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Quantum Mechanics Examples 5½ Examples in Quantum Mechanics IndexThe new theories, if one looks apart from their mathematical setting,are built up from physical concepts which cannot be explainedin terms of things previously known to the student, which cannoteven be explained adequately in words at all. Like the fundamental concepts(e.g., proximity, identity) which every one must learn on his arrivalinto the world, the newer concepts of physics can be mastered onlyby long familiarity with their properties and uses.P.A.M. Dirac (1930) Preface The Principles of Quantum MechanicsWe have always had a great deal of difficulty understanding theworld view that quantum mechanics represents. At least I do,because I'm an old enough man that I haven't got to thepoint that this stuff is obvious to me. Okay, I still getnervous with it... You know how it is, every new idea, it takesa generation or two until it becomes obvious that there's noreal problem... I cannot define the real problem, therefore I suspectthere's no real problem, but I'm not sure there's no real problem.R. P. Feynman as quoted in Genius (1992)PrefaceAs the above quotations suggest, quantum mechanicsis difficult, or perhaps impossible to understand(Steven Weinberg: "no one fully understands" [quantum mechanics]).Nevertheless, quantum mechanics is used on a daily basis bythousands of physicists, chemists and engineers. (E.g., Nobelprize winning chemistLinus Pauling: "There is no part of chemistry that does not depend,in its fundamental theory, upon quantum principles.")The ability to use quantum mechanics depends in parton mechanical mathematical skills ("doing the algebra",now made much easier with programs such as Mathematica)but more importantly on "physical intuition". Unfortunatelyour bedrock intuition -- based on classical mechanics --is often at odds with quantum mechanics. This does not meanyou must discard your hard-won classical intuition, rather you shouldtreasure those parts that survive into quantum mechanics.These pages try find common ground betweenNewton's explanations and Schrödinger's explanations.In addition because of the revolution in computeralgebra (e.g., Mathematica), these pages try toformulate problems in ways that allow easy translationinto Mathematica code. Often simple practices(like the use of dimensionless variables) make understandingeasier for both the computer and the human. Finally I shouldnote that these pages are aimed at folks who want to doquantum mechanics. Like Dirac, I believe that the apprenticequantum mechanic gains facility by practice. I hope thatyou will do the examples,not just read the examples.A set of problems at the end of each "chapter" provideextentions of the examples. Feel free to write meif you have questions or comments!Introduction: Classical Mechanics First!Usually books on quantum mechanics start with quantum mechanics.Instead I start with a brief review of examples in classical mechanics.Note that all of the below descriptions ofclassical motion report how the particle's position andvelocity change in time. Quantum descriptions must be quite differentbecause quantum mechanics asserts that a particle does not havea position and a velocity. Instead the particle has, in some sense, simultaneouslya range of possible positions and velocities. The particle has some chance of being found here,another chance of being found there, etc. Position is not a propertythat a particle has any more than a die has the property of the numberthat comes up. The die actually has is a rangeof possible outcomes (1-6) with a probability for each outcome (1/6).So too with position, but rather than just having a finite listof possibilities, usually a particle's position will be found withina continuous range of possibilities. Thus we seekthe probability density for a particle's position.The aim of quantum mechanics is to calculate this range ofpossible particle positions and the relative probability of thosepositions. This sounds nothing like classical mechanics!In classical mechanics if we say that the particle has aposition of 100±1, we mean that the particle has aposition in the range: 99-101, we're just not sure where. In quantummechanics if we say that the particle has a position of 100±1, wemean that the particle is simultaneously all over the range: 99-101.This ability to be "spread out" is not surprising for waves (in fact itwould make little sense to say the wave is localized to a regionsmaller than a wavelength... a wave needs at least a wavelength to"wave"); here we apply this wave property to things like electrons,which are traditionally called "particles". Almost all standardcourses in calculus-based introductory physics describe three examplesof motion: Constant Force F-- e.g., motion of anobject falling a few meters near the surface of the Earth (in whichcase the constant force depends on the particle's mass:F=-mg, resulting in all falling objects having thesame (downward) acceleration: g=9.8 m/s2)a=F/m=-g: acceleration is the result ofapplying the force; it can be calculated by the force divided by theparticle's massz(t)=z0+v0t+½at2:the height of the object (z) depends on the initial height(z0), the initial velocity (v0)and time (t)v(t)=v0+at: the velocitychanges uniformly in time from its initial valueU(z)=-Fz=mgz: the potential energy hasthe property that if you take minus the derivative of it w.r.t. position,you get the force. -Fz + constant works; we've set theconstant equal to zero.In the case of the ball falling near the surface of the Earth, the above described motioncannot continue indefinitely as the ball soon encounters the ground. In the case ofa perfectly elastic collision with the ground, the ground providesa force to exactly reverse the ball's velocity. It bounces forever between the ground(z=0) and some maximum height (zmax) that depends on its energy.Spring Force (Hooke's Law) F=-kx -- e.g., a force that alwayspulls the object back to the equilibrium position (x=0)...if x>0,the force is in the negative x direction, etc.x(t)=A sin( t+ ):the particle oscillates around equilibrium getting as far away as ±A. The period,T (the time it takes to make one complete oscillation), is determined by (the angular frequency): T=2 /. is in turn determined by the strength of the spring, k(called the spring constant) and the mass m: 2=k/m.Thus a strong spring connected to a light particle will oscillate quickly, i.e., with a short period.v(t)=A cos(t+):the velocity (v) of the particle also oscillates, i.e., sometimes the particles is movingto the right (positive v) sometimes it is moving to the left (negative v). Noticethat the particle has is maximum speed (of A) when the cosine termreaches its extremes of ±1. That happens only when sine is zero (because cos2 +sin2=1) and hence the particle is moving through theequilibrium position (x=0). Similarly, the particle is momentarily at rest (v=0)only when the particle is at an extreme position (±A; i.e., if cosine is zero,sine must be ±1).U(x)=½kx2: the potential energy has the property thatif you take minus the derivative of it w.r.t. position, you get the force. ½kx2 + constantworks; we've set the constant equal to zero.Orbital Motion: motion of a particle with a  /r2 central force applied.In intro physics the topic was the motion of planets under the influence of the Sun's gravitationalforce...orbital mechanics. In quantum mechanics the topic is the motion of an electron under the influence of theelectrostatic attractive force of the nucleus...atomic physics. Equivalent equations for the force (and hence "the same"as far a physics goes) but quite different distance scales. In addition energy emission bygravitational radiation is totally negligible in the Solar System, whereas energy emission byelectromagnetic radiation (light) should be important in an atom.Planets move in ellipses with the Sun at one focus. Ellipses can be described in termsof their semi-major axis, a, (basically the longest radius) and their eccentricity (basicallyhow squashed the ellipse is: e=0 is a circle, a fully squashed ellipse looks likea line and has e=1).r=a(1-e2)/(1+e cos()); rmin= a(1-e); rmax=a(1+e), the polar angle from closest approach is given the odd name:true anomaly. The timing of the motion (i.e., when the planet or electron hasa particular ) is a bit complex. The game is to express thetrue anomaly () in terms of the eccentric anomaly (u)and then find an expression relating time and the eccentric anomaly. For nearlycircular orbits it turns out that the true anomaly, the eccentric anomaly, and themean anomaly (t) are all approximatelyequal to each other. I apologize for this archaic nomenclature, but physics is stuck with thesenames. Below find the geometric construction that relates the true anomaly and the eccentric anomaly,the formula relating these two, and the formula relating time and theeccentric anomaly.   In the above picture, the yellow dot represents the Sun (or a nucleus); it isat a focus (F) of the ellipse. The red parts of the diagram have to do with the geometricconstruction for the eccentric anomaly which is measured from the center (C) of theellipse. There is nothing physically at the center of the ellipse, this is all justpart of a geometric construction. The blue parts of the diagram relate the semi-majoraxis (a) and the distance between the ellipse center and the focus(ae). The semi-minor axis, b, can be related to thesemi-major axis and the eccentricity:b = a(1-e2)½The black ellipse is the orbit of the particle, i.e., the setof positions the particle will traverse during a period (T=2/).The velocity vector is, of course, changing as the position vector is changing.The set of velocities the particle will have during a period is called thehodograph; it's just like an orbit, but for velocity rather than position.The hodograph is surprising: it's just a circle, but the center of the circle isnot v=0.   Note that at closest approach to the "Sun" (=0) the speed is a maximumand the velocity points in the y direction, whereas at the far point(=180°) the speed is a minimum and the velocity points in the -y direction. As the eccentricityapproaches 1, the maximum speed gets arbitrarily large and the minimum speedapproaches zero.The above figures have been drawn with a rather large eccentricity: e=.707. For eccentricitiessimilar to those of the planets it would be hard to distinguish the elliptical orbit froma circle (although for some planets--like Mars--the position of the Sun would looknoticeably "off-center"...because the Sun is at a focus rather than the center).If the (attractive) radial force is given by: Fr=-/r2, thepotential energy is U(r)=-/r, so that the minus derivative of U is the force. Notice that just like the spring (which also attracts to the origin), the potentialenergy is ever smaller as you approach the origin. Unlike the spring, this potential energy is numerically negative (i.e., U(r)<0).That is a result of our choice of additive constant. For the spring we choose the potentialto be zero at the origin, forcing the potential to be arbitrarily big as x approaches infinity.For this problem we take the potential energy to be zero as r approaches infinity, so the potentialat the origin must be infinitely less, i.e., negative infinity.Most introductory courses in quantum mechanics start out with problems that are not one of the above classical problems:"square well" potentials also known as particle-in-a-box problems, and delta functionpotentials. "Square well" potentials are very odd caseswhere the potential is constant except for a few steps up or down. Since the force isthe minus derivative of the potential, and the potential is flat everywhere you can take itsderivative, these problems correspond to cases where the force is zero (i.e., a "free" particle)almost everywhere. The only places where the force is not zero are the points wherethe potential takes a step up or down. A step up corresponds to a infinitely strong forceto the left; a step down corresponds to a infinitely strong force to the right. Again the particle would only feel those infinite forces at the point where the step happens,otherwise the force is zero. In thecase of a potential "well" (say, U(x)=0 if |x|<a andU(x)=U0 if |x|>a) the force is zeroexcept at x=±a. A particle, say starting at the origin andmoving in the positive x direction, wouldmove at a constant velocity until it was actually at x=a. At that pointthe particle would experience an infinite force pushing it to the left. If the particleis moving fast enough, the force only slows it down, and the particle moves on to theforce-free region x>a now moving at a constant but reduced speed.If the particle is not moving fast enough, the force is able to reverse its motion andthe particle heads back towards the origin with its velocity reversed and at an undiminishedspeed. It will soon hit x=-a, where it will again "bounce"...the particleis trapped between x=-a and x=a. In the "infinitesquare well" version of this problem, the jump in potential energy is itself infinite;no particle is able to "punch" through the forces at x=±a; allparticles are forced to forever bounce between x=-a and x=a.½ "Delta function" potentials are even odder. Like square wells, the potentialis almost everywhere a constant (0) and so, almost everywhere, there are no forcesacting on the particle. At a single point the potential undergoes a tremendous jump. So highis the jump that there is a finite area under this single point. With an attractivedelta function potential, a particle is attracted to the potential only whenit is actually on top of the single point of attraction; at all other points the forceis zero on the particle. Electrons in a lattice: While it is unsurprising that the exoticin physics (e.g., particle physics; cosmology) gets disproportionalpress, there is probably no single aspect of physics that has influencedthe world as much as the physics of everyday solids. The most typical problem insolid state physics is that of an electron moving through a crystal.We would not be properly doing our job if we left you with the impression thatquantum mechanics is used only for idealizations like square wells andharmonic oscillators. On the other hand, the aim of these pages is toprovide simple examples to play with -- not the necessarily complicatedreal-world applications.And in a real crystal there are nearly infinite complications (and hence a wealthof observations seeking an explanation). The first problemis thata crystal with 1023 electrons moving in it is clearly amany body problem whereas most of these pagesdeal only with one or two electron problems. Thus we consider here a "spherical cow":a 2D (!) array of potential wells (like nuclei) and electrons that interact only with those wells and not with each other.It should surprise you that such a distortion of reality is of muchinterest. When can we neglect electron-electron interactions andstill see reality in our calculations?But even to handle such a simple case we will be forcedto take on a new approach to "solution" of the problem: we willseek our "solution" using computer calculation rather than algebra. As a result much of the general will be lost, but I hope thatin presenting a wide array of particulars you'll see the natureof the general solution which cannot be written down as a compact formula. This willalso present you with a taste of how most physics calculationsare actually performed.Comments MailTo:tkirkman@guess.me |
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