The Electron in a Potential

Transcription

1 Te Electron in a Potential Edwin F. Taylor July, Stopwatc rotation for an electron in a potential For a poton we found tat te and of te quantum stopwatc rotates wit frequency f given by te equation: f E = (for te poton -- zero mass) (1) Tis is te frequency wit wic te stopwatc and rotates (rotations/second) as te poton explores alternative pats. For te free electron (te electron free of any forces or potentials) we postulated tat te and of te quantum cloc rotates at te following rate: f KE = (for te free electron) (2) Here KE is te inetic energy of te electron. Now suppose te electron as potential energy, possibly due to te electric attraction of a positively-carge nucleus. How do we combine inetic and potential energy to find a frequency for te stopwatc rotation wen an electron explores a pat in tis potential? We are tempted to use equation (2) also for tis case, wit inetic energy KE replaced by total energy E equal to inetic energy plus potential energy: E = KE + PE. WRONG! Wy wrong? Because if we sould tae f = E/ for te electron, substituting te total energy in tis formula, te minimum number of rotations would favor pats in wic PE, te potential energy, is low to mae E = KE + PE low and ence rotation rate f low. Between baseball pitcer and catcer (playing in a vacuum!), te trown electron would drop in eigt (lowering its PE and slowing its quantum cloc) and ten rise up again to eep its fixed appointment wit te catcer s mitt. By going down and ten rising, te pitced electron decreases its PE and minimizes te number of rotations of its quantum cloc between pitcer and catcer. But tis is absurd. Antigravity? Impossible! Wit tis WRONG coice for frequency, quantum pysics would not go over smootly into classical mecanics as te mass of te trown particle increases from tat of an electron to tat of a baseball. So it is WRONG to use te SUM of KE and PE in calculating te rate of rotation of te stopwatc as te electron explores pats. 43

2 But we now ow a baseball flies. We now te quantity tat is minimized in te classical pat. Te baseball moves so as to minimize te ACTION between fixed events of pitcing and catcing as described in te Feynman lecture on te Principle of Least Action, Lecture 19 of Volume II of Te Feynman Lectures in Pysics. and in te earlier section of tis manual titled Te Principle of Least Action. ALMOST ALL of te classical mecanics for a single particle can be derived from te Principle of Least Action. And tis principle involves summing up over time te contributions from te expression KE PE, wit a MINUS sign before te potential energy PE. Te quantum result, it turns out, simply adapts te ting tat we sum over time to get te classical action, namely KE PE. Te rate of rotation (in cycles/second) of te stopwatc and for an electron exploring a pat (for speeds muc less tan te speed of ligt) is: f KE PE = (for te electron) (3) We find te number of rotations along a worldline by a summing process: During a small increment of time dt, te number of rotations (or fraction of a rotation) is f dt = (KE - PE) dt/. We sum tis for all time increments along te worldline: rotations along a = worldline along ( KE PE ) dt worldline (4) But te numerator on te rigt side of (4) is simply te action S (defined in equation (1) in te preceding section on te Principle of Least Action). Hence te number of rotations along a worldline is just: rotations along a = worldline Action along tat worldline = S (for te electron) (5) Looing at equations (4) and (5), we see tat te number of revolutions between pitcer and catcer can be made minimum by increasing PE in order to reduce te frequency f of te quantum cloc since PE enters tese equations wit a minus sign. Increase PE by letting te pat rise. But it cannot rise too far, because ten KE must increase so tat te electron can rus along te longer pat to eep te fixed appointment wit te catcer s mitt. [Bot events pitc (emit) and catc (detect) are fixed in quantum as well as in classical descriptions.] Te actual pat is a 44

3 compromise between increasing PE (tat lowers te frequency f) and increasing KE (tat raises f). In a vacuum, ten, te greatest contribution to te resulting arrow for detection of te electron comes from trajectories near te classical pat of parabolic motion in a uniform vertical gravitational field. Wit increasing mass, te pat of minimum quantum rotations for te electron goes over smootly to te pat of minimum classical action for te baseball. We ave found a crucial connection between classical mecanics and quantum mecanics. Actually, we ave found someting vastly more important: We ave found te quantum mecanical basis for most of classical mecanics. Classical mecanics as no answer to te question, WHY does a particle moving in a potential (including zero potential) follow te pat predicted by te Principle of Least Action? Te answer true for bot quantum and classical worlds is, Because tis pat minimizes te number of rotations of te quantum cloc. So wat is te DIFFERENCE between te quantum world and te classical world? Te answer is embodied in Planc s constant in te denominator of te rigt-and sides of equations (3) tru (5) along wit te mass m of te particle idden in te numerator. Te electron as te smallest mass of any stable particle tat we now. A small mass in te numerator in (3) balances te tiny value of in te denominator. Te resulting frequency f (dividing numerator by denominator) can be low enoug so tat total cloc rotations along nearby pats are not too different. A sligtly different pat can ave a total rotation of, say, one-eigt turn more tan te pat for minimum total rotation. Ten to calculate te resulting arrow, you must tae account of tis nearby pat, along wit oters. Te electron sniffs out a fuzzy range of pats around te minimum-totalrotation pat; tis is wat we mean by quantum beavior! In contrast, for a large mass in te numerators in equations (3) tru (5) te mass of a baseball, for example te tiny value of in te denominator maes te frequency f extremely rapid. In tis case, a sligtly different pat will ave a total rotation undreds of turns more tan te pat for minimum total rotation. In tis case contributions from all nearby pats tend to cancel out. In te limit of large mass, only te single pat of minimum rotation needs to be taen into account, te pat predicted by te Principle of Least Action. Te baseball appears to follow a single pat; tat is wat we mean by classical beavior! Te electron can also explore all pats inside an atom or molecule. In tis case te nuclear carge provides potential energy tat influences te rotation rate of te electron cloc different rotation rates for different distances from te nucleus. Tis rotation rate canges as te electron explores regions of different potential PE 45

4 along eac pat. Tis simple story is complicated by spin and by te presence of oter electrons in te atom or molecule. You didn t really tin tat we would also cover all of cemistry, did you? 3. Tis Is a Derivation? Te above story line is NOT a derivation. Tere is no nown derivation of te fundamental laws of quantum mecanics. Certainly no derivation can come from classical pysics! No fundamental derivation appears possible. Neverteless, tere is powerful evidence for te correctness of te derivation. Long ago Feynman 1 sowed tat tis stopwatc way of tining leads rigorously to te usual macinery of quantum pysics expressed in te so-called Scroedinger equation. And te Scroedinger equation is te basis for our predictions about all non-relativistic quantum structures and experiments, including cemical bonding and te periodic table. Notice two limitations of te classical Principle of Least Action tat turn out to be advantages wen tis principle is applied to quantum mecanics: FIRST, te Principle of Least Action requires tat we fix in bot space time te two events of pitc and catc. But tis is an advantage in quantum mecanics, were we want to coose te event of emission and also coose te time and place were we will try to detect te particle. So te Principle of Least Action limits te description of motion to just tose conditions we want for our quantum description. SECOND, te Principle of Least Action does not apply classically wen tere is friction. You must be able to define a potential, wic you cannot do if friction is all te time robbing your moving stone of energy. But tis is OK in quantum mecanics, because tere is no friction at te atomic level: potential plus inetic energy is conserved rigorously in non-relativistic quantum pysics. Once again, te Principle of Least Action limits attention to just tose conditions we want for our quantum description. 4. Terminology Te function KE PE for a low-velocity particle is called te Lagrangian and given te symbol L (sometimes a script L). So we ave: L = KE PE (6) and from equation (3): 46

5 f = KE PE L = (electrons & oter particles wit mass) (7) Tose wo would lie to measure cloc rotation rate ω in radians per second instead of f in revolutions per second can use te usual relation between te two: ω π = L π = L 2 f 2 2π = L (electrons, etc.) (8) Here, written as wit a little diagonal strie across it, is pronounced -bar and stands for te expression: 2π So now we now ow fast te quantum cloc rotates for te electron. (9) Appendix: Some Formalism Leading Toward te Wave Function We try to avoid formalism. If matematics irritates and frustrates you, sip te following. But some people lie te clean condensation tat matematics can bring. And a big payoff in tis case is an understanding of te quantum mecanical wave function. Feynman develops a set of rules for reconing te final arrow at a detector. Te probability tat te detector will record te particle is proportional to te square of te lengt of tis final arrow. Te rules can be placed in a ierarcy derived from pages 37 and 61 of QED: GRAND PRINCIPLE: Te probability of an event is proportional to te square of te lengt of a resulting arrow, tis arrow called te quantum amplitude. (Feynman calls it by te confusing name probability amplitude. ) RULE FOR ALTERNATIVE PATHS: If an event can appen in alternative ways, draw an arrow for eac way, ten combine ( add ) te arrows by ooing te ead of one to te tail of te next. Te final arrow is ten drawn from te tail of te first arrow to te ead of te last arrow. Tis final arrow is te resulting arrow used in te GRAND PRINCIPLE RULE FOR SEQUENTIAL STEPS IN EACH PATH: Wen eac way tat an event can appen involves a series of steps in sequence, tin of eac step as a srin and turn of te little arrow. To find te arrow for tat complete pat, multiply te srins for all steps in te pat and add te angle canges (turns) for all steps in te pat. Te arrow 47

6 for tat complete pat is added to oters from ALTERNATIVE PATHS to give te resulting arrow used in te GRAND PRINCIPLE. Is tere a matematical quantity tat beaves in tese ways? First, one must be able to ADD suc quantities as arrows are added. Second, MULTIPLICATION of suc quantities must mean finding te product of teir magnitudes and te sum of teir angles of rotation. Suc a matematical quantity is te complex number, wic can be expressed in two forms tat are entirely equivalent: iθ Ae Acos θ + i A sin θ (10) Here e = is te base of natural logaritms and i 1 is te basis of imaginary numbers. Complex numbers combine real and imaginary numbers. REFERENCE: Te Feynman Lectures on Pysics, Volume I, Capter 22, especially pages 22-7 tru Adding two complex numbers means adding teir real parts and ten separately adding teir imaginary parts. Tis is equivalent to adding separately te x- components and te y-components of two arrows to obtain te components of te resulting arrow. For two complex numbers, designated by te subscripts 1 and 2, we ave: iθ iθ1 iθ2 1 2 Ae A e + A e = A1cos θ1 + ia1sin θ1 + A2 cos θ2 + ia2 sin θ2 = ( A cos θ + A cos θ )+ i A sin θ + A sin θ ( ) (11) Multiplying complex numbers is even easier: ( ) Ae i θ A e i θ A e i θ A A e i ( θ + θ ) = (12) Tis is a direct example of te rule for finding te arrow for a sequence of steps in one pat: MULTIPLY te magnitudes of te arrows for eac sequential step in te pat and ADD te angle canges. If A 2 is less tan unity, multiplying A 1 by A 2 corresponds to a srin. Adding te angles corresponds to a turn. Te probability for te final outcome is proportional to te square of te magnitude (te square of te lengt or A-value) of te resulting complex number. You may ave been told tat quantum mecanical quantities, suc as wave functions, are complex functions. Wy is tis so? Complex numbers and complex functions are just ways we combine te little arrows to form a resulting arrow, te 48

7 probability amplitude, wose square is proportional to te probability. Complex numbers are used to trac te srining, turning, and adding of tose little arrows tat lead to a resulting arrow and a final probability. Using tis notation, we can describe te motion of te electron along alternative pats. Te angles θ used in te complex notation are expressed in radians. Te rotation rate ω for te electron cloc in radians per second is given by combining equations (6) and (8). L KE PE ω = = (13) How many times will te little cloc and rotate along a given pat (call it pat ) from initial event 1 to final event 2? We can compute tis, starting wit te action S for tat pat: S = KE PE dt (14) pat ( ) Here eac pat is required to start at te same initial event 1 and to end at te same final event 2, so te time along te pat is te same for every alternative pat. Tis means tat te inetic energy is different for different pats, and total energy is not te same for alternative pats. (For an arbitrary pat, te total energy may not even be constant along tis single pat.) But for large-mass particles te little arrows from all pats point in very different directions and tend to cancel out except tose near te pat of least action, for wic te little arrows point in nearly te same direction. For tese pats te total energy is conserved, as we saw in te ACTION software. Te contribution to te probability amplitude for a single pat is given by: is Ae (15a) For tose of us wit failing eyesigt, tese exponentials may be too small, so we use te function exp() to represent te exponential wit e: A exp( is / ) (15b) Ten te probability amplitude for te given outcome is reconed from te sum of tese contributions for every alternative pat (indexed by ) between a fixed initial event 1 and a fixed final event 2: or Probability amplitude (from 1 to 2) = Ae all pats from event 1 is (16a) 49

8 Probability amplitude (from 1 to 2) = A exp( is / ) (16b) all pats from event 1 Tis is te probability amplitude tat te electron starts at a particular initial event 1 and arrives at a particular final event 2 placed at, say, x and t. We can call tis te wave function ψ(x,t) for a particle emitted from event 1: or ( ) = ψ xt, Ae all pats from event 1 is (17a) ( ) = ψ xt, A exp( is / ) (17b) all pats from event 1 Tis is te meaning of Feynman s statement in is abstract for te 1948 article in Reviews of Modern Pysics: 1 Te total contribution from all pats reacing x,t from te past is te wave function ψ(x,t). Reference 1 R. P. Feynman, Space-Time Approac to Non-Relativistic Quantum Mecanics, Reviews of Modern Pysics, Volume 20, Number 2, April 1948, pages Copyrigt 2000, Edwin F. Taylor 50