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3D Anharmonic Oscillator

The project goals are to calculate numerically the ground, first and second excited energy of the 3 dimensional anharmonic oscillator as a function of the perturbation parameter !. We tried two approaches to solve this problem; the first one uses the fact that the potential energy is a radial function and we can separate variables and reduce the problem to only one dimension only using the radial part. The second one was coding the full potential on Cartesian three dimensions. We found that the 1D version was much more complicated than first expected. We collected data from the 3D version of the harmonic oscillator and then will make variations to the potential to show the application for the anharmonic oscillator. The main goal was to achieve this by using Metropolis Hastings method in union with the Monte Carlo method.

I. INTRODUCTION

The Harmonic oscillator is one of the simplest systems to model when transitioning to a more complicated system. Since quantum motion is often quite complicated it is helpful to be able to model approximations. There is a potential energy, and that potential is dependent upon the square of the displacement from equilibrium. For the Harmonic Oscillator in 3 dimensions, with potential ! = !

!!!!!! the energies

are:

!! = 2! + ! + !! ℏ!!!!!!!!!!!! = 2! + !!(1)

With k the quantum number for the radial part, and l one

of the quantum numbers for the angular part (spherical harmonics !!!). A diatomic molecule vibrates much like two masses on a spring. For the diatomic molecule the natural frequency is of the same form as the classical simple harmonic oscillator. The main difference is the “zero point vibration” of the n = 0 ground state which implies that molecules are never completely at rest even at absolute zero temperature [1].

Approximations based on the Quantum harmonic

oscillator are the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. In real systems, energy spacings are equal only for the lowest levels where the potential is a good approximation of the “mass on a spring” type harmonic potential. The anharmonic terms, which appear in the potential for a diatomic molecule, are useful for mapping the detailed potential of such systems [1].

We will start solving the Harmonic oscillator because we

have all the analytic solutions and we can check if our code is working, after that, we basically change the potential and add the quartic term for the anharmonic oscillator.

We thought that it would be interesting to use the Markov

chain Monte Carlo to work on our project, since “smart” Monte Carlo methods work for both classical and quantum systems.

II. EXPERIMENT

In our experiment, we consider the anharmonic oscillator described by the potential !!!!

!!! + !!"!!. We worked on the harmonic oscillator first. To verify that

we are getting correct data, we plotted the histogram of the ground state ! ! !and compare it with the analytic wavefunction. We have been working with the same conditions as the first assignment, so we want to get an acceptance rate that is in the neighborhood of 50%. At this point we have been between 47-50%.

Our code was written to have 10! warm up sweeps, then

do 10! measurement sweeps with 100 sweep separation between measurements. Therefore our plots show every 100th sweep.

There are some differences between the 1-D code for the

1-D harmonic oscillator and the 3-D for the 3 dimensional oscillator. The 3-D code basically does the metropolis process on all the 3 dimensions, x, y and z one by one. The potential only depends on r, and the kinetic energy is just the sum of the changes between the 3 coordinates.

Just as the 1-D case, we only need to calculate the action

from 3 points, because we are changing only one point at a time, so the action is only a function of !! , !!!!and !!!!. Note that in this case, !! is a vector.

For the 1-D code, we separate variables, and end with the

radial equation:

− ! ℏ!!!!!!!! + !!"" ! ! ! = !!"(!) (1)

with !!"" ! = ! ! + !ℏ

!! !!!!!!! .

So everything reduces to the same 1-D code on the

assignment 2 by only changing the potential, at least until calculating the kinetic energy.

For the anharmonic oscillator we will use perturbation

theory on the harmonic oscillator and compare those values with the numerical ones.

To get the excited energies we used the correlator method,

for a code with a certain value of !, and plotted – ln!(! τ!!! τ ) v.s. τ and fitted a line to the linear part of the correlators, to

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get the ΔE. If we use <x(t)x(t+τ)> correlator we will get the first excited energy and if we use the <x2(t)x2(t+ τ )> correlator we will get the second excited energy. An example of the plot we use to

After that we made a code that changed the value of !,

and calculate ΔE for each value (it calculates the variance of the points and it fits a line to a collection of points with small variance, so that we use the linear part), going from ! =0.01to ! = 0.5!with 50 intervals.

III. RESULTS

The 3-D code works very well, but we dropped the 1-D

code because of the difficulties with the kinetic energy.

3.1 1-D code. We first made the code exactly the same (but with the

Veff) as the harmonic oscillator, but this was not correct when calculating the expectation values of the energy, because we are integrating over the whole space, so we had a missing 4!!! on every expectation value.

The kinetic energy is complicated because we have to take into account the movement on the angular directions, and also it’s not trivial the way to calculate it.

3.2 3-D code.

We ran a couple of analytic versions, both for the

harmonic and anharmonic through Mathematica to create a point of comparison to ensure that our program was working correctly.

The analytical radial part of the wavefunctions are plotted

on fig. 2, while on fig.1 is plotted only the ground state.

Fig. 1: Expected radial distribution for the ground state (l=0) of the harmonic oscillator.

Fig. 2: Unperturbed wave functions of the harmonic oscillator.

The code gives us the ground state energy and the

trajectories. We set ℏ, ! and the mass equal to 1. The numerical wavefunction is obtained from making a

histogram of the trajectories and the one we got is on fig. 3.

Fig. 3: Histogram of the radial wavefunction.

If we plot both the analytical and numerical distributions on the same graph, we can see that the code is generating the correct wavefunction. This is illustrated on fig. 4. The wavefunctions are not normalized because the main interesting thing is the shape of them.

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Fig. 4. Comparison of the numerical wavefunction (on green) with the

analytical wavefunction (on blue). Note that these wavefunctions are not normalized.

Analytically, for the ground energy, according to

equation (1), and with all the constants equal to 1, we should expect an energy value of 1.5.

The Results of the energy measurement every 100

trajectories are on fig. 5.

Figure 5: Computation of the energy for the ground state

We can take the running average of the energy, to clearly

visualize the value, as it’s shown on fig. 6.

Figure 6: Running average of the energy.

We can see it stabilizes around 1.5. We see now that our

code works, so we changed the potential and added the quartic term. We have to change the potential, but also we have to change the kinetic energy, because we used the virial theorem and that depends on the potential.

Since we know the code works, we obtained the first

excited energy for the Harmonic Oscilator as a function of lambda, plot that we show on fig. 7.

Figure 6: Change of the excited energy as a function of lambda.

And we did, with the method described on the experiment

section, a plot of the dependence of the energy as a function of ! for the ground, first and second exited energy, which we show on figure 7.

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Figure 7: Numerical change of the energies as a function of lambda.

We see that these results agree with our perturbation

analytical work shown on fig. 8, and in fig. 9 we plotted them together.

Figure 8: Theoretical change of the energies as a function of lambda.

Figure 9: Both theoretical and numeric change of the energies as a function of lambda.

As we can see, for the first excited energy we get correct results, but for the first exited energy as lambda gets bigger the theory differs with the numerical data.

IV. CONCLUSION

Our project will show the universality of the Markov

Chain Monte Marlo method in use with quantum systems even systems with no symmetry.

The 3D code works correctly and gives all the information

of the ground state from any quantum system. Then the correlator code gets the value of the first and second excited energy for each of the 50 lambdas.

The potential can be changed in the code, so the same

code can solve any potential, regardless of the potential.

V. ACKNOWLEDGEMENTS We would like to thank Professor xx and our teaching assistant xx for their support.

VI. REFERNCES [1] http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html [2] Barrow, G. M., The Structure of Molecules, W. A. Benjamin, 1963. [3] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, , Phys. Rev. 108 , 1175 (1957) [3] Blatt, Frank J.,Modern Physics, McGraw-Hill, (1992), Section 11.4 reference for liquid hydrogen. [4] Beiser, Arthur, Concepts of Modern Physics, 5th Ed., McGraw-Hill, 1995 [5]http://physics.ucsd.edu/students/courses/spring2013/physics142/Assignments/Assignment2.pdf [6]http://physics.ucsd.edu/students/courses/spring2013/physics142/Lectures/Lecture11/Lecture11.pdf [7] Samy H. Aly, Egypt Journal Sol., Vol. 23, page 217. [8] Mustafa, Odeh, Anharmonic Oscillators energies via artificial perturbation method, Eastern Mediterranean University, 2000. [9] Bell, Davidson, Warsop, The three-dimensional quartic oscillator, University of Dundee chemistry department, 1969. [10] Ikhdair, Sever, exact polynomial eigensolutions of the Schrodinger equation for the psuedoharmonic potential, department of physics near east university, 2008. [11] Floyd, Ludes, Moua, Ostle, Varkony, Anharmonic potentials, University of Nebraska, 2011. [12] Dan Longcope, Symmetry & Degeneracy, 2006.

Division of labor : (All in all we have worked very equally on all

tasks, if one is programming the others are trying to figure out how to solve the next portion of our project. We meet often and equally; if one cannot make it, they meet on another day. There has not been any imbalance of work.)

• xx o Main programming, running simulations, obtaining

data. o Handwork o Theory work. o Coding the 3-D changes and the calculation of the

energy. • xx

o Assist with programming and debugging. o Research on methods for analytic and computational

solutions in all versions of the oscillator. o Correlators. o Handwork

• xx o Assist with programming and debugging. o Degenerate perturbation theory (analytic) o Analytic coding and graphics.