Quantum Mechanics Using Back-of-the-envelope Calculations

Quantum Mechanics Using Back-of-the-envelope CalculationsYIP Chung OnINTRODUCTIONCalculations in quantum mechanism argon real a great deal prolonged and mathematically involved, and whatsoever problems be impossible to get an analytical solution. Our goal, preferably than obtaining an exact solution, we try to analyze a problem in quantum mechanics utilise dimensional analysis and provide a back-of-the-envelope aim. We carry the launch differentiate problem of a charitable- fourth power oscillator to put to death an analytical estimate, as it is a common and useful quantum mechanics problem. Then we use a figurer software, Mathematica to solve differential equations numericly, and compare the solutions with the back-of-the-envelope estimate.Above is the Schrdinger equation for a analogue tinge pathetic in a combination of a harmonical potential of frequency and a biquadratic potential of strength . The break down of cornerstone state problem of a harmonic-quartic problem is important, as it is a typical system in reality. There are two excess cases for a harmonic-quartic oscillator ane 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 heat content and nitrogen. It is one of the few quantum-mechanical systems which we are able to get an exact, analytical solution. Also, many another(prenominal) potentials can be approximated as a harmonic potential when the cleverness is very low, this provides a great help when field of honoring some very complicated systems.While in reality, it is unlikely that a system is rigorously 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-pot entials, 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 sinew of a harmonic-quartic oscillator making use of back-of-the-envelope calculations, which means that we wholly 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.METHODWe 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. inaugural we identify the principal unit of measurements of measurement for the problem, which means the minimal set up of units affluent to describe a ll the foreplay argumentations 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 home base 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 ofttimes greater or much less than unity. These include both the dimensionless parameters comprise 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 inpu t parameters can be narrowed down, but the order of magnitude of the unknown quantities cannot be determined. ultimately we express the unknown quantities as a multi-power-law of principal scales, times an despotic 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.SOLVEBefore we solve the harmonic-quartic oscillator problem, we would first-year go through the two special cases, the harmonic oscillator alone and the quartic oscillator alone.Harmonic oscillator alone realize the Schrdinger equation for one-dimensional particle moving in a harmonic potential of frequency ,where is the particles mass. comment the ground state energy.Principal unitsunit of length , unit of energy Input parameters and their unitswhere , and To arrive at the scale of length, let us make for the scale asThe units of areTo infer the scale of energy, let us repre sent the scale asThe units of areSolution for the unknownwhere const is a number of the order of unity. Its precise value isinaccessible for dimensional methods. discard that the exact value of this constant is 1/2.Finally,Quartic oscillator aloneConsider the Schrdinger equation for one-dimensional particle moving in a quartic potential of strength where is the particles mass.Find the ground state energy.Principal unitsunit of length , unit of energy Input parameters and their unitswhere To derive the scale of length, let us represent the scale asThe units of areTo derive the scale of energy, let us represent the scale asThe units of areSolution for the unknownFinally,Harmonic-quartic oscillatorConsider the Schrdinger equation for one-dimensional particle moving in a combination of harmonic potential of frequency and a quartic potential of strength where is the particles mass.Find the ground state energy.Principal unitsunit of length , unit of energy Input parameters and their unitswhere , and To derive the scale of length, let us represent the scale asThe units of areWe choose the scale associated uniquely withthe harmonic oscillator,To derive the scale of energy, let us represent the scale asThe units of areWe choose the scale associated uniquely withthe harmonic oscillator,There exists a dimensionless parameter expressed as a product of powers of principal scalesThe units of areAs is supposed to be dimensionless,There is an independent dimensionless parameterWe choose a scale of parameter in order that the system can be solvedSolution for the unknownwhere is an arbitrary function.Finally,SOFTWARE COMPARISON interchangeREFERENCESM. Olshanii, Back-of-the-Envelope Quantum Mechanics, 1st ed. (World Scientific, 2013)Quantum harmonic oscillator. Retrieved Feb 1, 2015, fromhttps//en.wikipedia.org/wiki/Quantum_harmonic_oscillatorQuantum Harmonic Oscillator. Retrieved Feb 1, 2015, fromhttp//hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html