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Journal of Applied Mathematics and Physics

Calculation of the Approximate Energy of Ground and Excited Stationary States in Quantum Mechanics Using Delta Method

Farrin Payandeh, T. Mohammadpour

Published Jan 1, 2016• 2 citations
Publisher

Abstract

In this paper, pursuing a new advised method called Delta method which is basically similar to variational method, we find the ground and excited states, according to a typical quantum Hamiltonian. Moreover, applying this method, the upper bound values for the eigenenergies of the so-called ground and excited states are estimated. We will show that this new method, is as beneficial as the traditional variational method which is common in deriving eigenenergies of some of the quantum Hamiltonians. This method helps physics students to broaden their knowledge about the possible mathematical ways; they can use to obtain eigenenergies of some quantum Hamiltonians. The advantage of Delta method to variational method is in its simplicity and reduction of the calculation procedures.

DOI
10.4236/jamp.2016.41016

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In this paper, pursuing a new advised method called Delta method which is basically similar to variational method, we find the ground and excited states, according to a typical quantum Hamiltonian. Moreover, applying this method, the upper bound values for the eigenenergies of the so-called ground and excited states are estimated. We will show that this new method, is as beneficial as the traditional variational method which is common in deriving eigenenergies of some of the quantum Hamiltonians. This method helps physics students to broaden their knowledge about the possible mathematical ways; they can use to obtain eigenenergies of some quantum Hamiltonians. The advantage of Delta method to variational method is in its simplicity and reduction of the calculation procedures.
The variational method in quantum mechanics plays a crucial role in estimating upper bound values of eigenenergies, particularly in the efficient determination of ground state energies. This paper introduces a formalism for the calculation of ground state energies in relativistic quantum systems employing the variational method. The proposed approach is applied to specific physical systems, allowing us to evaluate its effectiveness. Furthermore, we compare the results obtained through this method with existing literature, shedding light on its accuracy and applicability in the context of relativistic quantum systems.
Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
If the potential is not closely related to a potential that we already know how to solve exactly, the energies and states can still be approximated via the Variational Method. This method is simpler to use than to understand. A simple five-step procedure is given and then illustrated via an example. The principle, in a nutshell, is that if you guess at the ground state and are wrong, the energy you calculate will be higher than the ground state energy, so if you keep guessing and keep getting smaller energies, then you must be getting closer to the ground state wave function. This minimization procedure is called the Variational Method, and it was known prior to quantum mechanics.
Trial wave function was used to calculate ground state energy to obtain a relatively accurate value.After adding potential δ to one-dimensional harmonic oscillator,the odd parity stationary states of the harmonic oscillator still held and the even parity stationary states were no longer correct. Therefore,the perturbation theory and variational method were used respectively to calculate the approximate even parity stationary states,and variational method was used to solve Schrodinger equation.Results show that the calculated approximate even parity stationary states match well with the desired value,especially in excited state,the relative error is less than×10-3,and all the approximate values of energy for all levels are higher than desired value,which prove that it is simple and feasible by using perturbation theory or variational method to calculate approximate even parity stationary states with δ potential added.
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.