Chapter 8 The Simple Harmonic Oscillator

1 Chapter 8 The Simple Harmonic OscillatorA winter rose. How can a rose bloom in December? Amazing but true, there it is, a yellowwinter rose. The rain and the cold have worn at the petals but the beauty is eternal regardlessof season. Bright, like a moon beam on a clear night in June. Inviting, like a re in the hearthof an otherwise dark room. Warm, like a: : :wait! Wait just a MINUTE! What is Mickey Spillane would NEVER: : :Misery Street: : :that's more like it: : :a beautifulsecretary named Rose: : :back at it now: : :the mark turned yellow: : :yeah, yeah, all right: : :theelegance of the transcendance of Euler's number on a Parisian morning in 1873: : :what?: : :The in nite square well is useful to illustrate many concepts including energy quantizationbut the in nite square well is an unrealistic potential. The Simple Harmonic Oscillator (SHO),in contrast, is a realistic and commonly encountered potential.

2 It is one of the most importantproblems in quantum mechanics and physics in general. It is often used as a rst approximation tomore complex phenomena or as a limiting case. It is dominantly popular in modeling a multitude ofcooperative phenomena. The electrical bonds between the atoms or molecules in a crystal latticeare often modeled as \little springs," so group phenomena is modeled by a system of coupledSHO's. If your studies include solid state physics you will encounter phonons, and the description ofmultiple coupled phonons relies on multiple Simple Harmonic oscillators . The quantum mechanicaldescription of electromagnetic elds in free space uses multiple coupled photons modeled by simpleharmonic oscillators . The rudiments are the same as classical mechanics: : :small oscillations in asmooth potential are modeled well by the a particle is con ned in any potential, it demonstrates the same qualitative behavior asa particle con ned to a square well.

3 Energy is quantized. The energy levels of the SHO will bedi erent than an in nite square well because the \geometry" of the potential is di erent. Youshould look for other similarities in these two systems. For instance, compare the shapes of theeigenfunctions between the in nite square well and the 1 outlines the basic concepts and focuses on the arguments of linear algebra usingraisingand lowering operatorsand matrix operators. This approach is more modern and elegantthan brute force solutions of di erential equations in position space, and uses and reinforces Diracnotation, which depends upon the arguments of linear algebra. The raising and lowering operators,orladder operators, are the predecessors of the creation and annihilation operators used in thequantum mechanical description of interacting photons. The arguments of linear algebra providea variety of raising and lowering equations that yield the eigenvalues of the SHO,En= n+12 h!

4 ;and their eigenfunctions. The eigenfunctions of the SHO can be described usingHermite poly-nomials(pronounced \her meet"), which is a complete and orthogonal set of 2 will explain why the Hermite polynomials are applicable and reinforce the results ofpart 1. Part 2 emphasizes the method ofpower series solutionsof a di erential 5 introduced the separation of variables, which is usually the rst method applied in anattempt to solve a partial di erential equation. Power series solutions apply to ordinary di erentialequations. In the case the partial di erential equation is separable, it may be appropriate to solveone or more of the resulting ordinary di erential equations using a power series method. We will258encounter this circumstance when we address the hydrogen atom. You should leave this chapterunderstanding how an ordinary di erential equation is solved using a power series do not reach the coupled Harmonic Oscillator in this text.

5 Of course, the SHO is animportant building block in reaching the coupled Harmonic Oscillator . There are numerous physicalsystems described by a single Harmonic Oscillator . The SHO approximates any individual bond,such as the bond encountered in a diatomic molecule like O2or N2. The SHO applies to anysystem that demonstrates small amplitude Simple Harmonic Oscillator , Part 1 Business suit, briefcase, she's been in four stores and hasn't bought a thing: : :so this mallhas got to be the meet! Now a video store. She's as interested in videos as a cow is in eatingmeat. But, right in the middle of the drama section, suddenly face to face: : :\Sir, do you have acigarette?" and walks o more briskly than Lipton ice tea. Blown. Gone. Done. Just to tell meshe knows me: : :no meet for me. I've got to hang up my hat, but only my hat: : :She doesn't knowCharlie's face, and maybe the meet will happen in Part 2: : :1.

6 Justify the use of a Simple Harmonic Oscillator potential,V(x) =kx2=2 , for a particlecon ned to any smooth potential well. Write the time{independent Schrodinger equation for asystem described as a Simple Harmonic sketches may be most illustrative. You have already written the time{independent Schrodingerequation for a SHO in Chapter functional form of a Simple Harmonic Oscillator from classical mechanics isV(x) =12kx2:Its graph is a parabola as seen in the gure on the left. Any relative minimum in a smooth potentialenergy curve can be approximated by a Simple Harmonic Oscillator if the energy is small comparedto the height of the well meaning that oscillations have small 8 1: Simple Harmonic Oscillator :Figure 8 2:Relative Potential Energy Minima:Expanding an arbitrary potential energy function in a Taylor series, wherex0is the minimum,V(x) =V(x0) +dVdx x0(x x0) +12!}}

7 D2 Vdx2 x0(x x0)2+13!d3 Vdx3 x0(x x0)3+ de ningV(x0) = 0 ,dVdx x0= 0 because the slope is zero at the bottom of a minimum, and ifE the height of the potential well, thenx x0so terms where the di erence (x x0) has a259power of 3 or greater are negligible. The Taylor series expansion reduces toV(x) =12d2 Vdx2 x0(x x0)2whered2 Vdx2 x0=k :De nex0= 0)V(x) =12kx2. Sincek=m!2, this meansV(x) =12m!2x2:Using thispotential to form a Hamiltonian operator, the time{independent Schrodinger equation isH j >=Enj >) P22m+12m!2X2 j >=Enj > :Postscript:Notice that this Schrodinger equation is basis independent. The momentum andposition operators are represented only in abstract Hilbert Show that the time-independent Schrodinger Equation for the SHO can be written h! aya+12 j >=Enj > :Leta= m!2 h 1=2X+i 12m! h 1=2 Panday= m!2 h 1=2Xy i 12m! h 1=2Py:For reasons that will become apparent,ais called thelowering operator, andayis known astheraising operator.}

8 SinceXandPare Hermitian,Xy=XandPy=P, so the raisingoperator can be writtenay= m!2 h 1=2X i 12m! h 1=2P:Remember thatXandPdo not commute. They are fundamentally canonical, X;P =i h : h! aya+12 = h!(" m!2 h 1=2X i 12m! h 1=2P#" m!2 h 1=2X+i 12m! h 1=2P#+12)= h!" m!2 h X2+i 14 h2 1=2X P i 14 h2 1=2PX+ 12m! h P2+12#= h! m!2 hX2+12m! hP2+i2 h X P PX +12 = h! 12m! hP2+m!2 hX2+i2 hhX;Pi+12 = h! 12m! hP2+m!2 hX2+i2 hi h+12 = h! 12m! hP2+m!2 hX2 12+12 = 12mP2+m!22X2 260) 12mP2+m!22X2 j >=Enj >() h! aya+12 j >=Enj > :Postscript:The Schrodinger equation is P2+X2 j >=Enj >, when constant factors areexcluded. The sumP2+X2=X2+P2would appear to factor as X+iP X iP ;so that P2+X2 j >=Enj >) X2+P2 j >=Enj >) X+iP X iP j >=Enj > :This is only a quali ed type of factoring because the order of the \factors" cannot be changed;XandPare fundamentally canonical and simply do not commute.

9 Nevertheless, the parallel withcommon factoring into complex conjugate quantities is part of the motivation for the raising andlowering operators. In fact, some authors refer to this approach as the method of thataya=1 h!H 12:Notice also that thoughXandPare Hermitian,aandayare Show that the commutator a; ay = 1 .Problems 3 and 4 are developing tools to approach the eigenvector/eigenvalue problem of the want a; ay =a ay ayain terms the de nitions of problem 2. LettingC= m!2 h 1=2;andD= 12m! h 1=2to simplify notation; a; ay = CX+iDP CX iDP CX iDP CX+iDP =C2X2. iCDX P+iDCP X+D2P2. C2X2. iCDX P+iDCP X 2iCD P X X P = 2i m!2 h 1=2 12m! h 1=2hP;Xi=2i2 h i h = 1;sincehP;Xi= hX;Pi= i h :4. Show thatHay=ayH+ay h!.This is a tool used to solve the eigenvector/eigenvalue problem for the SHO though it should buildsome familiarity with the raising and lowering operators and commutator h!

10 Aya+12 )H h!=aya+12261hay;H h!i=hay; aya+12i=ayaya+ay12 ayaay 12ay =ay aya aya = ayha; ay = ay) ay;H = ay h!)ayH Hay;= ay h!) Hay=ayH+ay h! :Postscript:We will also use the fact thatHa=aH a h!, though its proof is posed to thestudent as a Find the e ect of the raising and lowering operators using the results of problem have written time{independent Schrodinger equation asH j >=Enj >to this the Hamiltonian is the energy operator, the eigenvalues are necessarily energy state vector is assumed to be a linear combination of all energy eigenvectors. If we speci callymeasure the eigenvalueEn, then the state vector is necessarily the associated eigenvector which canbe writtenjEn>. The time{independent Schrodinger equation written asH jEn>=EnjEn>is likely a better expression for the development that jEn>=EnjEn>whereEnis an energy eigenvalue, thenjEn>=jEn>) HayjEn>= ayH+ay h!}}