Ground-state energy eigenvalue calculation of the quantum mechanical well V(x)=\frac{1}{2}kx^{2}+\lambda {x^{4}} via analytical transfer matrix method

Abstract

We consider a fundamental quantum mechanical bound-state problem in the form of the quartic-well potential . The analytical transfer matrix method is applied. This yields a quantization condition from which we can calculate the phase contributions and ground-state energy eigenvalues numerically. We also compare the results with those obtained from other typical means popular among physics students, namely the numerical shooting method, perturbation theory and the standard WKB method.

Excited-State Energy Eigenvalue Evaluation of the Quantum Mechanical Potential <i>V</i>(<i>x</i>)=½<i>mω</i>²<i>x</i>²+<i>μx</i>³ via Numerical Shooting Method

“…There exist several means to study them, e.g. Wentzel-Kramers-Brillouin [1], perturbation theory [2], the quasilinearization method [3], the variational method, function analysis [3], the eigenvalue moment method [4], the analytical transfer matrix method [5][6][7] and numerical shooting method [8][9]. In this paper we consider approximation methods that deal with stationary states corresponding to time-independent Hamiltonian.…”

Evaluated Excited-State Time-Independent Correlation Function and Eigenfunction of the Harmonics Oscillator Cosine Asymmetric Potential via Numerical Shooting Method

“…A variety of such methods have been developed, and each has its own area of applicability. There exist several means to study them, for example, Wentzel-Kramers-Brillouin [1], perturbation [2], the quasilinearization method [3], the variational method [4], function analysis [5,6], the eigenvalue moment method [7], the analytical transfer matrix method [8][9][10], and numerical shooting method [11,12].…”

An Alternative Approach to Energy Eigenvalue Problems of Anharmonic Potentials

“…In the last decades, the study of anharmonic oscillator potentials (such as quartic and sextic anharmonic oscillators) has been taking attention because they are open for the theoretical perception of some recently discovered phenomena in different branches of physics [11][12][13]. On the other hand, these potentials are not exactly solvable which make them very popular for examining the validity of any approach [14][15][16][17]. In all these attempts, it is merely required to get a relatively effective and simple approach that gives the energy eigenvalues-and eigenfunctions-to a high degree of accuracy.…”

“…Comparison of the first few energy eigenvalues of potential in(17) with different values of . The percent errors for ATEM are also shown in the fifth and tenth columns.…”