Propagador

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The Free-Particle Propagator File: Propagator69

The Propagator for a Free Particle

Edw in F. Taylor

July 2000

I. EXPLORING ALL PATHSNow (finally!) we can show how to satisfy Feynman 's (and Natu re's) complete comm and to

the electron,Explore ALL worldlines. Up un til now w e have been just SAMPLING possible

worldlines, for examp le pointing and clicking to choose a few worldlines between emission

and detection. How can we sum the results of ALL path s that the electron can take between

one point on an initial wavefunction and another point on the later wavefunction? One way

to do this is to carry out the summ ation mathematicallya messy business. If you w ant to

see professionals at work on this task, look into the first two Advanced Texts referenced in

the Suggested Readings in Quantum Mechanics (Feynman and Hibbs page 42f and Schulman

page 31f). A way to avoid sum ming all paths is to GUESS the result, try it out, and improve

it until it works. That is what we w ill do here: a kind of reverse engineering, find ing a

technical way to achieve a prescribed end. We will use the results to describe a free pa rticle(moving in a region of zero p otential) as it explores ALL path s between an em ission point

on an initial wavefunction and a point on th e later wavefunction.

What is our best first guess about how the initial wavefunction propagates forward in time?

Well, the WAVEFUNCTION p rogram d id not d o too bad a job (Figure 1). What d id that

program d o? It rotated each arrow in the initial wavefunction along the directworldline to a

point on the final wavefunction, this rotation taking p lace at the ratef = KE/h. Between each

initial dot and each final dot, this rotation used only th e SINGLE direct world line, rather than

sum ming the results of all worldlines. Buthey!it worked pretty well. So go with it. Our

original trial process, used in the WAVEFUNCTION p rogram, could be wr itten as a w ord

equation. Read the following equa tion from right to left; the multiplication symbol meansapp lied to or m ultiplied by.).

little arrow

at detection

event via

DIRECT path

rotate at rate

f = KE/h along

DIRECT path

one arrow

at emission

event

=

(trial version) (1)

But th is is just ou r first crud e try. When this job is done righttaking into accoun t ALL

path swe expect the final little arrow to be not only rotated but also to be different in length

from the initial arrow. We want to summ arize the results of ALL path s with a singlemu ltiplier or procedure. A more general case might have the overall form (read right-to-left):

little arrow

at detection

event via

ALL paths

" finagle

factor"

rotate at rate

f = KE/h along

DIRECT path

one arrow

at emission

event

=

(2)

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Figure 1. The WAVEFUN CTION program (that created this figu re) uses the trial version

of the propagator (Eq. 1 above) to prop agate each initial arrow in the lower wav efunction

along the SINGLE DIRECT worldline to a yield a final little arrow at a d ot in the later

wavefunction. The program then adds head-to-tail the little arrows from all the dots in

the initial wavefunction to form th e resulting arrow shown here at each dot in the later

wavefunction. NEXT STEPS: Instead of using only the d irect worldline, we will find a

more accurate p ropagator (of the form g iven in Eqs. 2 and 3) that allows u s to mimic the

result of exploring ALL paths between each arrow on th e initial wavefunction an d a

point on the final wavefunction. The final wavefun ction is still assembled by sum ming

all the propagator-calculated little arrow s at each detection event head -to-tail to give

Nature's (correct) resulting arrow at that point.

In equation (2) the finagle factor is something (as yet undetermined) that we stick in there

to give the correct answer. We will see later that the finagle factor depends on the time

lapse between emission and detection event. It also depends on the mass of the particleand

on some other stuff. Turns out that it does NOT depend on the spatial separation between

emission and detection eventthat is taken care of by the procedu re in the bracket labeled

rotate at ratef = E/h along DIRECT path.

The square bracket on the right side of equation (2) constitutes the thing th at changes the

arrow at the emission event into the little arrow at the d etection event. Equation (2) can be

simplified by substituting a simpler-looking expression for the squ are bracket:

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little arrow

at detection

event D via

ALL paths

one arrow

at emission

event E

= ( )

K D E, (3)

(The words little arrow at detection event reminds us that the resulting arrow at thedetection event comes from sum ming all the little arrows from every p oint in the initial

wavefunction.) In equation (3) the expression K(D,E) is a fun ction that d oes two things to the

initial arrow (du plicating wh at wou ld hap pen if we summed the results of exploring ALL

path s between emissionEand detectionD):

1. It rotates the initial arrow to the direction of the final arrow.

2. It changes the length of the initial arrow to the length of the little arrow at the d etector.

The lettersD and Eare pa rt of the function K(D,E) to remind us that K(D,E) depend s on the

differences in position and time between th e Emission and Detection events. We want to

find the function K(D,E) that mimics for us the exploration along ALL PATHS which, takentogether, change the initial arrow into the final arrow for a free particle.

II. THE PROPAGATORThe function K(D,E) is called thepropagator. (Feynman and Hibbs call it the Kernel, wh ich is

wh y it has the symbol K.) The free-particle prop agator tells us how an arrow from a single

event in the initial wavefun ction is changed into a little arrow at a single event in the final

wavefunction w hen the p article explores ALL WORLDLINES between the tw o events.

The RESULTING arrow at the final point (the wavefun ction at that poin t) is not just th e result

of the arrow at ON E point in the initial wavefunction. No, it is the result of ALL arrows in

the initial wavefun ction. To get the RESULTING arrow at th e final point, we use theprop agator to prop agate EACH arrow in the initial wavefunction forward in time to yield a

little arrow at that p oint in the final wavefunction. Then the compu ter ADDS UP all the little

arrows at that final point, head-to-tail as usu al, to reckon the resulting arrow at that p oint.

Then the whole procedu re is repeated for every other point in the final wavefunction. The

WAVEFUNCTION program d id this crudely and inaccurately, using the rotation along the

single direct w orldline and no change in length of the initial arrows in the p rocess (Figure 1).

In contrast, the propagator does the job righttelling us th e result of exploring ALL path s

between each initial and each final point in the absence of a p otential.

III. THE UNIFORM WAVEFUNCTION

The FindProp software program helps you to try ou t various forms of the free-particleprop agator un til you find one that d oes the job. BUThow will you know when the job is

don e correctly? We test our trial propagator by asking it to move forward in time an initial

wavefunction that is the same everywhere in space. Our un iform initial wavefun ction

consists of a string of arrows of the same length all pointing u pw ard (Figure 2). Uniformity is

boring, because it does not change with time. And we will test our trial prop agator by its

ability to bore usto construct a later wavefunction that also consists of a string of arrow of

the same length as before, and all pointing up ward .

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Figure 2. Initial wavefun ction u sed to test our trial prop agator fun ctions. This

wavefunction is un iform over a wide sp an of space. We demand that a su ccessful

propagator move this w avefun ction forward in time without significant change.(The poor comp uter cannot d eal with a row of arrows that goes on forever, so we taper

the ends to create a gradu al match between the u niform arrow s and zero w avefun ction.

See below: Section XI Some Fine Poin ts of Interest )

WHY should an initial wavefunction with equal arrows all pointing up ward p ropagate

forward in time in such a way th at the arrows all remain the same? The equality of the

squared magn itudes of these arrows imp lies an initial probability distribution everywhere

the same in x. If the probability is everywhere the sam e in space, it has now here to gono

empty region into w hich it can d iffuse. Because of the very w ide extent of this initial

wavefunction along thex direction, we expect that any motion of p robability will leave

local probability near the center u nchanged for a long time. This analysis does not tell us thatthe arrow s will also stay vertical with time, but we p ostulate this result as w ell. The p rogram

helps you to ap ply a trial propagator fun ction between every dot in the initial wavefunction

and each d ot in the final wavefun ction, allowing you to mod ify the p ropagator u ntil the

wavefunction d oes not change significantly with time.

TRY THIS ANALOGY: Sup pose the entire un iverse is all at the same

temp erature (as it may be in the d istant futu re!). Then, we expect, from that time

forward the universe will remain at the same temperatu re. We cannot try this

with a universe, so try it with you r heated house, initially all at the same

temp erature. You open up the front and back doors on a windless, cold d ay.

How long w ill it be before you notice a temperatu re change in the midd le of thefirst floor? Guess: one minu te. Then for ten second s we expect the temperatu re

to remain th e same in the region near the mid dle of the first floor.

Temp erature in this example is like probability in quan tum m echanics. We

would like to start w ith a w avefunction that is initially uniform over ALL space.

But w e cannot d o that: the compu ter cannot d o calculations with an infinite

nu mber of initial arrow s! So instead think of a wavefunction that is initially

un iform over a RANGE of space. Initially un iform m eans that at t= 0 all

arrows are the same length and point in the same direction. Arrow s of equal

length mean (when th is length is squared ) equal initial probability of find ing the

electron in this region of space. How w ill this probability chan ge? By leakingaway at the edges, just as the heat in your hou se leaks out throu gh the front and

back doors. But in the center the probability will remain the same for a w hile.

So we expect the LENGTH of arrows in the center to remain the same for some time. Why

should their DIRECTION remain the same also du ring this time? I am still working on the

answer to this equation, but tell me what you th ink of the following one-word argum ent:

Symm etry! For a u niform initial wavefunction infinite in spatial extent, why shou ld the

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initial up ward-pointing arrows rotate clockwise, rather than coun terclockwise, with time?

You say: Because ALL the arrow s rotate clockwise with timethat is wha t ordinary

mechanical stopw atches do. Yes, but clockwise rotation is just a conventionwhat w e mean by

clockwise. We could just as well have chosen COUNTERclockwise rotation for the

quantu m stopw atch, and everything we h ave developed in this course would be the same.

So the bottom line is: There is no reason w hy the arrows in a un iform initial wavefunction

shou ld rotate either clockwise or counterclockwise. So they do not rotate at all! Does thisargument convince you? Me neither.

OBJECTION: Wait a minu te! You are asking that the wavefunction arrow s not

change with time. Yet to construct each arrow of the later wavefunction, your

prop agator rotates the arrows along the direct world line from each point on the

initial wavefunctionthen at each p oint in the later wavefunction ad ds h ead-to-

tail all the arrows from the initial wavefunction. Make up your mind! Are the

arrows rotating or not?

DISCUSSION: Think of your autom obile moving along th e street, behaving

smoothly in pu blic. Yet und er the hood its crankshaft rotates rapidly, eachpiston oscillating up and d own thousands of times a minu te. Similarly, our

un ique wavefunction does not change with time; that is the smooth way it

performs in public. But undern eath the hood the quan tum m achinery spins

along: The electron Explores all worldlines, its quantu m clock rotating as it

explores each worldline. The propagator summ arizes for us this infinite number

of explorations. But, lo and behold, when th e propagator finishes its task and w e

sum marize all the little arrows a t each point in the final wavefunction, the

wavefunction remains u nchanged for this particular uniform initial case. The

form of the prop agator that leads to THIS result is the one we seek.

After finding th e correct p ropagator K(D,E) for the free particle, we can app ly it to ANYinitial wavefunction (for a free particle) to tell us how this wavefunction changes w ith time.

We will find th at most w avefunctions DO change with time!

IV. FINDING THE PROPAGATORThe software program FindProp (Find Prop agator) starts with the simple prop agator used

in the WAVEFUNCTION programone that simply rotates the arrow from each p oint in the

initial wavefun ction along th e direct world line to the point on the final wavefunction,

without changing th e length of this arrow. We call this the UNITY PROPAGATOR.

Figure 3 shows the result of app lying the unity prop agator. The initial un iform w avefunction

displays itself along th e bottom on the horizontal line t= 0. (Why does this initialwavefunction taper d own to zero at either end ? See below: Section XI Some Fine Points of

Interest.) The up per three rows of arrows show the results of app lying the un ity prop agator

at three later times. The box across the up per right tells us tha t the arrows in these three later

wavefunctions are mu ch longer than th e arrows in the initial wavefunction. Notice also that

these later arrow s all point at an inclined angle to the right instead of straight up . Finally, see

that these arrows change length, getting longer for later times. We need to find a new

propagator, with a finagle factor chosen so that all three of these errors are fixed -- allowing

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Figure 4. (Modified Figure 2 from the u nit Worldlines for Quantum Particles) Resulting arrow for many

alternative path s. The resulting arrow for a (nearly) comp lete Cornu spiral (left) from man y wor ldlines is

simplified (right) using contributions from only those w orldlines along wh ich the nu mber of rotations differs

by one-half-rotation or less from th e direct worldline. The boxed small fat black arrow in the middle of the

right-hand diagram is from the direct wor ldline. Notice relative size and d irection of this and resulting arrow .

V. FIRST TASK: MAKE LATER ARROWS VERTICALFire up the p rogram and look at the initial wavefunction, wh ich is un iform across most of the

screen. (For now ignore the sloping tap ered p rofile at each end .) Click above the initial

wavefunction to see how well our UN ITY prop agator works, reprod ucing and extending the

display show n in Figures 2 and 3.

To get a cleaner idea of what is going on, click on CH ANGE DISPLAY and then on CENTER

DOT. (Turn the PYRAMID on or off, as you wish.) Then click on OK. Now again click on

later times to show only ONE resulting arrow of the later wavefun ction, the arrow in thecenter.

Your first task is to make the arrow s at the later wavefunction p oint in the right d irection,

nam ely straight up . You d o this by telling the computer to ap ply an initial offset angle to the

initial arrow before rotating it along the d irect w orldline.

Click on CH ANGE PROPAGATOR, then on CH AN GE TO TRIAL PROPAGATOR, then on

CHANGE OFFSET ANGLE. Choose some angle other than zero, click OK, and try the new

prop agator. Continue changing the offset angle un til arrows at later wavefun ction all point

up ward . Answer the following question for the uniform initial wavefun ction (red nu mber 1

in upp er left of screen):

Q1. For what value of the initial offset angle do the arrows in later wavefunction point in the

same d irection as the u niform arrows of initial wavefunction?

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VI. SECOND TASK: MAKE LATER ARROWS INDEPENDENT OF TIMEWe have found an initial angle offset that keeps all later arrow s pointing in the correct

direction (straight u p). KEEP this value of the initial angle offset in what follows.

The second task is to make the arrow s keep the sam e length, u nchanging w ith time, for this

particular un iform initial wavefunction. NOTE: The length d oes not have to be equal to 30,

which is the length of the initial wavefunction a rrows; that is the Third Task, below. For nowwe just w ant all LATER arrows to have app roximately the same length.

To make later arrows not change with time, give the propagator a magn itude that itself

changes with time so as to cancel the change in length of the final arrow that w ould

otherwise occur.

How shouldthe magnitude of the propagator change with time? We assum e that this change

is of the form

magnitude of propagator time TIMEexponent( ) ( )

You w ill try out var ious values of the TIMEexponent to find the value that leads arrows in

later wavefunctions to have a length that does not change with time. Click on the CHANGE

PROPAGATOR button and check that un der TRIAL PROPAGATOR the initial offset angle

has the value you determined in Section V. Then click and on th e CHAN GE TIME

FUNCTION bu tton. Choose a TIMEexponent and try out the resulting propagator.

Q2. What value of the exponent yields a w avefunction w hose central arrow has a constant

length as time p rogresses?

VII. THIRD TASK: MAKE LATER ARROWS THE RIGHT LENGTHSo far we have changed the free-particle prop agator so that the later arrows p oint in the right

direction and have a length that d oes not change w ith time.

The third task is to make arrows in later wavefun ctions not only un changing in time but also

equal in length to the arrows of the initial wavefunction. We do th is by chang ing the

coefficient that mu ltiplies the prop agator.

Click on CHANGE PROPAGATOR and check that the offset angle and time exponent h ave

the values that you determined in Sections V and VI. Then click on CHAN GE

COEFFICIENT and choose a value that you feel will make arrows the same length in laterwavefunction. Try this out u ntil later arrows are the same length as the initial arrows, at least

near the center of the wavefun ction for the uniform initial wavefunction (red num ber 1 in

up per left of screen).

Q3. For what coefficient of the p ropagator do arrows in the later wavefunctions have

(approximately!) the same length as the arrows in the initial wavefunction?

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VIII. FOURTH TASK: MAKE LATER ARROWS INDEPENDENT OFPARTICLE MASSSo far we have changed the free-particle prop agator so that the later arrows p oint in the

correct direction, do not change w ith time, and have the right length for the initial choice of

particle m ass.

Next, we wan t the prop agator to give us the correct later wavefunction no matter what themass of the particle. Of course, our p rimary client is the electron, but quantum m echanics

mu st be correct for particles of different m asses.

As before, we assum e that the prop agator changes with some pow er of the mass:

magnitude of propagator massMASSexponent

( ) ( )

where the MASSexponent w ill probably be different from the exponent found for the time

function.

Click on CH ANGE PROPAGATOR and check that all your ear lier results from Sections V,

and VI and VII are entered in th e TRIAL PROPAGATOR. Then click on CHANGE MASS

FUNCTION and choose a MASSexponent for the mass fun ction.

Change the pa rticle mass starting with the CHANGE WAVEFUNCTION button. Use only

values of the m ass between 15 m(electron) to 100 m(electron). Try out various values of

MASSexponent u ntil you find the exponent that m akes the later arrows (approximately!) the

same length as the initial arrow s, within this range of mass-values for the uniform initial

wavefunction (red num ber 1 in u pp er left of screen).

Q4. For what value of the MASSexponent d o arrow s in later wavefunctions have the same

length as those in th e initial wavefun ction, indep endent of particle mass?

IX. FIFTH TASK: APPLY THE PROPAGATOR TO OTHERWAVEFUNCTIONSIn deriving the free-particle propagator, we u sed an un realistic wavefunctionone that is

un iform over space. But now that we have derived the correct prop agator, we can app ly it to

ANY initial free-par ticle wavefunction an d wa tch this wavefunction develop in time. The

fifth task is to do this and report the results.

Choose DEFAULT VALUES from the DEFAULT menu at the top of the initial screen. Then

choose the CORRECT PROPAGATOR and ap ply it to wavefunctions 2 through 6. Changewavefunctions by clicking on the bu tton N EXT WAVEFUNCTION. (For more on the sharp -

end ed u niform wavefun ction #2, see the next section: FINE POINTS.)

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Q5. For wavefunction #4, describe QUALITATIVELY how the w avefun ction w ith the red

nu mber 4 prop agates forward in time. In particular:

Q5A. Does the AVERAGE position move with tim e? (YES or NO)

AVERAGE position? For each time, is there a point from w hich the wavefunction

looks sort of the sam e to the left of this point as looking to the right? That is the

average position. Does the average position move with time?

Q5B. Does the PROFILE spread w ith time? (YES or NO)PROFILE? As time passes, do the longish arrow s spread farther away from the

average position of question A?

Q5C. Do you notice any oth er effect wor th rep orting? (If yes, then p lease describe.)

Q6. Repeat qu estions Q5A, B, and C for wavefunction #5.

Q7. Repeat qu estions Q5A, B, and C for wavefunction #6.

X. SUMMARY OF RESULTSEquations (2) and (3) in Section I give a kind of recipe for the propagator K(D,E) between an

emission eventEand a detection event D:

K D E,( ) =

" finagle factor"

between events

E (emission) and

D (detection)

rotate at rate

f = KE/h along

DIRECT path

from E to D

(4)

The initial trial propagator, wh ich w e called the UNITY PROPAGATOR simply ap plied th e

procedu re in the second paren thesis at the right of Equation (4), ignoring any instructions

that might be contained in the first parenthesis. By trial and error, you now kn ow a LOT

more about the finagle factor in the first parenthesis of this equation. This finagle factorallows the single function K(D,E) to mimic the result of exploring ALL paths between Ean d

D.

Q8. Complete IN WORDS the following set of instructions, wh ich are ord ers to the

propagator K(D,E) in prop agating the initial arrow at the emission eventEto the little final

arrow at detection eventD. Your instru ctions should include your resu lts for all four tasks

that you carried ou t to find the correct prop agator. Be briefthe propagator is an imp atient

function!

Hey, Propagator! Here is what you gotta d o. Between each point on the initial wavefunction

and each point on the final wavefun ction:1. Before rotating the initial arrow along the direct world line from Eto D, give it an

initial "offset angle" equal to . . .

2. In the process, mu ltiply the length of the little arrow by the time-factor . . . .

3. . . . . .

4 . . . . . .

. . . . . . . .

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If you w ant to see the mathem atical equivalent to your word instructions, look at the

App endix below.

Q9. PREDICT wh at the CORRECT prop agator w ill do to an initial wavefun ction th at is

identical to the un iform wavefun ction w ith tapered ends (Wavefun ction #1) EXCEPT that

every initial arrow is tilted 30 degrees from the vertical. CHOOSE ON E of the following:

Q9A: The initial wavefunction will propagate forward in time unchanged (in the sameapp roximation that th e original Wavefunction #1 prop agates forward un changed). If so, give

an argument why this should be so. OR

Q9B: The initial wavefun ction will change with time. If so, give an argum ent why this

should be so.

XI SOME FINE POINTS OF INTERESTOn the way th rough the four tasks of find ing the free-particle propagator, you h ave probably

noticed som e strange things, a few of which we can explain here.

TAPERED WAVEFUNCTION. The flat initial wavefun ction that we have used to set upthe propagator has tapered ends. Without this taper, arrows in later wavefunctions show a

fine-grained unevenness. Observe this by choosing CORRECT PROPAGATOR, then click

once on the NEXT WAVEFUNCTION bu tton. The resulting w avefunction (red num ber 2 in

the box at the upper left of the screen) has sharp ends. When you click on later times, the

later wavefunctions have arrow s of un even length, as you can verify. The sharp

discontinu ities at the end s spread rapid ly through the wavefunction. This is a fun dam ental

result of quan tum mechanics: Sud den discontinu ities in slope of the wavefunction p rofile

lead to rap id changes of the wavefunction in time. (This is the basis of the Schroed inger

equation, which relates changes of slope of the wavefunction to its change with time.)

INVALID WAVEFUNCTION S. You may observe crazy wavefunctions at very short timesafter the initial wavefunction. This occurs especially when th e particle mass is large. This

effect is NOT fun dam ental physics. Rather, it comes from representing the wavefunction

with d iscrete, evenly-spaced ar rows.

The resulting ar row at the center of a later wavefunction (for examp le) is built up of

contributions from all the arrows in the initial wavefunction. We rotate each of these arrows

along the d irect worldline between the initial arrow and the center point of the later

wavefunction. Ord inarily these contributions add head-to-tail to form a Cornu Spiral, like

Figure 4. However, a smooth Cornu Spiral results only when the difference in rotation is

small between ad jacent ar rows in the initial wavefunction.

Now think of two worldlines stretching from tw o adjacent arrow s near one end of the initial

wavefunction to the central point in a later w avefunction, as shown in this figure.

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If the time is very short, then the tw o slanting worldlines are nearly horizontal (covering a

great d istance in a shor t time), which describes a high velocity. High velocity means a large

kinetic energy. (Greater mass also means a greater kinetic energy.) Large kinetic energy KE

means a high rotation ratef = KE/h. So the arrows at the edge of the initial wavefun ction

make m any rotations on th eir way to th e central position of a later wavefunction.

Now if the DIFFERENCE between the rotations of adjacent initial arrow s is great, the result is

a jagged Cornu Spiral, or the destruction of the Cornu Spiral altogether. You can observe this

directly, as follows:

In the CHAN GE WAVEFUNCTION d isplay, set the mass equal to 100 m(electron). Choose

CENTER DOT in the CHANGE DISPLAY window. Then use the CORRECT PROPAGATOR

and the flat initial wavefunction #1 (with tapered end s). You w ill see that, for short times,

the Cornu Spiral is destroyed.

In summ ary, to avoid crazy w avefunctions du e to the use of discrete arrows, we avoid setting

the time interval to very small values, especially w hen th e mass is large.

APPENDIX: The Free-Particle Propagator in Complex NotationSUGGESTION: Review the introdu ction to complex nu mbers in the App end ix of an earlier

un it in th is manu al, the one called The Electron in a Poten tial.

The arrow at the emission event can be represented as a complex number, a nu mber w hich

has magn itude and d irection, the direction sometimes described by the word p hase.

The little arrow at the d etection event that results from this emission arrow can also be

represented as a complex nu mber (magnitud e and p hase). And th e propagator is a third

comp lex number w hich tells the arrow at the emission event how to shrink and turn to

yield the little arrow at the d etection event.

As you have discovered, the propagator contains an initial offset angle (Task One above), is a

function of time (Task Two), and also is a fun ction of the mass of the particle (Task Four).Therefore the prop agator is not a comp lex nu mber bu t acomplex function that changes with

the time separation and par ticle mass. Equation (4) tells us that the prop agator also has a

factor that rotates the initial arrow along the d irect w orldline between emission and

detection.

All of these cond itions are satisfied by the following free-part icle prop agator :

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K D Em

hT

i i mX

hT, exp exp

/

( ) =

1 2 2

4

(5)

HereX is the spatial distance between emission an d detection events, and Tis the time lapse

between the two events. The function exp means e (the base of natural logarithms) with an

exponent given by the parenth esis that follows it. So for those with sharp eyesight, equation

(5) can be rewritten in the form:

K D Em

hTe e

i i mX

hT,/

( ) =

1 24

2

(6)

Now , have a look at equations (5) and (6). Those with some acquaintan ce with comp lex

nu mbers w ill notice the following:

1. The imaginary nu mber i in the exponents tell us th at these factors represent rotations

rather than an exponential increase or decrease of magnitud e.

2. The exponen t i/ 4 represents an offset angle of MINUS one-eighth tu rn, or minus 45degrees that you derived (did you?) in Task One.

3. The inverse square root T1/ 2

in the initial square bracket is the result you d erived in

Task Two.

4. The square root m+1/ 2

in the initial square bracket is the resu lt you d erived in Task Four.

5. The argument of the second exponential function can be written as the imaginary num ber

i times the quantity:

mX

hT

mX

hTT

mX

hTT

KE

hT f T

2 2

2

2

2

2

22 2 2= = =

= =

number of

rotations

along direct

worldline

(7)

Here KEis the classical kinetic energy of the p article along the d irect w orldline between

emission and detection. And KE/h = f is the frequencyf of rotation of the quantu m clock for

a free electron. (See equat ion 3 of the Free Electron u nit). Since Tis the time between

emission and detection, therefore fT is the num ber of rotations along this direct worldline.

Why is there a 2

in the last expression on the right of equation (7)? Because the imaginary

exponent of e measures angles in radians rather than nu mber of rotations. And th ere are 2radians per rotation.

NOTE: You can also show th at the exponent can be written as i times the quantity:

mX

hT

S

h

S

h

S2 2

2= =

( )=

/ h

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where S is the classicalaction compu ted along the direct world line and h ( )h/ 2 , calledh-bar, has wide use in p hysics. To demonstrate th is form, see equation (17) in the u nit on

The Principle of Least Action for th e case of zero potential (g = 0).

So the comp lex representat ion of the free-particle propagator in equations (5) and (6) fulfills

all the cond itions we d erived for this propagator by our trial-and -error method u sing theFind Prop software.

Copyright 2000 Edw in F. Taylor

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