Here is one computational problem I proposed to be solved as a final year project: Solve the energy spectrum of a particle subjected to a central potential V(are) using perturbative method you have learned.
In general, a particle subjected to a potential V(are) is governed by the Hamiltonin H = K + V(are), where K = -(d^2/dx^2)(hbar/2*m) is the kinetic energy operator.
To begin with, first, use perturbative method to write down the general expression of the energy spectrum in terms of a generic central potential, V(are). Once that is obtained, then proceed to solve the energy spectrum of the Kratzer oscillator [1,2] given by V(are) = -C_1/are + C_2/are^2, C_1, C_2 >0. Then, proceed to calculate the vibration-rotational spectral of diamtic molecules. The potential for such a molecule has, in addition to the Kratzer oscillator term, an extra contribution of (hbar^2)J(J+1)/(2*m*are^2), with J=0,1,2,...
The result should looks like the following:
epsilon_n = a function of (integer nunber n, the parameters of the Kratzer potential C_1,C_2, the mass m, and J from the rotational term the potential).
With the explicit expression of epsilon_n obtained, use a very simple mathematical program to evaluate the numerical values of set of the energy spectrum epsilon_n for a set of fixed parameters {m,C_1,C_2}.
In general, in QM, what one means when one says `solve the quantum problem, or the solution to the Hamiltonian', is the following: Given a hamiltonian, find out all the eigenvalues of the energies of this hamiltonian. The eigenvalues of the energy also goes by the name `the energy spectrum'. Given a hamiltonian, your job is simply to find out what the energy spectrum is for these hamiltonians.
The major difference between the kratzer oscillator and the simple harmoic oscillator (SHM) lies in their potentials. For SHM, the potential is V = (1/2)kx^2, whereas in the case here the V simply takes a different form, its V = (C_1)/are + (C_2)/are^2. To solve it perturbatively means to expand the potential into a suitable form of infinite series and then work out the coefficients of the series.
The solution to the Kratzer oscillator does not make use of the solution from SHO. The perturbative series of the Kratzer oscillator have to be first derived. However, if one knows how to treat SHM using perturbation method, then one can treat any other similar system, including the kratzer oscillator using similar method.
In principle, the first two terms in the taylor expansion of the potential can be considered as the 'non-perturbative' terms, and the next term down the series expansion of V is considered 'perturbed'. However, in this case, unlike the SHM case, one doesn't know the solution even to the unperturbed part. Operationally, in this case you dont really need to know which are is the 'perturbed part' and which is the 'unperturbed part' because the solution to the 'unperturbed' hamiltoinian is unknowned. This is unlike the SHM case where you know the solution to the 'unperturbed' energies, (n+1/2)h*omega.
Reference:
[1] ter Haar, D., Problems in Quantum M echanics, Pion, London, 3rd ed., 1975, pg. 357.
[2] Fernandez, F.M. and Castro, E.A., Algebraic Methods in Quantum Chemistry and Physics, CRC Press, Boca Raton, FL, 19996, pg. 94.
[3] Oglilvie, J.F., The vibrational and rotational spectometer of diatomic melecules, Academic Press, San Diego, 1998, pg. 87.
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