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The Morse Potential

The controls allow to change system parameters (the mass, linear and energy scale of the potential, energy) and explore dependency of wave-fuction of the system from energy, compare it with hamitltonian eigenfunctions.

Tools: AS1, FlaCon library

The Morse Potential
In this simulation, you will experimentally determine the allowed energy levels for an anharmonic oscillator. Rather than using the harmonic potential

, the more realistic Morse potential

is used to model the potential. A comparison of the harmonic and anharmonic Morse potentials is shown below.

Morse potential, V(x), (solid line) as a function of the bond length, x, for HCl, using the parameters from Example Problem 8.3. The zero of energy is chosen to be the bottom of the potential. The dashed curve shows a harmonic potential, which is a good approximation to the Morse potential near the bottom of the well. The horizontal lines indicate allowed energy levels in the Morse potential. D_e and D₀ are the bond energies defined with respect to the bottom of the potential and the lowest state, respectively, and x_e is the equilibrium bond length.

The energy levels in the Morse potential are given by

   Note from the figure that the energy spacing between adjacent energy levels is not equal. You will determine the allowed energy levels in a Morse potential by numerically integrating the Schrodinger equation, starting in the classically forbidden region to the left of the potential. The criterion that the energy you chose is an energy eigenvalue for the problem is that the wave function decays to zero in the classically forbidden region to the right of the potential.

   By experimentally finding the allowed energy levels in the potential how many bound states there are in the Morse potential with D_e = 9.00·10^-29 J and ν = 5.00·10¹³ s^-1.

   b) How do the total energy eigenfunctions differ from those for the harmonic potential?

   c) What is the likelihood of finding the harmonic oscillator compressed relative to being expanded? What is the likelihood of finding the Morse oscillator compressed relative to being expanded?

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