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1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 345 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 350 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "Wavepackets in quantum me chanics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 744 "We look at solutions to the free-particle Schroedinger wave equat ion in one spatial dimension. The first step is to look at exact solut ions to the equation for fixed momentum, i.e., for sharp energy. These plane-wave solutions are completely delocalized in space. Then we for m superpositions from these states which have the property of being lo calized in space, and having a spread in momentum (i.e., free-particle energy). These solutions can be imagined to represent an ensemble of \+ particles with a spread of momenta around an average value, i.e., to r epresent a beam of particles. The dispersion of such wavepackets in sp ace with time is demonstrated; it can be explained due to the presence of faster and slower components in the ensemble." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The one-dimensio nal time-dependent Schroedinger wave equation (TDSE) for free particle s is given as (eventually we choose units where hbar=" }{TEXT 257 1 "m " }{TEXT -1 11 "=1, we use " }{TEXT 19 2 "h_" }{TEXT -1 11 " for hbar) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "TDSE:=I*h_*diff(psi(x,t),t)=-h_^2/(2*m)*diff(p si(x,t),x$2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The LHS of the T DSE can be interpreted as the energy operator acting on the wavefuncti on psi(" }{TEXT 261 1 "x" }{TEXT -1 2 ", " }{TEXT 260 1 "t" }{TEXT -1 70 "), while the RHS represents the kinetic energy operator acting on \+ psi(" }{TEXT 259 1 "x" }{TEXT -1 1 "," }{TEXT 258 2 " t" }{TEXT -1 264 "). This wave equation is different from the one known in classica l electromagnetism: it represents waves of massive particles that obey the non-relativistic energy-momentum relationship (dispersion relatio n). The kinetic energy operator was formed on the basis of " } {XPPEDIT 18 0 "T = p^2/(2*m);" "6#/%\"TG*&%\"pG\"\"#*&F'\"\"\"%\"mGF)! \"\"" }{TEXT -1 41 " where the momentum operator is given as " } {XPPEDIT 18 0 "p = -I*h_*d/dx;" "6#/%\"pG,$**%\"IG\"\"\"%#h_GF(%\"dGF( %#dxG!\"\"F," }{TEXT -1 231 ". Our first task is to demonstrate the fo rm of the plane-wave solutions (the unfamiliar reader should look up t he solution to the relativistic wave equation for electromagnetic wave s in a first-year physics text before proceeding)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "The wave equatio n denoted as TDSE is obviously separable in position and momentum: the solution should be given as a product of a time-dependent and a posit ion-dependent factors. If we denote by " }{TEXT 262 1 "E" }{TEXT -1 36 " the energy of the particle, and by " }{TEXT 263 1 "p" }{TEXT -1 135 " its momentum value (a number, not to be confused with p-operator used above) we have the classical relationship for the free particle \+ " }{XPPEDIT 18 0 "E = p^2/(2*m);" "6#/%\"EG*&%\"pG\"\"#*&F'\"\"\"%\"mG F)!\"\"" }{TEXT -1 75 ". Keeping in mind that the LHS of the TDSE repr esents the energy operator (" }{XPPEDIT 18 0 "I*h_*d/dt;" "6#**%\"IG\" \"\"%#h_GF%%\"dGF%%#dtG!\"\"" }{TEXT -1 185 ") we are looking for a so lution where the time-dependent factor when acted upon by the LHS prod uces the energy as a factor, while on the RHS we want the two derivati ves with respect to " }{TEXT 264 1 "x" }{TEXT -1 131 " to do the same \+ with the position-dependent factor. The only choice (apart from an ove rall multiplicative normalization factor) is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "phi_p:=exp(-I/h_*p^2/(2*m)*t)*exp(I/h_*p*x);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We substitute" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(psi(x,t)=phi_p,TDSE);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "and divide out the solution to observe th at the answer is correct for any value of " }{TEXT 265 1 "p" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(%/phi_p );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We have meaningful solution s for any real-valued " }{TEXT 266 1 "p" }{TEXT -1 18 ". For each ener gy " }{TEXT 267 1 "E" }{TEXT -1 26 " there are two solutions: " }} {PARA 0 "" 0 "" {TEXT -1 31 "one which is right-travelling (" }{TEXT 268 1 "p" }{TEXT -1 39 ">0), and one which is left-travelling (" } {TEXT 269 1 "p" }{TEXT -1 4 "<0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 364 "The wavefunctions phi_p are labeled ther efore by their momentum value, indicating that we have found an infini te family of solutions. We think of the momentum value as being fixed \+ for the particle. The interpretation of phi_p is that it represents a \+ probability amplitude. The squared magnitude gives the probability to \+ find the particle at some location in space " }{TEXT 270 1 "x" }{TEXT -1 17 " at a given time " }{TEXT 271 1 "t" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalc(abs(phi_p)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "This, of course, is confusing. The result for the probability to find the particle at (" }{TEXT 274 1 "x" } {TEXT -1 1 "," }{TEXT 273 2 " t" }{TEXT -1 42 ") is independent of pos ition (at any time " }{TEXT 272 1 "t" }{TEXT -1 97 "). Moreover, since the summation over all allowed positions involves an integral over th e entire " }{TEXT 275 1 "x" }{TEXT -1 755 "-axis, there is a serious p roblem: we cannot find a normalization constant such that the sum over all probabilities remains finite. This represents a subtle problem th at is associated with the infinite volume of space. In the real world \+ this should not be a problem, and we can impose a finite volume (if wo rst comes to worst, the volume of the universe). For any finite volume the plane-wave solutions can be normalized such that the probabilisti c interpretation is salvaged, but there is a price to pay: the freedom to choose any real-valued momentum (or energy) for the free particle \+ will be sacrificed, as the boundary conditions (periodicity at the edg es of the volume) will impose a volume-dependent discreteness of allow ed momentum (energy) values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "For the sake of our discussion it will be adva ntageous at first to ignore this problem, and deal with solutions defi ned on the entire " }{TEXT 276 1 "x" }{TEXT -1 568 "-axis. We sacrific e the interpretation of the solution for the freedom to choose any ene rgy value, and to use the simple unnormalized solutions phi_p. These s olutions represent a complete basis set. Any function defined on the r eal axis that is a potential candidate for solving (satisfying) the TD SE can be expanded in this basis. For the sake of understanding the ba sis functions we look at the graphs of them, i.e., of their real and i maginary parts. Picking out the real (or imaginary) part can be done b y hand, but one has to take into consideration both factors." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Wr:=unapply(evalc(Re(phi_p)) ,p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Wi:=unapply(evalc(I m(phi_p)),p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h_:=1; m:= 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Let us observe the evolution of th e real and imaginary parts of the plane wave for the case of " }{TEXT 277 1 "p" }{TEXT -1 89 "=1/2. We generate a sequence of snapshots in t ime, and graph over a fixed position range:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "PLf:=s->plot([subs(t=s,Wr(0.5)),subs(t=s,Wi(0.5))] ,x=-10..10,color=[red,blue]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "PLs:=seq(PLf(it*0.2),it=0..50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(PLs,insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "As expected, the real and imaginary parts of the pl ane wave remain in phase. The crests (and zeroes) travel with a speed \+ that depends on the chosen momentum. The reader can verify this by cho osing different values of " }{TEXT 278 1 "p" }{TEXT -1 25 ". For the c ase chosen of " }{TEXT 279 1 "p" }{TEXT -1 9 "=1/2 and " }{TEXT 280 1 "t" }{TEXT -1 62 "_fin = 10 units the phase of the wave traveled a dis tance of " }}{PARA 0 "" 0 "" {TEXT -1 9 "Dx = 0.5 " }{TEXT 281 1 "p" }{TEXT -1 1 " " }{TEXT 282 1 "t" }{TEXT -1 4 "_fin" }}{PARA 0 "" 0 "" {TEXT -1 147 "Therefore the phase velocity is only one half of what on e might have expected naively. It is the result of the definition of t he phase velocity as " }{XPPEDIT 18 0 "v[ph] = omega/k;" "6#/&%\"vG6#% #phG*&%&omegaG\"\"\"%\"kG!\"\"" }{TEXT -1 31 " . We have for the wave number " }{XPPEDIT 18 0 "k = p/h_;" "6#/%\"kG*&%\"pG\"\"\"%#h_G!\"\"" }{TEXT -1 27 " (in our units h_=1, i.e.," }{TEXT 285 2 " k" }{TEXT -1 1 "=" }{TEXT 284 1 "p" }{TEXT -1 59 "), and the circular frequency \+ is related to the energy as " }{XPPEDIT 18 0 "E = h_*omega;" "6#/%\"E G*&%#h_G\"\"\"%&omegaGF'" }{TEXT -1 16 " (in our units: " }{XPPEDIT 18 0 "E = omega;" "6#/%\"EG%&omegaG" }{TEXT -1 71 "). Thus, we have fo r the phase velocity of a Schroedinger matter wave: " }{XPPEDIT 18 0 " v[ph] = p/(2*m);" "6#/&%\"vG6#%#phG*&%\"pG\"\"\"*&\"\"#F*%\"mGF*!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 546 "It is important to realize that contrary to electromagne tic waves in vacuum (where all plane waves move with the velocity of l ight) the matter waves propagate with phase velocities that differ in \+ magnitude for different momentum values. However, one does know that e lectromagnetic waves in a medium (when the Maxwell equations are coupl ed to an ionic plasma) follow a different dispersion relation which re sults in a slowdown compared to propagation in vacuum, and dispersion \+ of wavepackets (for an elementary discussion see Richard W. Robinett: \+ " }{TEXT 286 17 "Quantum Mechanics" }{TEXT -1 42 ", Oxford Unversity P ress 1997, chapter 2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 11 "Wavepackets" }}{PARA 0 "" 0 "" {TEXT -1 416 "Clearly the phase velocity can not be the physically important qu antity. Also, the complete delocalization of the plane waves appears t o make them inappropriate for the description of a localized bunch of \+ particles. Nevertheless, we will observe shortly that a superposition \+ (linear combination) of plane waves can describe a localized wave. We \+ begin by superimposing just a few plane waves, and graphing the result ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 435 "In \+ fact, when choosing the superposition we will have the freedom to admi x waves of certain momenta with pre-determined mixing coefficients. It is this momentum profile which we are free to choose that will lead t o physically understandable propagation properties. Let us do the simp lest-possible thing: we choose a window of a few discrete momentum val ues with plane waves at these momenta superimposed with equal mixing c oefficients." }}{PARA 0 "" 0 "" {TEXT -1 72 "For the average momentum, the spread, and the number of waves we choose:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "p_avg:=1; Dp:=1/2; Np:=5;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 36 "The chosen discrete momentum values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p_d:=i->p_avg+Dp*(i-(Np+1)/2)/Np;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(p_d(i),i=1..Np);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "We choose real mixing coefficient s, and can therefore superimpose the real and imaginary parts of the p lane waves separately to form the real and imaginary parts of the wave packet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "WPr:=add(Wr(p_d( i)),i=1..Np):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "WPi:=add(W i(p_d(i)),i=1..Np):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "We procee d as before to obtain an animation of the time evolution, except that \+ we add a graph of the magnitude squared (scaled down to fit on the sam e graph):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "PLf:=s->plot( [subs(t=s,WPr),subs(t=s,WPi),0.3*(subs(t=s,WPr)^2+subs(t=s,WPi)^2)],x= -15..15,color=[red,blue,green]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "PLs:=seq(PLf(it*0.1),it=0..50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(PLs,insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The magic seems to work:" }}{PARA 0 "" 0 "" {TEXT -1 53 "1) We have obtained a localized object that can move. " }}{PARA 0 "" 0 "" {TEXT -1 113 "2) The object moves at the expexted \+ speed [5 distance units in 5 units of time for an average momentum of \+ 1 unit]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Several questions remain, which should be explored by the reader \+ (using modifications to the lines above):" }}{PARA 0 "" 0 "" {TEXT -1 72 "1) What happens for narrower/wider momentum distributions (variati on of " }{TEXT 19 2 "Dp" }{TEXT -1 2 ")?" }}{PARA 0 "" 0 "" {TEXT -1 59 "2) Check the travel distance for different average momenta." }} {PARA 0 "" 0 "" {TEXT -1 80 "3) Does the wavepacket preserve its shape (unlikely due to the momentum spread)?" }}{PARA 0 "" 0 "" {TEXT -1 36 "4) What happens at larger distances " }{TEXT 287 1 "x" }{TEXT -1 119 " ? Can the problem be ameliorated by choosing finer momentum disc retizations (more superimposed plane-wave components)?" }}{PARA 0 "" 0 "" {TEXT -1 149 "5) What are the signatures of a larger momentum spr ead? What can one say about the structures in the oscillations ahead o f the centre, and behind it?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 288 20 "Gaussian wavepackets" }}{PARA 0 " " 0 "" {TEXT -1 714 "Some of the concerns/questions raised above are r esolved by considering a 'perfect' superposition: instead of summing o ver discrete momentum values we can integrate over the entire range of allowed momenta and weigh the plane waves with a momentum profile. A \+ practical issue that arises is whether we can carry out the integral i n closed form. For a Gaussian momentum profile, which is characterized by an average momentum and a width (standard deviation) the integral \+ can be carried out, and the wavefunction (packet) can be calculated in the coordinate representation. The result is the so-called Gaussian w avepacket, which is localized in real space. It can be extended also t o more than one spatial dimenstion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 1199 "The Gaussian (or other, e.g., Lorentzi an) wavepacket maintains its momentum profile, which is consistent wit h free-particle propagation (no external forces). It spreads in positi on space due to the dispersion of the different plane-wave components \+ (they move with different phase velocities). The wavepacket conspires \+ to achieve this: the product of momentum and position uncertainties gr ows as a function of time (and does not violate Heisenberg's principle , as it involves an inequality). This dispersion is entirely consisten t with classical dispersion. As long as one sticks to the ensemble int erpretation of quantum mechanics there is no conceptual problem with t his growth in uncertainty (delocalization due to dispersion). One does not attach meaning to the wavepacket as far as individual particles a re concerned, one remains on the safe side by the statement that the S chroedinger wave equation describes an ensemble (and not individual pa rticles). The ensemble, of course, can be realized sequentially (an ex ample from a multiple-slit experiment: dots appear on a screen randoml y one after the other, forming eventually an interference pattern that can be calculated by quantum mechanics)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We proceed to carry out the integral whic h calculates psi(" }{TEXT 292 1 "x" }{TEXT -1 1 "," }{TEXT 293 1 "t" } {TEXT -1 70 ") from a plane wave superposition with a Gaussian profile centered at " }{TEXT 291 1 "k" }{TEXT -1 1 "=" }{TEXT 290 1 "q" } {TEXT -1 33 ". We need the following integral:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "Int(exp(-a^2/2*(k-q)^2+I*k*x-I*h_*k^2/(2*m)*t) ,k=-infinity..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Here \+ the first argument in the exponential represents the superposition coe fficient in which " }{TEXT 342 1 "a" }{TEXT -1 60 " controls the momen tum spread (it has dimensions of length)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "The argument in the exponential to be integrated is give n as (the calculation tries to follow S. Fluegge: " }{TEXT 343 14 "Pra ctical QM I" }{TEXT -1 12 ", Springer):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "arg:=-a^2/2*(k-q)^2+I*k*x-I*h_*k^2/(2*m)*t;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(arg);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "That doesn't do the job to complete the s quare." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "arg1:=completesquare(arg,k );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(b>0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "I1:=unapply(int(exp(-b^2*z^2 ),z=-infinity..infinity),b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "I1(b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This integral can \+ be used (without checking the positivity of " }{TEXT 289 1 "b" }{TEXT -1 39 "). We realize that a constant shift in " }{TEXT 341 1 "z" } {TEXT -1 124 " does not affect the result (the integration range remai ns the same). A stretching of the variable has the following effect:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "arg11:=op(1,arg1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k0:=solve(arg11,k);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "For some reason " }{TEXT 19 5 "sol ve" }{TEXT -1 77 " repeats the solution. We will need to lift out one \+ of them later, i.e., use " }{TEXT 19 5 "k0[1]" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "One can use the change of variable s procedure that is part of the student package: " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "assume(a>0,h_>0,t>0,m>0); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "changevar(k-k0[1]=y,Int(exp(arg1),k=-infi nity..infinity),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "simp lify(%): #Maple won't do it due to the shift of b into the complex pla ne" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We use a cheat to reach the goal: our integral I1(" }{TEXT 294 1 "b" }{TEXT -1 48 ") has been sim plified under the assumption that " }{TEXT 295 1 "b" }{TEXT -1 33 ">0. It is, in fact, valid for Re(" }{TEXT 296 1 "b" }{TEXT -1 98 ")>0, im plying a complex integration contour parallel to the real axis. Thus, \+ we can carry out the " }{TEXT 297 1 "k" }{TEXT -1 48 " integration and need only to keep track of the " }{TEXT 298 1 "x" }{TEXT -1 30 "-depe ndent exponential factor." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "b0:=simplify(sqrt(arg11/(k-k0[1])^2));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "I1(b0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "arg12:=op(2,arg1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a rg12cs:=completesquare(arg12,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "psi:=I1(b0)*exp(arg12cs);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Note that psi still looks like a mess: it is not obvious \+ what it describes, since " }{TEXT 299 1 "x" }{TEXT -1 174 " is not cen tered on a real value. The result appears to agree with the textbook c alculation. The starnge form of psi is needed to accomplish the growin g uncertainty in Dx Dp." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " factor(psi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The density is ea sier to understand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rho: =evalc(simplify(abs(psi)^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "completesquare(rho,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "N ow it is evident that the density represents a Gaussian centered on th e expected location: " }{TEXT 301 1 "p" }{TEXT -1 4 "=h_ " }{TEXT 300 1 "q" }{TEXT -1 117 " is the average momentum of the GWP, therefore th e distance travelled by the position expectation value (starting at " }{TEXT 303 1 "x" }{TEXT -1 6 "=0 at " }{TEXT 302 1 "t" }{TEXT -1 8 "=0 ) is: " }{TEXT 304 2 "s " }{TEXT -1 2 "= " }{TEXT 305 3 "v t" }{TEXT -1 3 " = " }{TEXT 306 2 "pt" }{TEXT -1 1 "/" }{TEXT 307 2 "m " }{TEXT -1 2 "= " }{TEXT 308 1 "t" }{TEXT -1 4 " h_ " }{TEXT 309 1 "q" }{TEXT -1 1 "/" }{TEXT 310 3 "m ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Let us check the normalization integral. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(rho,x=-infinity..in finity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The normalization st ays constant in time, but for a correct probabilistic interpretation w e would need to adjust it by multiplying it with the" }{MPLTEXT 1 0 0 "" }{TEXT -1 8 " factor " }{TEXT 19 20 "sqrt(a/(2*Pi^(3/2)))" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 102 "For a graph we set a few obv ious constants, and choose the average momentum, and the spread parame ter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rho0:=subs(h_=1,m=1,a=1,q=2, (a/(2*Pi^(3/2)))*rho);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "a nimate(rho0,x=-2..18,t=0..5,color=green,numpoints=200,frames=25);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "WPr:=evalc(Re(subs(h_=1,m=1, a=1,q=2,sqrt(a/(2*Pi^(3/2)))*psi))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "WPi:=evalc(Im(subs(h_=1,m=1,a=1,q=2,sqrt(a/(2*Pi^(3/2 )))*psi))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "PLf:=s->plo t([subs(t=s,WPr),subs(t=s,WPi),2*subs(t=s,rho0)],x=-5..20,color=[red,b lue,green],title=cat(\"time= \",convert(s,string))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "PLs:=[seq(PLf(it*0.2),it=0..25)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(PLs,insequence=true, view=[-5..20,-0.6..1.2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "We \+ observe the asymmetry of the oscillatory structure at the end of the p ropagation: the fast components ahead of the wavepacket are oscillatin g more quickly than the slow components left behind. This is not appar ent at t=0. We explore this behaviour further by looking at the flux ( current density)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We can calculate the flux of the wavepacket." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "j:=phi->(h_/(2*I*m))*(conjugate(phi )*diff(phi,x)-phi*diff(conjugate(phi),x));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "assume(q,real); assume(x,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(diff(psi,x)/psi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "What does this imply? The derivative of \+ psi is proportional to psi and it is possible to factor out the densit y in " }{TEXT 313 1 "j" }{TEXT -1 1 "(" }{TEXT 312 1 "x" }{TEXT -1 1 " ," }{TEXT 311 1 "t" }{TEXT -1 48 "), as the same holds for the complex conjugate. " }}{PARA 0 "" 0 "" {TEXT -1 39 "This is why it makes sens e to simplify " }{TEXT 320 1 "j" }{TEXT -1 1 "(" }{TEXT 319 1 "x" } {TEXT -1 1 "," }{TEXT 318 1 "t" }{TEXT -1 6 ")/rho(" }{TEXT 317 1 "x" }{TEXT -1 1 "," }{TEXT 316 1 "t" }{TEXT -1 62 "). Exploration of the l ine below shows why the assumptions on " }{TEXT 314 1 "q" }{TEXT -1 2 ", " }{TEXT 315 1 "x" }{TEXT -1 13 " were needed." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "simplify(diff(conjugate(psi),x)/conjugate(ps i));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "res:=simplify(j(psi )/rho); # This doesn't work without the assumptions on x and q!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 284 "According to classical flows (in \+ hydrodynamics, or charge flows in electrodynamics) the current density can be interpreted as particle density times a local (position-depend ent) velocity. Thus, we have calculated a position-dependent velocity \+ field for the matter wavepacket. At time " }{TEXT 344 1 "t" }{TEXT -1 18 "=0 it is constant:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s implify(subs(t=0,res));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "At tim e zero the flux is given by a constant velocity times the density prof ile. At " }{TEXT 321 1 "t" }{TEXT -1 153 "=0 all particles described b y the ensemble move with the same velocity irrespective of their locat ion! At later times this changes as the result becomes " }{TEXT 322 1 "x" }{TEXT -1 52 "-dependent. The faster particles in the inital GWP ( " }{TEXT 323 1 "k" }{TEXT -1 1 ">" }{TEXT 324 1 "q" }{TEXT -1 155 ") m ove further ahead, while the slower particles fall behind the average \+ position. By how much the velocity profile changes depends on the widt h parameter " }{TEXT 325 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The velocity profile is r eal-valued:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "res:=simplif y(evalc(res));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "This is equival ent to Fluegge's eq. (17.12)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "res1:=simplify(subs(h_=1,m=1,a=1,q=2,res));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 208 "P0:=plot(subs(t=0.1,res1),x=-5..10,color=bl ack): P1:=plot(subs(t=1,res1),x=-5..10,color=red): P2:=plot(subs(t=2,r es1),x=-5..10,color=blue): P3:=plot(subs(t=3,res1),x=-5..10,color=gree n): display(P0,P1,P2,P3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "The se straight lines illustrate the dispersion: we can find the location \+ of the average momentum value is space at a given time:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "l1:=simplify(subs(t=1,res1)); l2:=s implify(subs(t=2,res1)); l3:=simplify(subs(t=3,res1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve(l1=2,x); solve(l2=2,x); solve (l3=2,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The velocity " } {TEXT 345 1 "v" }{TEXT -1 3 " = " }{TEXT 346 1 "p" }{TEXT -1 1 "/" } {TEXT 347 1 "m" }{TEXT -1 80 " is maintained at the center position of the WP at all times: for our choice of " }{TEXT 348 1 "q" }{TEXT -1 4 "=2 (" }{TEXT 349 1 "m" }{TEXT -1 14 "=1) and times " }{TEXT 350 1 " t" }{TEXT -1 37 "=1,2,3 this happens at the locations " }{TEXT 351 1 " x" }{TEXT -1 74 "=2,4,6. Furthermore, it is evident that the velocity \+ dispersion is linear!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The graphs illustrate that something dramatic happens \+ to the velocity profile early on (0<" }{TEXT 326 1 "t" }{TEXT -1 51 "< 1), and that it becomes less interesting later on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "animate(res1,x=-5..10,t=0..5,frames=30,colo r=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Note that despite of the fact that the WP moves forward with an appreciable velocity on av erage, and has a constant 'local velocity' at " }{TEXT 327 1 "t" } {TEXT -1 47 "=0, the Gaussian distribution of velocities at " }{TEXT 328 1 "t" }{TEXT -1 355 "=0 does imply the presence of some negative v elocity components. A small part of the WP moves backwards. In the ens emble interpretation of the WP it means that there are some particles \+ that move to the left. The velocity distribution alone, of course, doe s not reveal what fraction of particles described by the WP carries ou t this 'non-intuitive' motion." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "If we had been more careful to normalize at " }{TEXT 329 1 "t" } {TEXT -1 119 "=0 we would be getting a conserved probability of unity, but the main point is that our WP maintains its normalization!" }} {PARA 0 "" 0 "" {TEXT -1 229 "The animation of the WP motion performed further above doesn't show flux moving to the left. We can make it mo re evident by plotting the logarithm of the density (the natural logar ithm calculates much faster since it simplifies)." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "rho0l:=simplify(log(rho0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "animate(rho0l,x=-5..15,t=0..5,view= [-5..15,-10..0],frames=30,color=red,numpoints=200);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "What is the interpretation of these results? " }}{PARA 0 "" 0 "" {TEXT -1 120 "Initially there can be no correlation \+ between the particle position and its momentum. The density at negativ e values of " }{TEXT 330 1 "x" }{TEXT -1 101 " is dominated by right-m oving components. The simultaneous knowledge of a constant velocity pr ofile (" }{TEXT 331 1 "v" }{TEXT -1 1 "(" }{TEXT 332 1 "x" }{TEXT -1 1 "," }{TEXT 333 1 "t" }{TEXT -1 2 ")=" }{TEXT 334 1 "j" }{TEXT -1 1 " (" }{TEXT 335 1 "x" }{TEXT -1 1 "," }{TEXT 336 1 "t" }{TEXT -1 6 ")/rh o(" }{TEXT 337 1 "x" }{TEXT -1 1 "," }{TEXT 338 1 "t" }{TEXT -1 166 ") ) and a momentum distribution prescribed by Fourier analysis (which is a consequence of localization of the GWP in coordinate space!) is puz zling, to say the least. " }}{PARA 0 "" 0 "" {TEXT -1 69 "Eventually t he dominating right-moving components move away from the " }{TEXT 339 1 "x" }{TEXT -1 100 "<0 region, and we can see the appearance of left- moving parts. These appear also to arrive from the " }{TEXT 340 1 "x" }{TEXT -1 279 ">0 region. Wherever the original density profile was 's queezed' (curvature in psi), there was a large momentum uncertainty, w hich implies also noticable left-moving contributions in this case. Th e log-plot allows one to observe what is masked in the regular graph o f the density." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 2 0" 56 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }