Wigner functions and Weyl transforms for edestrians William B. Case a Deartment of Physics, Grinnell College, P.O. Bo 85, Grinnell, Iowa 5 Received 8 February 7; acceted 4 June 8 Wigner functions and Weyl transforms of oerators offer a formulation of quantum mechanics that is equivalent to the standard aroach given by the Schrödinger equation. We give a short introduction and emhasize features that give insight into the nature of quantum mechanics and its relation to classical hysics. A careful discussion of the classical limit and its difficulties is also given. The discussion is self-contained and includes comlete derivations of the results resented. 8 American Association of Physics Teachers. DOI:.9/.957889 I. INTRODUCTION: WHY WIGNER FUNCTIONS? In the standard formulation of quantum mechanics the robability density in osition sace is given by the square of the magnitude of the wave function, =. Thus knowing it is easy to visualize the distribution. Obtaining the distribution in momentum is also straightforward. The wave function in is found by = h e i/ d =, where all integrations are understood to be over the entire sace. The quantity gives the robability density in the momentum variable. Although straightforward, the momentum distribution is difficult to visualize if one only has. It would be desirable to have a function that dislays the robability distribution simultaneously in the and variables. The Wigner function, introduced by Wigner in 93, does just that. Wigner s original goal was to find quantum corrections to classical statistical mechanics where the Boltzmann factors contain energies which in turn are eressed as functions of both and. As is well known from the Heisenberg uncertainty relation, there are constraints on this distribution and thus on the Wigner function. Another reason for a reresentation of a quantum state in hase sace is to eamine the connection between quantum and classical mechanics. Quantum mechanics inherently deals with robabilities, while classical mechanics deals with trajectories in hase sace. If we wish to comare the two, we can consider ensembles of trajectories in hase sace for the classical case and density distributions in and or Wigner functions for the quantum case. Given the wave function the standard way to obtain the eectation value of a quantity A is by A = *  d = Â, where  is the oerator corresonding to A. The oerator  is a function of the osition and momentum oerators ˆ and ˆ, Â=A ˆ, ˆ. We would like to think of the state as describing some robability distribution in hase sace, call it P,, which is everywhere ositive and such that P, A, dd gives the eectation value of A,. In general, it is not ossible to find such a robability distribution in quantum mechanics, and so the Wigner function cannot be a simle robability distribution. For this reason, it is often called a quasidistribution. Of course, a simle robability distribution determining eectation values is ossible in the classical world. A main goal of quantum mechanics is to obtain eectation values for hysical observables. If the Wigner function is to be a comlete formulation of quantum mechanics, it must also be able to reroduce these eectation values of all functions of and. When using Wigner functions the eectation values are obtained in conjunction with the closely associated Weyl transforms of the oerators corresonding to hysical observables. As shown in Sec. IV the correct Weyl transform is critical for obtaining the sread of the energy of a state; without it, the Wigner function is little more than a visual aid for understanding quantum states. The literature on Wigner functions is etensive. There are several fine review articles,3 and chaters in books 4,5 on the Wigner function, Weyl transforms, and related distributions. Several articles on the Wigner function alone without the accomanying Weyl transform have aeared in this journal. 6 8 A aer by Snygg 9 is similar in aroach to the resent aer, but more formal and abstract. My goal is to give a shorter and more focused resentation of these toics with an eye on the relation between quantum and classical hysics. I will also oint out a few features that have not been emhasized reviously. Secial emhasis will also be given to the Wigner Weyl descrition s ability to shed light on the classical limit of quantum mechanics. The Weyl transform and Wigner function are introduced in Sec. II. Other characteristics are eamined in Sec. III. Section IV considers the harmonic oscillator as an eamle and also contains some warnings. In Sec. V we find the time deendence of the Wigner function. U to this oint the resentation is devoted to ure states. Section VI considers the generalization to mied states. The relation between the Wigner Weyl formulation and other distributions is also discussed. Section VII eamines the classical limit of the Wigner Weyl descrition of quantum mechanics. Finally, Sec. VIII discusses the advantages and disadvantages of the Wigner Weyl descrition in comarison to the standard Schrödinger equation aroach. The Aendi contains a few derivations to allow the main oints of the resentation to flow more freely. 937 Am. J. Phys. 76, October 8 htt://aat.org/aj 8 American Association of Physics Teachers 937
II. THE WEYL TRANSFORM AND THE WIGNER FUNCTION W, d = *. The Weyl transform à of an oerator  is defined by Ã, = e iy/ + y/  y/ dy, where the oerator has been eressed in the basis as the matri Â. The Weyl transform will be indicated by a tilde. The Weyl transform converts an oerator into a function of and. As shown in the Aendi it can also be eressed in terms of matri elements of the oerator in the momentum basis, Ã, = e iu/ + u/  u/ du. A key roerty of the Weyl transform is that the trace of the roduct of two oerators  and Bˆ is given by the integral over hase sace of the roduct of their Weyl transforms, Tr ÂBˆ = h Ã, B, d d. The derivation of Eq. 5 is straightforward, but is left for the Aendi. To reresent the state, we introduce the density oerator ˆ. For a ure state it is given by ˆ =, which eressed in the osition basis is ˆ = *. One of the virtues of the density oerator and thus the Wigner function is that it is easily generalized to mied states. If we form the trace of ˆ with the oerator corresonding to the observable A, we have for the eectation value Tr ˆÂ =Tr  =  = A. Thus using Eq. 5 we have A =Tr ˆÂ = h Ãd d. 3 4 5 6 7 8 9 Equation gives the robability distribution for. A similar integral over gives W, d = *. 3 Equation 3 gives the robability distribution for the momentum variable. Thus we seemed to have achieved our goal. The Wigner function reresents the distribution in hase sace reresented by. The rojection of W, onto the ais gives the robability distribution in, and the rojection on the ais gives the distribution in. Eectation values of hysical quantities are obtained by averaging Ã, over hase sace. We will see that the interretation of W, as a simle robability distribution is soiled by a number of features. III. CHARACTERISTICS OF THE WEYL TRANSFORMATION AND WIGNER FUNCTION A direct consequence of the definition of the Wigner function in Eq. is that it is real, as can be seen by taking the comle conjugate of Eq. and changing the variable of integration from y to y. Using Eq. 4 we can also eress the Wigner function in terms of the momentum reresentation of, W, = /h = h e iu/ + u/ u/ du 4a = h e iu/ * + u/ u/ du. 4b The Weyl transform of the identity oerator ˆ is because = e iy/ + y/ ˆ y/ dy = e iy/ + y/ y/ dy =. We use Eqs. 5,, and 5 and find that 5 The Wigner function is defined as W, dd =Tr ˆ =. 6 W, = /h = h e iy/ + y/ * y/ dy, and the eectation value of A is given by Thus W, is normalized in, sace. Also from the definition of the density oerator we see that for ure states ˆ = ˆ, and thus Tr ˆ =Tr ˆ =. From this relation and Eqs. 5 and we see that A = W, Ã, d d. W, d d = h. 7 We see that the eectation value of A has been obtained by what looks like the average of the hysical quantity reresented by Ã, over hase sace with robability density W, characterizing the state. If the Wigner function is integrated over alone and use is made of e i/ d=h, we have The Wigner functions have a reasonable translation roerty. If the wave function gives the Wigner function W,, then the wave function b will give W b,. Shifts in the wave function lead to corresonding shifts in the Wigner function in the osition variable. Also, if the original wave function is relaced with 938 Am. J. Phys., Vol. 76, No., October 8 William B. Case 938
e ib /, the new Wigner function becomes W, b. Shifts in momentum of the original wave function lead to corresonding shifts of the Wigner function in the momentum variable. Both of these roerties follow directly from the definition of the Wigner function, Eq.. The signs in these shifts might be a little disturbing. If is concentrated about, then b will be concentrated about +b. If has a certain momentum distribution, then e ib / will have the same distribution shifted by +b. Each of the shifts shift their resective distribution by +b or +b, resectively. Consider two density oerators, ˆ a and ˆ b, from different states a and b, resectively. We can form the combination Tr ˆ a ˆ b = a b. The Weyl transform of Eq. 8 using Eqs. 5 and is W a, W b, d d = h a b. 8 9 The roduct of Wigner functions integrated over hase sace is the square of the inner roduct of the original wave functions divided by h. The left-hand side of Eq. 9 acts as a ositive inner roduct of the original states. If we now consider orthogonal states where a b =, we have W a, W b, d d =. Thus some and indeed most Wigner functions must be negative for some regions of, sace. The definition of the Wigner function, Eq., can be eressed as the inner roduct of two wave functions. First, note that y/ * y/ dy = y/ * y/ d y/ =. Thus we may define the two normalized functions of y, y =e iy/ +y/ / and y = y/ /, and eress the Wigner function as Thus W, = /h y * y dy. Given the Wigner function W, we can recover the original wave function. We multily the definition of the Wigner function in Eq. by e i / and integrate over to obtain W, e i / d = * / + /. 4 We set =/ and then = in Eq. 4 and recover u to an overall constant with = W /, e * i/ d. 5 The constant reresented by * can be determined u to a hase by normalization of. Note that not all functions of and which obey the reviously listed constraints given in Eqs. 6, 7, and 3 are accetable Wigner functions. For ure states a test would be to first use the W, to find the wave function using Eq. 5. Then use this in Eq. to determine if the original Wigner function W, is recovered. Now consider the Weyl transform of the oerators corresonding to the observables. Suose that the oerator  is only a function of the oerator ˆ, which allows us to write Â=A ˆ. The Weyl transform in this case is articularly simle. From Eq. 3 we have à = e iy/ + y/  ˆ y/ dy = e iy/ A y/ y dy = A. 6 We see that if the oerator  is urely a function of ˆ, then its Weyl transform is just the original function with the oerator with ˆ relaced by. If we start with an oerator deendent only on the momentum oerator ˆ and Eq. 4, we find a similar result. If the oerator Bˆ is urely a function of ˆ, then its Weyl transform is simly the original function with the oerator with ˆ relaced by. We can etend this argument to sums of oerators where each term is urely a function of ˆ or ˆ. Thus the Weyl transform of the Hamiltonian oerator Ĥ ˆ, ˆ =Tˆ ˆ +Û ˆ becomes H, =T +U, where T and U are the kinetic and otential energies. The eectation values of,, T, U, and H are given by W, /h, 3 and the distribution W, cannot take on arbitrarily large values as would be allowed in a classical distribution in hase sace. From the definition of the Wigner function in Eq. we see that all even wave functions reach +/h at, =,, and all odd wave functions reach /h at the same oint. Thus a symmetric wave function with widely searated eaks will have a Wigner function with the maimum ossible value, +/h, at, =,. A similar antisymmetric wave function will give /h at the same oint. The Wigner function will take on these etreme values even if the original wave function is zero in the region of =. = W, d d, = W, d d, T = W, T d d, U = W, U d d, 7a 7b 7c 7d 939 Am. J. Phys., Vol. 76, No., October 8 William B. Case 939
-..5 W.5 - - (a) W (b) H = W, H, d d. 7e These results could also have been obtained from Eqs. and 3. The Wigner function acts like a robability distribution in hase sace ecet for the fact that W can be negative. The eectation values of other quantities will not be as simle. IV. AN EXAMPLE: THE HARMONIC OSCILLATOR We aly the develoments in the revious two sections to the harmonic oscillator. Its Hamiltonian and two lowest energy states are given by Ĥ = ˆ m + m ˆ, 8 = - - - -. -. 4 a e / a, 9 = 4 a a e / a, 3 where a = / m. The corresonding Wigner functions for and can be found using Eq., W, = h e a / /a, W, = h + a/. Fig.. Plots of the Wigner functions for the two lowest energy states of the harmonic oscillator; a n= and b n=. For these lots a and h are set equal to. 3 + /a e a / /a. 3 Plots of W and W with a= and h= are shown in Fig.. We see that both functions obey the inequality in Eq. 3. W equals + at, =, and W equals at the same oint. We now take a closer look at the lowest energy state, Eq. 3. The eectation value of the energy can be determined using Eq. 7e, and we find H = W, m + m d d =. 33 Although Eq. 33 is correct, the way it is obtained is a little disturbing. This result should shout out / because this value is the only value the energy can take. We would eect all of the nonzero oints of the distribution in hase sace to lie on an ellise corresonding to the energy /. However, the eectation value is obtained by taking an average of combinations of and corresonding to different energies with robability W,. We would eect such a distribution to imly a sread in energy leaving us with an aarent contradiction. The energy sread is determined by = H H. The second term is / ; for the first term we must have H = W, Ĥ d d. 34 The Weyl transform of Ĥ will not simly be H, because Ĥ is no longer a sum of terms urely deendent on ˆ or ˆ, but involves cross terms. The Weyl transform is given by = Ĥ / m ˆ 4 + m 4 /4+ ˆ4 ˆˆ + ˆ ˆ /4. 35 The first two terms on the right are 4 / m +m 4 4 /4. The last term is determined with the hel of ˆˆ + ˆ ˆ =, 36 which is derived in the Aendi. When this eression is included we find = Ĥ H,. 37 4 Thus we find after carrying out the integration that H =, 38 and the resulting sread is equal to zero as it should. We see that even for the Wigner function of Eq. 3, which is ositive everywhere, quantum behavior is still resent. It is the way the hysical results are etracted, using not only the Wigner function, but also the Weyl transform of the desired oerator, which give this system its quantum behavior. It is widely believed that a Wigner function which is ositive everywhere can ehibit only classical henomena. As we see from this eamle, this belief is incorrect. 94 Am. J. Phys., Vol. 76, No., October 8 William B. Case 94
V. TIME DEPENDENCE OF THE WIGNER FUNCTION By taking the derivative of Eq. with resect to time we have W = h iy/ e * y/ + y/ + y/ + * y/ dy. 39 The artial derivatives on the right-hand side can be eressed using the Schrödinger equation,,t =,t im + i U,t. We use Eq. 4 to write Eq. 39 as W where W T W U = W T 4 + W U, 4 = e 4 im iy/ * y/ + y/ * y/ + y/ dy, 4 = ih e iy/ U + y/ U y/ * y/ + y/ dy. 43 We consider each of these terms in the Aendi and find W T = W,, 44a m W U = s s= s +! s s+ U s+ s+ W,. 44b Equation 4 with Eq. 44 is equivalent to solving the Schrödinger equation, as can be seen by the following argument: Consider the wave function, and its corresonding Wigner function W,,. As we saw in Eq. 5 the relation between and W is one to one ecet for an overall constant hase. We then use the Schrödinger equation to determine,t and Eqs. 4 and 44 to find W,,t. Because both equations are linear and first order in t, these solutions will be unique once the initial functions are given. Since Eqs. 4 and 44 were derived from the Schrödinger equation, these solutions must have the same one-to-one relation as the original wave function and Wigner function. Thus the two methods must be equivalent. Note that the result in Eq. 44a is entirely classical in that it contains no, while Eq. 44b is more comlicated. If all derivatives of U higher than the second order are zero, as for a free article, a constant force, and a harmonic oscillator, then Eq. 44b becomes W U = U W,. 45 With this assumtion the eression governing the evolution of the Wigner function becomes W, = W, m + U W,. 46 Equation 46 is the classical Liouville equation. In such a regime the motion of the Wigner function in hase sace is eactly that of classical hysics under the influence of the otential U. If higher derivatives of U are resent, then the additional terms will give a diffusion-like behavior. For a harmonic oscillator the motion in, sace is urely classical. The time evolution of the classical harmonic oscillator is described by = cos m sin, 47a = cos + m sin, 47b where and are the values of the osition and momentum at time t, and and are the values at t=. We need only to require that each oint of the Wigner function moves in ellitical aths in hase sace. Thus if the Wigner function at time t= is W,,, the Wigner function at a future time t is given by W,,t = W cos sin, cos m + m sin,. 48 For the harmonic oscillator all the analysis with Hermite olynomials and eonentials has only to do with the sort of state that can be reared and not with the hysics of its time evolution. As an alication of the time evolution of a harmonic oscillator state we take the Wigner function at time t= to be the lowest energy state of the harmonic oscillator shifted by b in the direction. If we use the rule for translation given in Sec. III, W,, is easily obtained from Eq. 3, W,, = h e a / b /a. 49 From the revious discussion we see that the time evolution of the state is motion in an ellise in the, lane centered about,. The Wigner function at other times becomes W,,t = h e a cos + m sin a cos sin b. 5 m We note that although the source of the Wigner function was the ground state of the harmonic oscillator and thus a roer Wigner function, the oscillator dictating the motion may be a different one; the a in Eq. 5 can be taken as arbitrary and need not be constrained by the relation a = / m. Ifwedo add the requirement that this state be the shifted ground state of the same oscillator, we may use a = / m and find 94 Am. J. Phys., Vol. 76, No., October 8 William B. Case 94
A similar treatment can be given of the free article. Here each oint would move in a straight line arallel to the ais in hase sace as dictated by its osition and momentum. The Wigner function evolves as.5 W.5 (a) -5 5 - W,,t = W m t,,. 5 Equation 5 corresonds to a shear of the distribution. Parts of the Wigner function above the ais move to the right in roortion to how far above the ais they lie. Points below the ais move to the left in a similar fashion. Many of these results would be difficult to obtain starting from the Schrödinger equation. VI. MIXED STATES AND OTHER DISTRIBUTION FUNCTIONS.5 W.5 (b) -5 Fig.. Plots of the Wigner functions of a a coherent state and a b squeezed state. The state starts at the right and moves in a clockwise fashion about,. Both are shown at times t=, t=t/4, and t=t/, where T is the eriod for the harmonic oscillator. In generating these lots the following values were used: h=, =, m=, b=5; for the coherent state, a= as follows from a = / m. For the squeezed state, a=. sin W,,t = h e a + b a a b cos. 5 Because the and deendencies have now factored, we see that * = Wd will be a Gaussian of constant width a moving back and forth with amlitude b and angular frequency. This state is the coherent state. Coherent states were introduced by Glauber 3,4 in the study of quantum otics as the closest quantum descrition of a classical electromagnetic wave. These states lay a arallel role in the study of the harmonic oscillator. As the Wigner function moves around its ath in hase sace, its rojection on the ais moves back and forth with unchanging rofile. We now return to Eq. 5 and consider its imlications without the restriction on a, a = / m. The initial state is no longer the shifted ground state of the harmonic oscillator describing the motion. It will have a different ratio of sread in the and directions from the coherent state. This state is the squeezed state. 4 Equation 5 now describes the time evolution of this state where m and refer to the harmonic oscillator. Based on this evolution in hase sace we can imagine how this state will evolve and how its rojection on the ais will differ from that of the coherent state. Although it will still oscillate with angular frequency, its width in will vary during the motion. The coherent state and the squeezed states are shown in Fig. at t=, t=t/4, and t =T/, where T is the eriod of the oscillator. 5 - There is an interesting comlementarity between classical hysics and quantum hysics. In quantum mechanics, linear combinations of wave functions,t that satisfy the Schrödinger equation are also solutions to the Schrödinger equation. This roerty is the usual linearity of quantum mechanics. When the transformation is made to the corresonding Wigner functions and the, sace, this linearity is lost. Suose that = +. As can be seen from Eq., we will not have W =W +W. For classical systems there is linearity in hase sace. If we have one distribution D a, and add another D b,, we obtain the roer reresentation of the sum of the two by taking D, = D a, +D b, /. Classical distributions are linear in hase sace. For mied states the definition of the density oerator, Eq. 6, is generalized by relacing it with ˆ = P j j j. 53 j The robability of each state P j will be ositive and j P j =. The eectation values will still be given by A =Tr ˆÂ. The Wigner function is calculated as before with Eqs. and 3, W, = /h = P j W j,, 54 j where W j, is the Wigner function obtained for j alone. Thus there is a linearity of mied states in hase sace. In this way, the quantum system of mied states takes on some of the character of a classical system. Of the relations that were found in Sec. III only Eq. 7, which deends uon ˆ = ˆ, is no longer alicable. The Wigner function can be inverted along the lines of Eq. 5 to recover the density oerator if it eists. We now give an eamle comaring ure and mied states. For the ure state we take the sum of two coherent states, one centered at =+b, the other at = b. This state can be formed from two ground states of the harmonic oscillator in Eq. 9 shifted by a distance b in oosite directions in, = A b + + b, 55 where A is a normalization constant. The Wigner function for this state can be found using Eq. giving 94 Am. J. Phys., Vol. 76, No., October 8 William B. Case 94
Ã,,...,,,... = e i y + y + / W (a) - - -5 -.5.4. -..5 -.4 5 + y /, + y /,...  y /, y /,... dy dy. 58 The defining equation for the Wigner function for a ure state,,... becomes W,,...,,,... = h n e i y + y + / W - - -5 -.5 (b) W, = h +e b /a e a / e b /a + e + b /a +e /a cos b/. 56 For the mied state of the same two coherent states we can just sum the Wigner functions of the shifted ground states, Eq. 3, inserting a factor of / reresenting the equal robability of obtaining each, W, = W b, + W + b, 57a = h e a / e b /a + e + b /a. 57b The Wigner functions given in Eqs. 56 and 57b are shown in Fig. 3. As we can see, the mied state has two eaks centered at = b, and the ure state is similar to the mied state with nonclassical behavior between the two eaks where the wave function is small. This behavior near, for symmetric or antisymmetric states was discussed in Sec. III. As was indicated in Sec. I, the Weyl transform and the Wigner function can be easily generalized to many dimensions. For the Weyl transform of oerator  we relace Eq. 3 with. -..5 -.4 5 Fig. 3. Plots of the Wigner functions for a a ure state and a b mied state consisting of two coherent states. In generating these lots the following values were used: h=, a=, and b=4..4 + y /, + y /,... * y /, y /,... dy dy, 59 where n is the dimension of the system. The Wigner function is not the only candidate that gives a distribution in, sace and a reresentation of eectation values for quantum mechanics in the form of Eq.. The other candidates reresent trade-offs between the distribution function and the transformed oerators, some making the distribution look tamer at the eense of the transform of the oerator. 3 All are caable of giving all of the quantum details. It is not surrising that this ambiguity eists and that there is no unique choice. As ointed out in Sec. I, finding a roer robability distribution in hase sace to reresent quantum mechanics is imossible. Thus it is not surrising that there are many ways of aroimately erforming this task, each falling short of the goal. VII. CLASSICAL LIMIT At this oint it might seem that we could take the limit and obtain classical hysics. We might insist that we begin with a ositive distribution in hase sace at t= either by the classical nature of the rearation or by using some smoothing scheme based on the inability to observe details in the distribution in hase sace coarse graining. The equation of evolution of W,,t, Eqs. 4, 44a, and 44b, reduces to the classical case when. The difference between the Weyl transform, Ã,, of the oerator  =A ˆ, ˆ and the function A, disaears when. This difference is due to the fact that ˆ and ˆ do not commute ˆ, ˆ =i, which led to the etra term in Eq. 37. Thus in this limit we can calculate eectation values in the usual way as an integral of W, A, over and. We might try to give such an argument but we cannot. The roblem with these naive assumtions can be seen easily. Droing the higher order terms in in Eq. 44b is susect. Note that from the definition of the Wigner function Eq., / s+ will bring a factor of / s+, which will more than offset the s factor unless some hel can be found in the density oerator. This issue has been considered by Heller, 5 who reached similar conclusions to those eressed here. As an eamle, consider the ground state of the harmonic oscillator given in Eq. 3. Although we are using states of the harmonic oscillator, we are not assuming that the Hamiltonian dictating the motion is that of the harmonic oscillator. This construction is simly a way of getting a 943 Am. J. Phys., Vol. 76, No., October 8 William B. Case 943
roer Wigner function. If we take / s+ of this Wigner function and the limit, we obtain a / s+ e a / for the deendence on and. As the eonential becomes narrower with significant values only for /a. With this constraint W / s+ goes as / s+. Thus the terms in Eq. 44b that aear to go as s actually go as / and revent their neglect in the limit. Part of the roblem is that as the Wigner function of the ure state becomes very narrow in the direction because for a ure state the width in is tied to the width in via. It might be argued that we should have taken the width in as fied in the limit, allowing the Wigner function to become narrow in the distribution. It might seem that this limit would avoid the roblems with the higher order derivatives in. In general, such an aroach would not be satisfactory. As we saw in Fig. for the harmonic oscillator, initial distributions can become rotated in hase sace interchanging the widths of the distributions in and. It is eected that in the classical limit we cannot determine distributions with higher and higher recision. We want to control the widths in and indeendently. Once the width in is fied we cannot make the width in arbitrarily small, but we can construct a mied state that has an arbitrarily wide distribution in. This construction is done by forming a mied state of ground states of the harmonic oscillator in Eq. 3 shifted in the direction with normalized robability density, P = e c /c, 6 where c is a ositive constant. The Wigner function of the mied state is W, = W, P d = /a e / c + /a. 6 a c + e We now have a Wigner function with width a in the direction and width c + /a in. The limit can be taken of this Wigner function and the neglect of the derivatives with resect to beyond the first in Eq. 44b can now be justified. If we aly this same rocedure to the double coherent ure state in Eq. 56, we obtain in the limit, W, = ac +e b/a e /c e b /a + e + b /a +e /a e b /a. 6 Again we see that the behavior is considerably smoother than the ure state and closely resembles the eression for the mied state of two coherent states given in Eq. 57b. All that remains of the nonclassical behavior near, shown in Fig. 3 is a eak suressed by a factor of e b /a. Again the neglect of terms containing derivatives with resect to beyond the first order is justified. Can we carry out this rogram of introducing mied states and successfully taking the for all cases? That is not so clear. Note the concern is not just that W might become negative, although that would be a roblem, but the raid variation of W. It is well known that many classical nonlinear systems ehibit very comlicated behavior which can evolve into distributions that we would eect to have large higher order derivatives. 6 The issue of the incomatibility of classical mechanics and quantum mechanics has been ointed out by Ford 7 based on information theory arguments and is a central question in quantum chaos. This question is still oen. 8 A recent aer 9 analyzed the nonlinear Duffing oscillator as a classical system and as a quantum system. Their resective evolutions in hase sace are then shown side by side and clearly show a classical system that develos fine structures in hase sace, while the quantum system develos negative regions in the corresonding Wigner function. We may also consider systems that start off classically but evolve into the quantum regime. In the eeriments of Arndt and co-workers a beam of C 6 molecules from an oven asses through a grating with a grating constant of nm. This eeriment reveals an interference attern at a distance of. m. The interference attern is clearly a quantum effect. The grating sacing, although fine, is much larger than the de Broglie wavelength of the molecules, which is about.8 m. The reason that the attern is revealed is due to the growth of transverse coherence with distance from the grating. Thus we have a seemingly classical system that evolves into a quantum system. We could argue that making very small would reduce the effect, but roagation over a greater length would bring the attern back. In the end, what can we say about the classical limit? As we can see the limit is subtle and involves not just the time deendence of the states, but the oerators and nature of the states themselves. If we can say that the initial distribution in hase sace is ositive and smooth everywhere and if the Hamiltonian is such as to leave the distribution sufficiently smooth so as to allow us to neglect the higher order terms in with their high order derivatives with resect to, then the oening argument of this section should hold. In the end, we believe that the quantum descrition correctly describes our world. We note that the distinction between classical and quantum is not simly the distinction between large and small, but the etent to which we know the distribution. If we in down the distribution in hase sace, either due to the details of rearation, details of evolution, or fineness of measurement, to details aroaching =, the quantum nature will emerge. VIII. SUMMARY: WIGNER WEYL VERSUS SCHRÖDINGER Almost all quantum mechanics tets resent the subject based on the Schrödinger equation, wave functions, and oerators. The Wigner Weyl aroach resented here is comletely equivalent. The question of which aroach is to be used deends on the system under consideration and the questions asked. Our initial goal was to find a hase sace reresentation of the quantum state. This reresentation was given by the Wigner function. It was found to ossess some disaointing features due to the quantum character of the system, which is comletely described by the formulation. What was accomlished beyond a visual descrition of a quantum system? Two were discussed in the aer. For the harmonic oscillator and the free article the time evolution of 944 Am. J. Phys., Vol. 76, No., October 8 William B. Case 944
the system is simle in this formulation and is identical to that of the classical system. These features are hidden in the standard Schrödinger aroach. The second virtue of the Wigner Weyl aroach is its ability to naturally include mied states. The Schrödinger equation is written in terms of the wave function, and is limited to a descrition of ure states. As we saw in Sec. VI the Wigner Weyl descrition easily moves from the descrition of a ure state to that of mied states. We saw an alication of this descrition in Sec. VII. As an added benefit the time evolution of the Wigner function given in Eqs. 4 and 44 was in a form that heled us understand the classical limit. There are still oen questions, but we gained a clearer icture of the roblem. Had we started with the Schrödinger equation and taken we would conclude that we should dro the kinetic energy. It is also ossible to obtain the Wigner functions corresonding to the energy eigenstates directly from the time evolution of the Wigner function, Eq. 4. 5 Thus for the harmonic oscillator, Eq. 3 could be obtained directly without the use of the result given in Eq. 9. Should we give u the Schrödinger equation in favor of the Wigner Weyl aroach? Certainly not. The simlicity of the Schrödinger equation makes it easier to solve directly and ideal for finding aroimate solutions. However, there is much to be gained by studying the Wigner Weyl descrition. ACKNOWLEDGMENTS I would like to thank Markus Arndt and Anton Zeilinger for their hositality during my time in Vienna. I would also like to thank Caslav Brukner and Wolfgang Schleich for several useful conversations. APPENDIX: DETAILS OF THE DERIVATIONS In the following we will fill in some details of derivations that were omitted in the main tet. Equation 4. The first task is to eress the Weyl transformation in terms of momentum eigenstates. The definition of the Weyl transformation is given in Eq. 3. The identity oerator can be eressed as ˆ = d, where the states obey the orthogonality condition =. We substitute this eression on both sides of the oerator of the oerator  on the right-hand side of Eq. 3 giving à = e iy/ + y/  y/ dyd d. A Net we note that =h / e i/ see, for eamle, Ref. and use e iy/ dy = h A to carry out the y integration giving + /  e i / d d. A3 Net we change variables u=, v= +, and dudv =d d. With this change of variables Eq. A becomes à = v v + u /  v u / e iu/ du dv, A4 where we have used the relation y/ = y. Carrying out the v integration, we have the desired result à = e iu/ + u/  u/ du. A5 Equation 5. We net derive the key relation between the trace of two oerators and their resective Weyl transforms. Suose we have two oerators  and Bˆ and their Weyl transforms Ã, = e iy/ + y/  y/ dy, B, = e iy / + y / Bˆ y / dy. A6a A6b We form the roduct of these two and integrate over all of, sace and find Ã, B, d d = e i y+y / + y/  y/ + y / Bˆ y / d d dy dy. A7 The integration is done using Eq. A, giving a delta function which is used to do the y integration: Ã, B, d d = h + y/  y/ y/ Bˆ + y/ d dy. A8 Then we erform the change of variables u= y/, v= +y/, and du dv=d dy, giving Ã, B, d d = h v  u u Bˆ v du dv = h Tr ÂBˆ, A9 which is Eq. 5, the desired relation between the trace of two oerators and their Weyl transforms. Equation 36. From the definition of the Weyl transform in Eq. 3 we have ˆ ˆ + ˆˆ = e iv/ + v/ ˆ ˆ + ˆˆ v/ dv Aa = e iv/ + v / + v/ ˆ v/ dv. Ab We insert the identity ˆ = d just after ˆ and find 945 Am. J. Phys., Vol. 76, No., October 8 William B. Case 945
ˆ ˆ + ˆˆ = e iv/ + v / + v/ v/ dv d Aa = h e iv/ + v / v ei v/ dv d. Ab We net erform two integrations by arts on the v variable and obtain the desired result, ˆ ˆ + ˆˆ = v v eiv/ + v / dv =. A Equation 44a. We note that the integral in the first term of Eq. 4 can be written as e iy/ * y/ + y/ dy = e iy/ * y/ + y/ dy. y = s= A3 Equation A3 can be integrated by arts to give i e iy/ * y/ + y/ dy + e iy/ * y/ + y/ dy. A4 If combined with the integral in the second term in Eq. 4 rewritten in a similar fashion, we obtain the result in Eq. 44a, W T = e hm iy/ * y/ + y/ dy = W m. A5 Equation 44b. For the W U / art of W/ we assume that U can be eanded in a ower series in and write U + y/ U y/ = y n U n n! n y A6a s +! s s+ U s+ y s+. A6b When Eq. A6 is incororated into Eq. 43, we find Eq. 44b. a Electronic mail: case@grinnell.edu E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 4, 749 759 93. M. Hillery, R. F. O Connell, M. O. Scully, and E. P. Wigner, Distribution functions in hysics: Fundamentals, Phys. Re. 6, 67 984. 3 H. Lee, Theory and alication of the quantum hase-sace distribution functions, Phys. Re. 59, 5 995. 4 S. R. degroot and L. C. Suttor, Foundations of Electrodynamics North- Holland, Amsterdam, 97. 5 W. Schleich, Quantum Otics in Phase Sace Wiley-VCH, Berlin,. 6 R. A. Camos, Correlation coefficient for incomatible observables of the quantum harmonic oscillator, Am. J. Phys. 66, 7 78 998. 7 D. F. Styer, M. S. Balkin, K. M. Becker, M. R. Burns, C. E. Dudley, S. T. Forth, J. S. Gaumer, M. A. Kramer, D. C. Oertel, L. H. Park, M. T. Rinkoski, C. T. Smith, and T. D. Wothersoon, Nine formulations of quantum mechanics, Am. J. Phys. 7, 88 97. 8 M. Belloni, M. A. Doncheski, and R. W. Robinett, Wigner quasirobability distribution for the infinite square well: Energy eigenstates and time deendent wave ackets, Am. J. Phys. 7, 83 9 4. 9 J. Snygg, Use of oerator functions to construct a refined corresondence rincile via the quantum mechanics of Wigner and Moyal, Am. J. Phys. 48, 964 97 98. H. Weyl, The Theory of Grous and Quantum Mechanics Dover, New York, 93. 75. Weyl s original goal was to find a transformation that would give the quantum oerator starting from the classical function. Based on that our Eq. 3 should be called the inverse Weyl transformation. L. Cohen, Time-Frequency Analysis Prentice Hall, Uer Saddle River, NJ, 995,. 7. J. Bell, Seakable and Unseakable in Quantum Mechanics Cambridge U.P., Cambridge, 987, Cha.. In fairness to Bell it should be ointed out that he seemed to be equating quantum behavior with nonlocality. Recent work suggests that nonlocality can be found even for Wigner functions that are everywhere ositive. See K. Banaszek and K. Wodkiewicz, Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner reresentation, Phys. Rev. A 58, 4345 4347 998. 3 R. J. Glauber, The quantum theory of otical coherence. Phys. Rev. 3, 59 539 963. 4 M. O. Scully and M. S. Zubairy, Quantum Otics Cambridge U.P., Cambridge, 997, Cha.. 5 E. J. Heller, Wigner hase sace method: Analysis for semiclassical alications, J. Chem. Phys. 65, 89 98 976 ; 67, 3339 335 977. 6 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Sringer, New York, 983, Chas. and. 7 J. Ford and M. Ilg, Eigenfunctions, eigenvalues, and time evolution of finite, bound, undriven, quantum systems are not chaotic, Phys. Rev. A 45, 665 673 99. 8 See, for eamle, E. J. Heller and S. Tomsovic, Postmodern quantum mechanics, Phys. Today 46, 38 46 993. 9 I. Katz, A. Retzker, R. Staub, and R. Lifshitz, Signatures for a classical to quantum transition of a driven nonlinear oscillator, Phys. Rev. Lett. 99, 444-444-4 7. M. Arndt, O. Nairz, J. Voss-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Wave-article duality of C 6 molecules, Nature London 4, 68 68 999 ; O. Nairz, M. Arndt, and A. Zeilinger, Quantum interference eeriments with large molecules, Am. J. Phys. 7, 39 35 3. D. J. Griffiths, Introduction to Quantum Mechanics Pearson Prentice Hall, Uer Saddle River, NJ, 5, nd ed.,. 3. The careful reader may be concerned about the droing of the terms at infinity that occur during the integration by arts. This neglect can be justified by relacing e iy/ in the definition, Eq. 3, with e iy/ y, where is a small ositive constant. The calculation can now be carried out and allowed to go to zero afterward giving the same result as was found reviously. 946 Am. J. Phys., Vol. 76, No., October 8 William B. Case 946