set will include only the parameters that are generally either much greater or much less than unity. These include both the dimensionless parameters present in the problem and the dimensionless combinations of the dimensionful input parameters. If the set is empty, the unknown quantities can be determined almost completely, i.e. up to a numerical prefactor of the order of unity. If some dimensionless parameters are present, the class of possible relationships between the unknowns and the input parameters can be narrowed down, but the order of magnitude of the unknown quantities cannot be determined. Finally we express the unknown quantities as a multi-power-law of principal scales, times an arbitrary function of all dimensionless parameters, if any. If no dimensionless parameters are present, the arbitrary function is replaced by an arbitrary constant, presumed to be of the order of unity. SOLVE Before we solve the harmonic-quartic oscillator problem, we would first go through the two special cases, the harmonic oscillator alone and the quartic oscillator alone. Harmonic oscillator alone Consider the Schrà ¶dinger equation for one-dimensional particle moving in a harmonic potential of frequency à °Ã‚ Ã…“†, where à °Ã‚ Ã¢â‚¬ËœÃ… ¡ is the particle’s mass. Find the ground state energy. Principal units: unit of length [], unit of energy [ Input parameters and their units: [ [ where , and To derive the scale of length, let us represent the scale as The units of are [ To derive the scale of energy, let us represent the scale as The units of are [ Solution for the unknown: where const is a number of the order of unity. Its precise value is inaccessible for dimensional methods. Recall that the exact value of this constant is 1/2. Finally, Quartic oscillator alone Consider the Schrà ¶dinger equation for one-dimensional particle moving in a quartic potential of strength where is the particle’s mass. Find the ground state energy. Principal units: unit of length [], unit of energy [ Input parameters and their units: [ [ where To derive the scale of length, let us represent the scale as The units of are [ To derive the scale of energy, let us represent the scale as The units of are [ Solution for the unknown: Finally, Harmonic-quartic oscillator Consider the Schrà ¶dinger equation for one-dimensional particle moving in a combination of harmonic potential of frequency and a quartic potential of strength where is the particle’s mass. Find the ground state energy. Principal units: unit of length [], unit of energy [ Input parameters and their units: [ [ [ where , and To derive the scale of length, let us represent the scale as The units of are: [ We choose the scale associated uniquely with the harmonic oscillator, To derive the scale of energy, let us represent the scale as The units of à ¢Ã¢â‚¬Å¾Ã‚ ° are: [ We choose the scale associated uniquely with the harmonic oscillator, There exists a dimensionless parameter expressed as a product of powers of principal scales: The units of are: [ As is supposed to be dimensionless, There is an independent dimensionless parameter We choose a scale of parameter in order that the system can be solved Solution for the unknown: where is an arbitrary function. Finally, SOFTWARE COMPARISON DISCUSSION REFERENCES M. Olshanii, Back-of-the-Envelope Quantum Mechanics, 1st ed. (World Scientific, 2013) Quantum harmonic oscillator. Retrieved Feb 1, 2015, from https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator Quantum Harmonic Oscillator. Retrieved Feb 1, 2015, from http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

Social Inequality to Kill a Mockingbird

Social Inequality Think of social inequality. Is it fair for people to be ranked and divided simply because of their social status? Maybe you’re thinking of the high school social rankings but it’s more than that. People are being treated according to their social classes by the colour of their skin or their family background or financial status, all aspects that we do not have a choice in. Take the Royal family for instance. They are on the top of the pyramid, simply because of their status and wealth; people will treat them differently to how they would treat us commoners.They are deemed superior simply because they are the Royals, but it is only because they were born into that family with a silver spoon in their mouths. What about those homeless people on the streets? Would you treat them the same way you would treat the royals? In To Kill a Mockingbird, the Finches were a pretty well off family with Atticus being related to nearly everyone in town and his job as a lawyer provided sufficient money to support his family. The Cunningham family on the other hand, are poor famers that try to scrape along with what they have.They are ridiculed because of their lack of wealth, and because they were ‘Cunninghams’. Although the Cunninghams are poor, they are respectable because they find other ways to pay instead of money. Scout is unaware of such things as social standings, but is however, aware that the Cunninghams are poor; and points out rather helpfully to her first grade teacher Miss Caroline, that Walter Cunningham is ‘a Cunningham’ so he would never be able to pay back her quarter if she had lent it to him.Later Scout is disgusted by Walter’s eating manners, which leads to her being scolded by Calpurnia to not judge someone because of whether they were poor or not. Even if the Cunninghams were poor, they were nothing compared to the Ewells. They were typical ‘white trash’ characters that lived with horrible hygiene, uneducated and did not abide by the law. The Ewells were considered the worst of the worst with their uncouth manners and language. Most people avoided them but felt sorry for them just like Tom did for Mayella. However he helping her only led to landing him a court case and ultimately death.But even below the Ewells on the social pyramid is the black community. They are deemed even worse than the Ewells because they are Negroes. The colour of their skin decided their social status and people judged them simply because of that. They put up with far worse treatment than the Ewells or the Cunninghams. Social inequality just goes to show if people are labelled ‘Cunningham’ or ‘Ewell’ or ‘Negro’. It is pigeonholing people into someone might not be. Social status does not decide the people we are and people should be treated equally no matter what.

The Story of an Hour Essay Topics - What Is It?

The Story of an Hour Essay Topics - What Is It? The Secret to the Story of an Hour Essay Topics So I will just supply you with an extremely speedy demonstration. Inside this essay, you can demonstrate that you're so far more than just your grades and test benefits! As all booklovers should have the very same opinion here is a brief analysis of the story. We are going to take a look straight away. This is the ideal way you'll be able to prepare your very own unique academic paper and score the grades you have earned. There is an assortment of patterns that writers can use if writing an argumentative essay even though there isn't any particular model of organization. So far as design goes, the C310a is wholly gorgeous. Experience contributes to knowledge and wisdom. Follow this URL to find out more about what the Friends do and ways to get involved. Other folks argue it doesn't exist in any way. Check the many music forums. Ultimately, our objective is have 10,000 Crystal Earth Seeds in the ground to be able to create the most effective energetic grid possible. Go at your own pace and do some comparison shopping as if you would for any other item. We never understand what's going to happen next. When you go to a holy location, it's wonderful. Much of her writing is dependent on her own sufferings and is frequently full of irony that's very subtle. As an issue of fact, the entirety of the story would be quite different if it was written in the first-person viewpoint. Although it is really short, it is very rich and complete, and every word in it carries deep sense and a lot of meaning. Chopin's story is a significant instance of the literary device called situational irony. The greatest short story idea on the planet won't help you in the event that you don't write this, and a mediocre idea can be produced into an award winning story if it's written well. Finally there's dramatic irony which occurs whenever there's a contrast between the readers knowledge and the wisdom of the characters in the job. Oh, and should you join, here's the URL to mine if you would like to share your feedback! With the huge selection of themes presented throughout the story, hope is hard to identify. The Most Popular Story of an Hour Essay Topics Mrs. Mallard's heart condition plays a huge part of the entire story. He or she is said to suffer from heart trouble or heart disease. He or she dies at the end of the story. He or she knows that she can begin to live for herself. Mother Earth has been in the procedure for a wonderful Awakening. Death is employed as a theme within this story. The Story of An Hour appears to be among the brightest works, to find the feeli ng of Kate Chopin's writing style. The Story of an Hour is mainly a story of excellent irony. The Basic Facts of the Story of an Hour Essay Topics Here we should remind you to be sincere when responding to this question! The chart is a great chance to practice and go over the lesson's stated objectives. To the peak of the porch, to the peak of the wall! Ask yourself these very same questions! Louise's life provides no refuge for this type of joy, and the remainder of society won't ever accept it or understand it. What's more, you should describe the way in which this individual has shaped your personality. The woman becomes killed as a consequence of a heart failure due to over-excitement. At the exact same time, the sensation of obedience to be able to avoid divorce lives in her only at the start of their marriage. At the start of the story, her heart trouble is the first thing is mentioned about her. The exact first sentence wouldn't have been possible, and so the knowledge of her heart malady wouldn't have been revealed. Self awareness and identity is also a theme to consider in this quick story. She has heart issues, but they're presented very vaguely and without an authentic label of disease. What Everybody Dislikes About the Story of an Hour Essay Topics and Why This can demonstrate the reader a woman at this time may not even be mindful of exactly how much of herself she has to hold back when married. She'd have rather died young then to need to obey her husband for the remainder of her life. Unexpectedly, the feeling of joy was born inside her soul. Therefore, she's pretending inside her relationship.

Justinian And The Roman Empire - 1477 Words

The Byzantine Empire, though often forgotten among better-known empires, such as the Roman Empire, ran for nearly one thousand years, occupying the eastern half of what once was the Roman Empire. It engaged in trade, expansion, and decades of warfare. It also gave women a better status then men, valued and preserved Greek history, and eventually fell to the Ottomans. During the year 535, Justinian overthrew the Ostrogothic Kingdom, which had occupied Italy after the split and decline of the Roman Empire. Justinian was said to be the most ambitious emperor since Constantine, whose rule ended in 337. His goal in overthrowing the Ostrogothic Kingdom was to eventually reunify the Roman Empire. Belisarius, Justinian’s general, had already†¦show more content†¦Under the first emperor, Heraclius, the new empire struggled to keep its territory. Persia was conquering land rapidly, taking the empire’s territories in Syria and Palestine, along with Jerusalem. It took unti l 627 for Heraclius to gain a victory, recapturing Jerusalem from the Persians. It was not long, however, before Islam had sparked a desire for expansion in the Arabs. They quickly conquered Jerusalem, the entire Persian Empire, and headed to claim North Africa. In 677, the Arabs attempted to take Constantinople, the city along the port Byzantium was named for. Unlike Heraclius, the emperor during the second attempt on Constantinople had a better strategy, known as the â€Å"Greek Fire†. Leo III was able to not only defeat the Arabs at Constantinople, but also reclaim almost all of the Asia Minor, which was known as the Imperial Heartland for nearly three hundred years. Due to a stable government and a properly managed economy, the Byzantine Empire was able to last for quite a while. Emperors used their power correctly and there was a centralized government that helped keep the empire functioning even during rough times. Constantinople’s bureaucracy made sure everythin g ran smoothly. They regulated prices and wages, had systems for licensing, controlled the trade, oversaw schools, churches, observation of holidays, and even controlled the military and courts. The