We investigate the perturbative expansions of the ground-state wavefunctions of the quantum anharmonic oscillators. With an appropriate change of spatial scale, the weak-coupling Schrödinger equation is transformed to an equivalent strong-coupling one. The Friedberg–Lee–Zhao method is applied to obtain the improved perturbative expansions. These perturbative expansions give a correction to the WKB results for large spatial distances, and reproduce the conventional weak-coupling results for small spatial distances.
We investigate the perturbative expansions of the ground-state wavefunctions of the quantum anharmonic oscillators. With an appropriate change of spatial scale, the weak-coupling Schrödinger equation is transformed to an equivalent strong-coupling one. The Friedberg–Lee–Zhao method is applied to obtain the improved perturbative expansions. These perturbative expansions give a correction to the WKB results for large spatial distances, and reproduce the conventional weak-coupling results for small spatial distances.
Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.
Dark states are excited quantum states that decouple from their environment in such a way that they do not emit or absorb external photons. These states are found in a variety of different open quantum systems and can be derived from the collective interactions of individual quantum emitters interacting with one another. One of the simplest model where these states exist is in a pair of dissipatively coupled harmonic oscillators described under the Bose-Hubbard model. When on-site interactions are included, these states can no longer be classified as genuine dark states since dissipation is induced in them. In this paper we study the origin of this dissipation in dark states by using weak anharmonicity as a perturbing factor. In our analysis, we find the first and second order corrections to the wavefunction and apply these corrections to the master equation in order to track the dynamics.
We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville–Green transformations of the Schrödinger equation. For anharmonic oscillators, first-order semiclassical quantization is seldom accurate and the higher order expansions eventually break down given the asymptotic nature of the series. A quantum phase that allows us in principle to retrieve the exact quantum mechanical quantization condition and wavefunctions is given along with an iterative scheme to compute it. The arbitrariness surrounding quantum phase functions is lifted by supplementing the phase with boundary conditions involving high order semiclassical expansions. This allows us to extend the definition of oscillation numbers, which determine the quantization of the harmonic oscillator, to the anharmonic case. Several illustrations involving homogeneous as well as coupling constant dependant anharmonic oscillators are given.
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.