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American Journal of Physics

Quantum solution for the biharmonic oscillator

W. Edward Gettys, H. W. Graben

Published Jul 1, 1975• 13 citations
Publisher

Abstract

The quantum-mechanical solution for the biharmonic oscillator is presented. This one-dimensional asymmetric oscillator is described by the potential energy function V(x) = mω12x2/2, x ≳ 0, and V(x) = mω22x2/2, x < 0. One of the more interesting features of the solution is that the energy levels are very nearly equally spaced. The biharmonic oscillator provides a relatively simple model for systems which lack the symmetry of the harmonic oscillator.

DOI
10.1119/1.9763

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The quantum-mechanical solution for the biharmonic oscillator is presented. This one-dimensional asymmetric oscillator is described by the potential energy function V(x) = mω12x2/2, x ≳ 0, and V(x) = mω22x2/2, x < 0. One of the more interesting features of the solution is that the energy levels are very nearly equally spaced. The biharmonic oscillator provides a relatively simple model for systems which lack the symmetry of the harmonic oscillator.
The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both x and p, and calculation of the corresponding quantum-mechanical solutions which give |?n(x)|2 and |?n(p)|2 for comparison to their classical counterparts in the?classically allowed regions. We then focus on the behavior of ?n(p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-p limit of the momentum-space wavefunction.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
So far we have computed path integrals for potentials V which are linear in the variable x: V(x) = cx for some constant c. Consider now a quadratic potential: V(x) = cx2. Specifically, consider V(x) =C9; 2 mx 2/2 where m is the mass of a simple harmonic oscillator with frequency v = ω/2π. As discussed in Chapter 2 the equation of motion is 1 x(t)=Acos⁡(ωt−δ)=Asin⁡(ωt+Φ),x(t) = A\cos (\omega t - \delta ) = A\sin (\omega t + \Phi ),x(t)=Acos(ωt−δ)=Asin(ωt+Φ),