determination of the quantum mechanical energy eigenstates...

Determination of the Quantum Mechanical Energy Eigenstates of a Nondegenerately

Perturbed Harmonic Potential Well

Kevin M. Berg PHYS 472

7 December 2011

ABSTRACT

A numerical program was created to evaluate the probability distributions of energy

eigenstates using a Numerov algorithm. Using this program, and by observing the

behavioral trend of probability functions about the energy eigenstates, the first six

eigenstates of the potential V(x) = mw2x2/2, |x| > 1; mw2/2, |x| ≤ 1 were determined to be

.69535190(1)wh, 1.5478178(1)wh, 2.57075103(1)wh, 3.54959056(1)wh,

4.54322461(1)wh, and 5.54342249(1)wh. These eigenstates, based on their probability

distributions, were determined to correspond to the original energy eigenstates, given by

En = (n + ½)wh, associated with n = 0, 1, 2, 3, 4, and 5. We therefore conclude that the

perturbation is nondegenerate in nature.

I. INTRODUCTION

This project aimed to analyze the energy eigenstates of a nondegenerate perturbation of a

more traditionally well-known potential function, the harmonic potential. We have

chosen to evaluate a potential function of the form

V(x) = mw2x2/2, |x| > 1; mw2/2, |x| ≤ 1 (1)

where m is the mass of the particle, w is the frequency constant, and x is the distance from

the center of the potential. By construction, this potential function is continuous for all

possible values of x.

Using the Numerov algorithm, we were able to calculate the first six energy eigenstates

of this potential to be .69535190(1)wh, 1.5478178(1)wh, 2.57075103(1)wh,

3.54959056(1)wh, 4.54322461(1)wh, and 5.54342249(1)wh, where h is the reduced form

of Planck’s constant. These solutions correspond to the first, second, third, fourth, fifth,

and sixth solutions of the unperturbed harmonic potential, which is consistent with the

expected results of a nondegenerate perturbation.

II. BACKGROUND

The general formulation of a nondegenerate perturbation – that is, a perturbation that

does not result in two energy eigenstates with the same value – is well known and can be

explicitly calculated to a high degree of accuracy given sufficient time and resources.2

However, such calculations quickly become tedious as they rely on increasingly

complicated equations to compute.3

The time-independent Schrödinger equation can be solved numerically using the

Numerov algorithm, a sixth-order numerical technique which can be combined with a

half-integer mesh to evaluate symmetric potentials.4 As the harmonic potential is

symmetric by construction1, it should be possible to evaluate a symmetrically perturbed

harmonic potential by the techniques outlined in Forcrand and Werner.4 Using this

algorithm, it is possible to determine the energy eigenstates of a degenerately perturbed

potential while avoiding the very lengthy calculations that would ordinarily be required.

The harmonic potential is one of the standard potentials taught in introductory quantum

mechanics courses and is often used to approximate the local minima of arbitrary

potential functions.1 Via perturbations, this approximation can be made more accurate,

and can even be tailored to suit the exact nature of the region about the minima. Our

specific perturbation accounts for a region of symmetrical flatness in the center of the

approximated potential, such as might be seen in a more realistic approximation of a

potential well.

The energy eigenstates of an unperturbed harmonic potential have the form

En = (n + ½)wh (2)

where h is the reduced form of Planck’s constant. It was predicted that the energy

eigenstates of the perturbed function will also have an explicit w dependence, and will be

close to the original energy eigenstates. The wave functions and probability distributions

associated with these eigenstates have been found analytically.5

III. COMPUTATIONAL WORK

A. Preliminary

Based on the graphical behavior of the probability functions, it was determined that an

intelligent guess and check method would be ideal for locating the energy eigenstates of a

particular problem. The computer on which the calculations were performed had

sufficient RAM for step sizes as small as .00001 and thus smooth approximations were

possible without taxing the hardware.

B. Computation Tests

A “bare-bones” setup was created to use the Numerov algorithm in an infinite square well

(See Appendix A). The output was then tested for a square well of dimension “4.0”

relative to the chosen beta value, 1.0. These values were chosen to simplify calculations.

Eigenvalues were then of the form n2π2/16. Even functions were designated the initial

condition ψ(-.5Δx) = ψ(.5Δx) = 1, while odd functions were provided with ψ(0) = 0,

ψ(+Δx) = Δx. Because of the manner in which the algorithm is calculated, only the right

side of the resulting wavefunction is shown. However, by construction the wavefunctions

are symmetrical about the x-axis.

The graphical results obtained for the first four eigenstates are provided in Fig. 1 – 4.

Notably, the sign is flipped on half the results as a result of our choice of initial

conditions, this problem, however, can be eliminated via squaring the wavefunctions.

Fig. 5 demonstrates that wave functions will form improperly given incorrect even/odd

conditions, which is useful as the energy eigenstates of the wavefunctions for the chosen

potential will be unknown. In addition, these functions are not normalized, but this is not

necessary to obtain qualitative information about the probability distributions.

C. Theory Tests

The program was then altered (See Appendix B) so as to evaluate the energy eigenstate

wavefunctions associated with the potential given in (1). These eigenstates were located

by a sophisticated means of guess and check – it was noticed that an under-approximation

of an eigenstate energy would result in the wavefunction eventually tending towards

infinity, and that an over-approximation would result in the wavefunction tending

towards negative infinity. Because of this, it became possible to sieve eigenstate guesses

so as to close in on solutions. Using this technique, we were able to calculate the energy

eigenstates to within eight decimal places of accuracy. In addition, it was noticed that the

energy would flip between an under-approximation and an over-approximation at those

states where the wavefunction is expected to have an overall negative sign relative to the

Numerov predictions – i.e. where an even eigenstate is expected to have a negative value

at the origin.

The value of beta is set to .26246 (eV angstroms2)-1 and thus the mass of the particle to 1

electron mass. To simplify computations, w is set to 2 eV1/2 angstroms-1 electron mass-1/2.

The first six energy eigenstates were determined to be .69535190(1)wh, 1.5478178(1)wh,

2.57075103(1)wh, 3.54959056(1)wh, 4.54322461(1)wh, and 5.54342249(1)wh. Graphs of

the wavefunctions associated with these eigenstates are provided in Fig. 6 – 11. The

graphs of the probability distributions are provided in Fig. 12 – 17.

IV. DISCUSSION OF RESULTS

Based on the fact that no energy eigenstates appear to have folded into other eigenstates,

it can be safely concluded that the perturbed potential (1) is a degenerate perturbation. In

addition, the probability functions located are identical to the original probability

functions associated with the n = 0, 1, 2, 3, 4, and 5 states, with associated eigenstates

within .2wh of the original eigenstates. This indicates that those states have been

perturbed slightly upward, as would traditionally be expected.

It would theoretically be possible to increase the accuracy of the results given using the

techniques used, in a manner that would only be limited by the computational power of

the hardware used. Our results were calculated to eight decimal places of accuracy as a

result of RAM usage becoming an issue at around the tenth decimal position of

calculations due to the size of the arrays that would be required for very accurate results.

V. CONCLUSIONS

The first four energy eigenstates of the perturbed potential given in (1) are indeed

.69535190(1)wh, 1.5478178(1)wh, 2.57075103(1)wh, 3.54959056(1)wh,

4.54322461(1)wh, and 5.54342249(1)wh as there is no reason to suspect an error in

calculation, nor that an eigenstate has been overlooked. This perturbation is indeed

nondegenerate as no energy eigenstates have collapsed into other eigenstates.

Due to the complicated shape of these probability functions, and the general complexity

of the original wavefunctions associated with the harmonic potential, it is unlikely that an

analytical solution to the perturbed potential presented would be feasible to calculate

excepting a long period of computation time. There is no apparent reason to suspect that

such a calculation would be inconsistent with the results presented here.

It would be pertinent to apply the Numerov algorithm calculation and energy eigenstate

finding method presented here to other symmetric potentials in the future. We suspect

that the majority of trivially symmetric potentials would be solvable using such a

technique, as the Numerov algorithm is quite powerful. It would also be pertinent to

discover a means of predicting whether a function should have a positive or negative

slope, so that the actual wavefunctions can be found using this technique.

VI. SOURCES CITED

1Griffiths, D. J. (2005). Introduction to quantum mechanics. (Second ed., pp. 40-59).

Upper Saddle River, NJ: Pearson Prentice Hall.

2Griffiths, D. J. (2005). Introduction to quantum mechanics. (Second ed., pp. 249-257).

Upper Saddle River, NJ: Pearson Prentice Hall.

3Griffiths, D. J. (2005). Introduction to quantum mechanics. (Second ed., p. 256). Upper

Saddle River, NJ: Pearson Prentice Hall.

4de Forcrand, P. (2009). Computational quantum physics. Retrieved from

http://www.itp.phys.ethz.ch/education/lectures_fs09/cqp/Script1

5Griffiths, D. J. (2005). Introduction to quantum mechanics. (Second ed., p. 58). Upper

Saddle River, NJ: Pearson Prentice Hall.

FIGURES Fig. 1 – Test potential; energy eigenstate 1, nonnormalized, no sign correction

Fig. 2 – Test potential; energy eigenstate 2, nonnormalized, no sign correction

Fig. 5 – Test potential; energy eigenstate 1 evaluated as an odd solution to demonstrate

explosive properties of the algorithm

Fig. 6 – Perturbed potential; energy eigenstate .69535190(1)wh, nonnormalized

wavefunction. X-axis units are in angstroms.

Fig. 7 – Perturbed potential; energy eigenstate 1.5478178(1)wh, nonnormalized






Fig. 12 – Perturbed potential; energy eigenstate .69535190(1)wh, nonnormalized

probability function. X-axis units are in angstroms.

APPENDIX A

TEST POTENTIAL PROGRAM

/* Kevin Berg qmproject.c Last modified on 11/23/2011 Approximates the appearance of the wavefunction for a provided potential function at a provided energy eigenstate for symmetric potential functions. Change the value of is_even if an odd or even function is suspected. to compile: gcc -c qmproject.c Then gcc qmproject.o -lgsl -lgslcblas to run: ./a.out to create a graphable output file ./a.out > [desired name] */ #include <stdio.h> #include <stdlib.h> #include <math.h> #include <gsl/gsl_sf.h> #include <gsl/gsl_complex.h> #include <gsl/gsl_complex_math.h> const int XGRID = 5000; int main(void){ double phi[XGRID+1]; double V[XGRID+1]; //Potential double E; //Energy state double beta = 1.0; //2*m/h_bar^2 double leftside = 0.0; //Calculation starts from center double stepsize = 2.0/XGRID; //X ranges from 0 to 1 int n; //Counting variable int is_even = 0; //1 for even, 0 for odd.

for(n=0;n<=XGRID;n++){ V[n] = 0.0; //Infinite Square Well } E = 1.0*M_PI*M_PI/16.0; //Energy state guess if(is_even==1){ phi[0]=1.0; phi[1]=1.0; } //phi(0)=1.0 for even else{ phi[0]=0.0; phi[1]=stepsize; }//odd starting condition double coefficient_1, coefficient_2, coefficient_3; //Coefficients from evaluation steps for(n=1;n<XGRID;n++){ coefficient_1 = (1.0 + ((stepsize*stepsize)/12.0)*beta*(E-V[n+1])); coefficient_2 = 2.0*(1.0 - 5.0*((stepsize*stepsize)/12.0)*beta*(E-V[n])); coefficient_3 = -1.0*(1.0 + ((stepsize*stepsize)/12.0)*beta*(E-V[n-1])); phi[n+1] = (coefficient_2*phi[n] + coefficient_3*phi[n-1])/coefficient_1; //Numerov algorithm } for(n=0;n<=XGRID;n++){ if(is_even==1){ printf("%le %le\n",(n-0.5)*stepsize,phi[n]); } //Even else{ printf("%le %le\n",n*stepsize,phi[n]); } //Odd } }

APPENDIX B

PERTURBED POTENTIAL PROGRAM

/* Kevin Berg qmproject.c Last modified on 12/5/2011 Approximates the appearance of the wavefunction for a provided potential function at a provided energy eigenstate for symmetric potential functions. Change the value of is_even if an odd or even function is suspected. to compile: gcc -c qmproject.c Then gcc qmproject.o -lgsl -lgslcblas to run: ./a.out to create a graphable output file ./a.out > [desired name] */ #include <stdio.h> #include <stdlib.h> #include <math.h> #include <gsl/gsl_sf.h> #include <gsl/gsl_complex.h> #include <gsl/gsl_complex_math.h> const int XGRID = 5000; int main(void){ double phi[XGRID+1]; double V[XGRID+1]; //Potential double E; //Energy state double beta = .26246; //2*m/h_bar^2, .26246 (eV angstroms^2)^-1 double m = 1.0; //mass, electron mass double w = 2.0; //Frequency, 2.0 eV^.5 angstroms^-1 electron mass^-.5 double stepsize = 8.0/XGRID; //X ranges from 0 to 8 angstroms int n; //Counting variable int is_even = 0; //1 for even, 0 for odd.

for(n=0;n<=XGRID;n++){ if(n*stepsize<=1.0){ V[n] = .5*m*w*w; // V(x) = .5mw } else{ V[n] = .5*m*w*w*((n*stepsize)*(n*stepsize)); // V(x) = .5mwx^2 } } E = 2.57075103*w/sqrt(beta/(2.0*m)); //Even //3.54959056*w/sqrt(beta/(2.0*m)); //Odd //5.54342249*w/sqrt(beta/(2.0*m)); //Odd //4.54322461*w/sqrt(beta/(2.0*m)); //Even //1.54781718*w/sqrt(beta/(2.0*m)); //Odd //.69535190*w/sqrt(beta/(2.0*m)); //Even //Energy state guess if(is_even==1){ phi[0]=1.0; phi[1]=1.0; } //phi(0)=1.0 for even else{ phi[0]=0.0; phi[1]=stepsize; }//odd starting condition double coefficient_1, coefficient_2, coefficient_3; //Coefficients from evaluation steps for(n=1;n<XGRID;n++){ coefficient_1 = (1.0 + ((stepsize*stepsize)/12.0)*beta*(E-V[n+1])); coefficient_2 = 2.0*(1.0 - 5.0*((stepsize*stepsize)/12.0)*beta*(E-V[n])); coefficient_3 = -1.0*(1.0 + ((stepsize*stepsize)/12.0)*beta*(E-V[n-1])); phi[n+1] = (coefficient_2*phi[n] + coefficient_3*phi[n-1])/coefficient_1; //Numerov algorithm } for(n=0;n<=XGRID;n++){ if(is_even==1){ printf("%le %le\n",-1.0*(n-0.5)*stepsize,phi[n]*phi[n]); } //Even else{ printf("%le %le\n",-1.0*n*stepsize,phi[n]*phi[n]);

} //Odd } for(n=0;n<=XGRID;n++){ if(is_even==1){ printf("%le %le\n",(n-0.5)*stepsize,phi[n]*phi[n]); } //Even else{ printf("%le %le\n",n*stepsize,phi[n]*phi[n]); } //Odd } }

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