Quantum Mechanics Using Back-of-the-envelope Calculations

Quantum Mechanics Using Back-of-the-envelope CalculationsYIP Chung OnINTRODUCTIONCalculations in quantum mechanics be very often lengthy and mathematically involved, and any(prenominal) problems argon 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 auspicate. We choose the ground state problem of a harmonic- biquadratic oscillator to perform an analytical enter, as it is a common and routineful quantum mechanics problem. Then we use a computer softw ar, Mathematica to solve differential equations numerically, and compare the solutions with the back-of-the-envelope estimate.Above is the Schrdinger equation for a one-dimensional particle moving in a combination of a harmonic probable of frequency and a quartic potential of strength . The study of ground state problem of a harmonic-quartic problem is important, as it is a typi cal 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, some other 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 tall(a) 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.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.First we identify the jumper lead units of measurement for the problem, which means the minimal compulsive of units enough to describe all the commentary 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 racing shell 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 determine d.Finally we extend the unknown quantities as a multi-power-law of principal weighing machines, 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.SOLVEBefore 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 believe the Schrdinger equation for one-dimensional particle moving in a harmonic potential of frequency ,where is the particles mass.Find the ground state energy. lead story unitsunit of length , unit of energy Input parameters and their unitswhere , and To derive the surmount of length, let us represent the scale asThe units of areTo derive the scale of energy, let us represent the scale asThe units of are response for the unknownwhere const is a number of the order of unity. It s precise value isinaccessible for dimensional methods. Recall 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 s cale 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 COMPARISONDISCUSSIONREFERENCESM. 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