As a simple illustration of this, we consider the harmonic oscillator, with a potential vx 1 2 kx2 9 px s 2m e 1 2 kx2 10 in this case, the turning points are x 1 r 2e k 11 x 2 r 2e k 12 and. Chapter 32 wkb quantization t he wave function for a particle of energy e moving in a constant potential v is. The wkb approximation in this section our aim is to construct approximate solutions of the static schr odinger equation 4. Adding anharmonic perturbations to the harmonic oscillator equation \ \ref 5.
Finite differences solution of the harmonic oscillator. Finite differences solution of the harmonic oscillator clear. Here we generalize this solution to the case where the potential varies slowly over many wavelengths. Hitoshi does this problem in his wkb notes, on page 8. The wkb approximation is also called the semiclassical. Wkb approach to calculating the lifetime of quasistationary. Anharmonic reflects the fact that the perturbations are oscillations of the system are not.
To illustrate it, two examples of the potentials are studied. Phys 771, quantum mechanics, final exam, fall 2011. It is clearly seen that the wkb approximation breaks down near. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. The simplicity of the wkb method is that we can directly right down the form of the wavefunction, and the relevant quantization conditions imposed by the form of the boundary. And vx for the sho is a quadratic function, the next best to a linear function. Chapter 6 the linear harmonic oscillator principles of quantum 9. Wentzel,kramers,brillouin thismethodallowstoobtainapproximatesolutions tothetimeindependentschrodinger.
Approximation methods in quantum mechanics science academies. In the case of the harmonic oscillator, theres no nonperturbative effect, so one would expect the wkb expansion is enough. Wkb approximation of the harmonic oscillator physics pages. Because the energy is much higher, the approximation is far more accurate i. For the harmonic oscillator, this takes the form s0x p 2e. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Having derived these basic quantities, we can express the frequency of orbital motion in state n as. The classical turning points a, b are where ke0, i. Wkb approach to calculating the lifetime of quasistationary states. Anharmonic oscillation is described as the restoring force is no. Wkb approximation of the harmonic oscillator 3 x 2 x1 pdx p 2m p 2ek p 2ek r e kx2 2 dx. Wkbsolutionsforquantummechanicalgravitationalpotentialplusharmonicoscillatorpotential. Furthermore, it is one of the few quantummechanical systems for which an exact.
Linear harmonic oscillator in quantum mechanics pdf. D1, we follow merzbacher treatment, where he considered the wentzelkramersbrillouin wkb. See, for example, the onedimensional eigenfunctions of the harmonic oscillator illustrated in figs. Anharmonic oscillation is defined as the deviation of a system from harmonic oscillation, or an oscillator not oscillating in simple harmonic motion. Phys 771, quantum mechanics, final exam, fall 2011 instructor. The chapter discusses the concept of asymptotic matching the wkb solution with the exact solution very close to. The wkb approximation is a semiclassical calculation in quantum mechanics in which the wave function is. The primary focus is the groundstate splitting, but many other levels, including the wavefunctions, can be. Wentzelkramersbrillouin wkb approximation the wkb approximation states that since in a constant potential, the wave function solutions of the schrodinger equation are of the form of simple plane waves, if the potential, uux, changes slowly with x, the solution of the schrodinger equation is of the form, where.
Lecture 8 wkb approximation, variational methods and the. Lecture 8 wkb approximation, variational methods and. The simple harmonic oscillator sho, in contrast, is a realistic and commonly encountered potential. Harmonic oscillator in a polynomial perturbation j. The classical analog of the system in question is a half oscillator, which bounces from a perfectly elastic wall placed at the equilibrium position.
Wkb approximate solution of the schrodinger equation fails at classical turning points. In fact, we shall soon be confronted with situations where an exact analytic solution is unknown. The oscillation occurs with a constant angular frequency \ \omega \sqrt\dfrackm\. We use the formula that we derived in the previous lecture.
Classical thermodynamics of particles in harmonic traps. Approximation methods in quantum mechanics science. The rain and the cold have worn at the petals but the beauty is eternal regardless of season. This chapter presents a simple way to derive the semiclassical approximation of wenzel, kramers, and brillouin wkb. The classical hamiltonian of a simple harmonic oscillator is 389 where is the socalled force constant of the oscillator. Wkb approximation for abruptly varying potential wells. The application of the wkb approximation gives see, for example, ref.
Gravitational potential, bohr sommerfeld wkb approximation. Chapter 8 the simple harmonic oscillator a winter rose. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, ac circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter. The halfoscillator oscillates with natural frequency 2 the quantum of energy ein eq.
The solution of this problem is expressed in terms of hermite polynomials. Bohrsommerfeld approximation for excited states partial wave approximation in potential scattering. A simple derivation of the wkb approximation notes by sergei winitzki draft october 7, 2005 1 motivation the wkb approximation is ordinarily introduced in courses of theoretical physics when one begins studying quantum mechanics. To establish the basic wkb ansatz, we work with the schrodinger equation for an. I wanted to see if theres any simple way to reconstruct the wave function from wkb. Ofcourse, in the vicinity of the turnings the wkb approximation breaks down. Ive been trying to dig through the approximations we make in deriving the wkb expressions. This turns out to be the exact spectrum for the bound states of the simple harmonic oscillator potential. It consists of a mass m, which experiences a single force f, which pulls the mass in the direction of the point x 0 and depends only on the position x of the mass and a constant k. A simple harmonic oscillator is an oscillator that is neither driven nor damped. As another example of applying the wkb approximation to a 3d problem with a spherically symmetric potential, well look at the radial equation for the hydrogen atom. Unlike the harmonic oscillator, this problem does not a have simple solution.
Request pdf wkb approach to calculating the lifetime of quasistationary states. We need to cook up a trial wavefunction which we think might look something like the. The wkb approximation is a method for solving the onedimensional. The general wave function can be written as the product of a radial function rr and a spherical harmonic y. Wkb harmonic oscillator a energy levels hitoshi does this problem in his wkb notes, on page 8. It is one of the most important problems in quantum mechanics and physics in general. Harmonic oscillator approximation 3 if the coordinates are enumerated so that 1 1 2 1. The hookes atom is a simple model of the helium atom using the quantum harmonic oscillator. And now, you want to add an extra term that will make this anharmonic. However, the wkb approximation can be seen as a purely mathematical technique that has littletodowithquantumphysics. Sep 30, 2019 this is known as simple harmonic motion and the corresponding system is known as a harmonic oscillator.
We want our solutions to decay at in nity and so we expect that the family. Amazing but true, there it is, a yellow winter rose. Let us consider the anharmonic oscillator, which means that youre taking the unperturbed hamiltonian to be the harmonic oscillator. Borns series and approximation in scattering theory. A charge q \displaystyle q, with mass m \displaystyle m, in a uniform magnetic field b \displaystyle \mathbf b, is an example of a onedimensional quantum harmonic oscillator. So far, we have concentrated on problems that were analytically solvable, such as the simple harmonic oscillator, the hydrogen atom, and square well type potentials. Another example is the simple harmonic oscillator in which the potential is parabolic in distance. Harmonic oscillator in a polynomial perturbation a simple and straightforward wkb approach to calculating the. Jan 25, 2020 adding anharmonic perturbations to the harmonic oscillator equation \ \ref 5. The half oscillator oscillates with natural frequency 2 the quantum of energy ein eq.
For an arbitrary potential barrier as shown in fig. Wkb for the harmonic oscillator physics stack exchange. Solution weusetheformulathatwederivedinthe previouslecture. S pe, the action of a periodic orbit p at energy e, equals the area in the phase space traced out by the 1dof trajectory. Harmonic oscillator energy wkb approximate energy why. Nonetheless, it is easy to solve numerically where one. The exact solution is accessible with elementary mathematics, though numerical. It is the quantum analog of the sommerfeldrunge procedure for the transition from wave optics to ray optics and hence also called eiconal approximation. We shall then proceed to investigate the rules of quantum mechanics in a more systematic fashion in chapter 4. Apr 26, 2009 the quantization rule that gives the correct eigenvalue for the harmonic oscillator is derived under the condition that near the turning points, the potential can be expanded as a linear function. In many cases it is preferable to have approximate explicit formulas, that can be further used to calculate other quantities, than to have exact numerical solutions.
The formulation of a wkb quantization rule is proposed. The wkb, or bwk, or wbk, or bwkj, or adiabatic, or semiclassical, or phase integral approximation or method, is known under more names than any confidence man. As in the case of a quantum harmonic oscillator the wkb approximation replicates the exact eigenvalues. Stefan institute, po box 3000, 1001 ljubljana, slovenia 2 racah institute of physics, the hebrew university, jerusalem 91904, israel 3 department of physics and astronomy, university of pittsburgh. The classical analog of the system in question is a halfoscillator, which bounces from a perfectly elastic wall placed at the equilibrium position.547 889 831 197 748 1285 1561 270 294 1172 658 1561 715 411 309 1589 1098 854 386 1368 624 342 275 928 906 984 459 1532 585 388 939 648 50 532 880 292 364 941 1343 30