Quantum Mechanics Using Back-of-the-envelope Calculations

• YIP Chung On

1. 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 Schrödinger equation for a one-dimensional particle moving in a combination of a harmonic potential of frequency 𝜔 and a quartic potential of strength 𝛽. 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.

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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.

2. 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.

3. 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.

1. Harmonic oscillator alone

Consider the Schrödinger equation for one-dimensional particle moving in a harmonic potential of frequency 𝜔,

where 𝑚 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,

2. Quartic oscillator alone

Consider the Schrödinger 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,

3. Harmonic-quartic oscillator

Consider the Schrödinger 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 â„° 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,

4. SOFTWARE COMPARISON

5. DISCUSSION

6. REFERENCES

1. M. Olshanii, Back-of-the-Envelope Quantum Mechanics, 1^st ed. (World Scientific, 2013)
2. Quantum harmonic oscillator. Retrieved Feb 1, 2015, from

https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

3. Quantum Harmonic Oscillator. Retrieved Feb 1, 2015, from

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html