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Fig. 1 is 2π-periodic and contiguous. Encoding the phase as a color allows to plot even relatively strongly oscillating wave functions. It requires, however, high-quality color print- ing for good reproduction. Thaller1,2 has also generalized this approach to multi-component wave functions.
3. Quantum mechanics in phase space
3.1. The Wigner function
The Wigner distrubution function8–10 was introduced by Eu- gene Paul Wigner to take into account quantum corrections to classical statistical mechanics in 1932. Today, it finds ap- plications in current research in various branches of physics ranging from quantum optics11 and quantum chaos12 to nu- clear and solid state physics. The Wigner function allows for a mathematical description of (non-relativistic) quantum mechanics in phase space. But it is more than a sheer mathematical object, it may be reconstructed by a series of measurements13–17. The Wigner function formalism is math- ematically equivalent to the standard language of quantum mechanics in terms of wave functions and the Schrödinger equation but it is more closely related to classical mechanics because it formulates quantum mechanics in phase space as Hamiltonian mechanics does. The Wigner function of a (one-dimensional) pure state wave function Ψ(x, t) is defined as
1 isp/~ w(x, p, t) = Ψ∗(x + s/2, t)Ψ(x s/2, t)e ds , (3) 2π~ −
where Ψ∗(x, t) denotes the complex conjugate of Ψ(x, t). Note that due to the symmetry of the integrand in (3), the Wigner FIG. 1: (Color online) Three different ways of visualizing the Gaus- function w(x, p, t) is always a real function. The expression sian wave packet (2) at time t = 0 withx ¯ = 1,p ¯ = 3, and σ = 1/2 ~ = = (dimensionless units with 1 and m 1); a) visualization by 1 the wave-function’s real part and imaginary part, b) visualization by = Ψ˜ + Ψ˜ isx/~ w(x, p, t) ~ ∗(p s/2, t) (p s/2, t)e− ds (4) the wave-function’s modulus and its phase using arrows (phasors), c) 2π − visualization by plotting the wave-function’s squared modulus and encoding the phase in a periodic color scheme. is equivalent to (3) if Ψ(x, t) and Ψ˜ (p, t) are linked via (1). Further important mathematical properties9,11 of the Wigner function include the following relations. had been used, for example, by Feynman7. It is beneficial in situations where advanced printing methods, color printingfor • If Ψ(x, t) is normalized to one then also w(x, p, t), viz. example, are not available. However, it is difficult to picture fast oscillatory wave functions in this manner. w(x, p, t)dp dx = 1 , (5) Another way of presenting a wave function by its modu- lus and its phase has been popularized by Thaller1,2. This method is utilized in part c of Fig. 1. It shows the squared furthermore, the equation modulus of the wave function (2) plus its phase which is en- coded by the color between the graph of the squared modulus 1 w(x, p, t)2 dp dx = (6) and the horizontal axis. Plotting the squared modulus allows 2π~ for a direct read off of the probability distribution in position space. Note that the color scheme for the phase in part c of holds for all pure states Ψ(x, t). 3
• Quantum mechanical probability densities in position After several algebraic manipulations that include expanding space and momentum space may be obtained from the the potential V(x, t) into a Taylor series around x = 0 and in- marginals of the Wigner function tegrating (13) by parts one finally finds the so-called quantum Liouville equation ρ(x, t) = Ψ(x, t) 2 = w(x, p, t)dp , (7a) | | ∂w(x, p, t) = ρ(p, t) = Ψ˜ (p, t) 2 = w(x, p, t)dx . (7b) ∂t | | 2l+1 ∞ (i~/2)2l ∂ ∂ ∂ ∂ H(x, p, t)w(x, p, t) , (2l + 1)! ∂pw ∂xH − ∂pH ∂xw • The state overlap of two wave functions Ψ1(x, t) and l=0 Ψ2(x, t) in terms of their Wigner functions w1(x, p, t) and (14) w2(x, p, t) reads which is a partial differential equation of infinite degree. The ∂ ∂ ∂ ∂ ff Ψ Ψ = operator ∂x ∂p ∂p ∂x in (14) consists of di erential op- ∗(x, t) 2(x, t)dx w H − H w 1 erators that act on the Hamilton function H(x, p, t) (indicated by the index H) and differential operators that act on the 2π~ w1(x, p, t)w2(x, p, t)dp dx . (8) Wigner function w(x, p, t) (indicated by the index w). The function H(x, p, t) in (14) is the classical Hamilton function • The Wigner function obeys the reflection symmetries p2 H(x, p, t) = + V(x, t) (15) Ψ(x, t) Ψ∗(x, t) w(x, p, t) w(x, p, t) , (9) → ⇒ → − 2m Ψ(x, t) Ψ( x, t) w(x, p, t) w( x, p, t) . (10) → − ⇒ → − − that corresponds to the Hamilton operator Hˆ of the • The Wigner function is Galilei-covariant, that is Schrödinger equation (12). For systems where magnetic fields are absent as in (12), the Ψ(x, p, t) Ψ(x + y, p, t) Hamilton function H(x, p, t) can be written as a sum of a ki- → ⇒ netic energy term K(p) = p2/(2m) that depends only on the w(x, p, t) w(x + y, p, t) , (11) → momentum p and a potential energy term V(x, t) that depends but not Lorentz-covariant. However, Lorentz-covariant only on the position x (and possibly the time t). In this case, generalizations of the Wigner function have been consid- the quantum Liouville equation (14) may be written as a con- ered in the literature18–20 which may be applied in relativis- tinuity equation 21 tic quantum optics . ∂ ∂w(x, p, t) ∂x • The Wigner function is not gauge invariant. The definition = j(x, p, t) , (16) ∂t − ∂ (3), however, may be generalized22 such that it becomes in- ∂p dependent of the gauge in the presence of a magnetic field. where we have introduced the two-dimensional vector field The Wigner function w(x, p, t) is called quasi probability p function because it is real and normalized to one but not m w(x, p, t) strictly non-negative. In fact, a sufficient and necessary con- = j(x, p, t) ∞ (i~/2)2l ∂ ∂ ∂ 2l ∂V(x,t) , Ψ + w(x, p, t) dition that w(x, p, t) is strictly non-negative is that (x, t) is (2l 1)! ∂p ∂xV ∂xw ∂x − l=0 − a minimum uncertainty wave packet, that is a Gaussian wave (17) 23 packet . As a consequence of (6), the Wigner function can called the Wigner function flow. not be arbitrarily highly peaked, which is a reminiscence of Let Oˆ x denote some operator function of the quantum me- the quantum mechanical uncertainty relation. chanical position operator and O(x) the corresponding classi- 9,11,24,25 The evolution equation of the Wigner function may cal function, then it follows from (7b) that be obtained by calculating the Wigner-function’s time deriva- tive. Taking into account the Schrödinger equation O (t) = Ψ(x, t) Oˆ Ψ(x, t) = w(x, p, t)O(x)dp dx . x x ∂Ψ(x, t) ~2 ∂2Ψ(x, t) i~ = Hˆ Ψ(x, t) = + V(x, t)Ψ(x, t) (12) (18) ∂t −2m ∂x2 This means one can calculate the quantum mechanical expec- ˆ for the wave function moving in the potential V(x, t) we get tation value Ox (t) of an operator Ox by averaging the clas- sical function O (x) over the Wigner function w(x, p, t). For ˆ ∂w(x, p, t) momentum dependent operators Op and their corresponding i~ = classical functions one finds using (7a) a similar statement ∂t 1 Ψ Ψ Ψ Ψ isp/~ Hˆ (x, t) (x, t)∗ (x, t) Hˆ (x, t) ∗ e ds . = Ψ Ψ = 2π~ Op (t) (x, t) Oˆ p (x, t) w(x, p, t)O(p)dp dx . − (13) (19) 4
For general operators Oˆ x,p and classical functions O(x, p) that where H(x, p, t) denotes the Hamilton function that also gov- depend on the position as well as on the momentum one can erns the motion of the ensemble’s single particles. The equa- show9,24 that tion (26) may be written as a continuity equation (16) by in- troducing the probability flow Ox,p (t) = Ψ(x, t) Oˆ x,p Ψ(x, t) = p w(x, p, t) m w(x, p, t)O(x, p)dp dx (20) j(x, p, t) = . (27) ∂V(x,t) w(x, p, t) − ∂x ˆ provided that the relation between the operator Ox,p and the The quantum Liouville equation (14) reduces in the limit function O(x, p) is established by ~ 0 to the classical Liouville equation (26). Note, how- → i(ξxˆ+πpˆ)/~ ever, that also if the third spatial derivative of the potential Oˆ x,p = O˜(ξ,π)e dξ dπ, (21) V(x, t) and all its higher derivatives vanish the quantum Liou- ville equation (14) also reduces to the classical Liouville equa- ˜ where O(ξ,π) denotes the Fourier transform of O(x, p) tion (26). This means that the quantum mechanical evolution i(xξ+pπ)/~ of the Wigner function for a particle in a homogeneous con- O˜(ξ,π) = O(x, p)e− dx dp (22) stant potential or in a harmonic oscillator (or a combination of both) follows the same dynamical law as the probability andx ˆ the position operator andp ˆ the canonical momentum density of an ensemble of classical particles in the same po- operator, respectively. tential. Nevertheless, non-classical features may be present in The Wigner function w (x, p) of an energy eigen-state E the Wigner function. wave function of the Hamiltonian H(x, p) = p2/(2m) + V(x) In summary, quantum theory in phase space in terms of the is time independent and therefore it follows from (14) that Wigner function features formal parallels to the classical the- + + p ∂ ∞ (i~/2)2l ∂2l 1V(x) ∂2l 1 ory for an ensemble of classical non-interacting particles. The w (x, p) = 0 . m ∂x − (2l + 1)! ∂x2l+1 ∂p2l+1 E state of both kinds of systems is completely characterized by l=0 a real-valued phase space distribution function w(x, p, t). The (23a) This equation, however, is not sufficient to specify the en- evolution of w(x, p, t) is determined by the classical Liouville ergy eigen-value E, which is determined by the eigen-value equation (26) or the quantum Liouville equation (14), respec- equation11 tively. The quantum Liouville equation equals its classical counterpart plus some additional quantum terms. For classi- p2 ~2 ∂2 ∞ (i~/2)2l ∂2lV(x) ∂2 cal systems, w(x, p, t) may be any non-negative normalizable + wE(x, p) = function. The quantum mechanical Wigner function, however, 2m − 8m ∂x2 (2l)! ∂x2l ∂p2l l=0 has to be compatible with the laws of quantum mechanics, EwE(x, p) . (23b) e.g., to respect the uncertainty relation, and may be negative. Thus, proper quantum characteristics of a quantum dynami- The two equations (23a) and (23b) together are equivalent to cal process are governed the additional terms in the quantum the time independent Schrödinger eigen-value equation Liouville equation or the negative parts of the Wigner func- ~2 ∂2Ψ (x, t) tion. In fact, the negative parts of the Wigner function are a E + V(x, t)Ψ (x, t) = EΨ (x, t) . (24) − 2m ∂x2 E E common measure for non-classicality14,16,29,30.
3.2. Relations to classical mechanics 4. Visualizing quantum dynamics in The motion of an ensemble of classical non-interacting phase space particles may be characterized by a probability distribu- 26 tion w(x, p, t) in phase space. The probability distribution After our short review of the Wigner-function’s properties in w(x, p, t) determines how likely it is that a particle in the en- the previous section, we are now prepared to visualize quan- semble is at position x and has momentum p at time t. It tum mechanics in phase space. This may be accomplished allows to calculate expectation values of observables for the by picturing the Wigner function itself or quantities that are classical ensemble by an integral over the phase space derived from the Wigner function, e.g., the Wigner function flow (17). Ox,p (t) = w(x, p, t)O(x, p)dp dx (25) In contrast to the wave function, the Wigner function is a real-valued quantity. It assigns just a single real value to likewise for a quantum system by (20). The evolu- each point in phase space, which simplifies the visualization tion of w(x, p, t) is determined by the (classical) Liouville 27,28 of Wigner functions to some degree as compared to the visu- equation alization of wave functions, which assign two real values to ∂w(x, p, t) ∂H(x, p, t) ∂w(x, p, t) ∂H(x, p, t) ∂w(x, p, t) each point in configuration space. The dimension of the phase = , ∂t ∂p ∂x − ∂x ∂p space—which is twice the configuration-space’s dimension— (26) turns out to be a limiting factor when visualizing Wigner func- 5
FIG. 2: (Color online) Visualization of the Wigner function (28) of a Gaussian wave packet by false color plots at time t = 0 (left part) and t = 1 (right part) withx ¯ = 1,p ¯ = 3, and σ = 1/2 (dimensionless units with ~ = 1 and m = 1). The panels above the false color plots show the probability distribution in position space (7a) while panels right next to the false color plots show the probability distribution in momentum space (7b). Values of the Wigner function are indicated by the color-bar at the left. tions. Therefore, we restrict ourselves in this contribution to H(x, p, t) = p2/(2m). Thus, the Wigner function of a free par- systems with a two-dimensional phase space. Displaying two- ticle satisfies the relation dimensional real-valued functions is a standard task of scien- w(x, p, t) = w(x pt/m, p, 0) . (29) tific visualization and may be accomplished by various meth- − ods, e.g., by two-dimensional false color plots (sometimes As a consequence of (29), we find the equation called heat maps), by contour line maps, which show lines of constant values of the Wigner function, or by (pseudo) three- ρ(x, t) = w(x, p, t)dp = w(x pt/m, p, 0)dp , (30) dimensional surface plots. − Systems with two or more spatial dimensions have a phase space with at least four dimensions which prohibits visualiz- which states that the probability distribution in position space ing the full Wigner function. In this case one may in order to ρ(x, t) at position x and time t is given by an integral over the reduce the dimensionality, for example, picture lower dimen- Wigner function at time zero along phase space strips which are rotated against the p-axis by an angle of arctan t/m. sional cuts of the Wigner function through the phase space − or projections of the Wigner function onto lower dimensional Thus, measuring the distribution ρ(x, t) of a function of surfaces. the position x and the time t allows to reconstruct the full Wigner function w(x, p, 0) at time zero by means of an inverse Radon transformation11,31. This has also been demonstrated experimentally13–17. 4.1. A free wave packet
The Wigner function of the Gaussian wave packet (2) is given 4.2. Characterizing eigen-states by the strictly positive bivariate normal distribution The Wigner function eigen-state of some time independent 1 (x pt/m x¯)2 2σ2(p p¯)2 w(x, p, t) = exp − − − , potential V(x) is determined by the two equations (23a) and π~ − 2σ2 − ~ (23b). For the harmonic oscillator potential V(x) = mω2 x2/2 (28) we find9,11 the eigen-states which is illustrated in Fig. 2 by two false color plots for two ff = = di erent points in time, for t 0 in part a and for t 1 in wn(x, p) = part b, respectively. At time t = 0, the Wigner function of ( 1)n p2 p2 the Gaussian wave packet is symmetric in phase space with − exp (xκ)2 L 2 + 2(xκ)2 , (31a) ~ ~ 2 n ~ 2 symmetry axes parallel to the x-axis and the p-axis. During π −( κ) − ( κ) the evolution of the wave packet, the Wigner function gets ro- and eigen-energies tated and stretched along the x-axis in phase space. This cor- responds to the familiar wave packet broadening in position 1 En = ~ω n + (31b) space, see also the probability distribution in position space 2 (7a) as shown above the false color plots in Fig. 2. As we pointed out in section 3.2, the quantum Liouville with κ = √mω/~, n = 0, 1,... and Ln(x) denoting the nth equation (14) of a free quantum particle’s Wigner function Laguerre polynomial. reduced to the classical Liouville equation (26) for an en- In Fig. 3 we depict the Wigner function of the harmonic semble of free non-interaction particles with the Hamiltonian oscillator eigen-state with energy E = ~ω(3 + 1/2) together 6 with its Wigner function flow (17). Because the nth Laguerre the Schrödinger ground state wave functions have been calcu- polynomial has n positive roots, the Wigner function of the lated by imaginary time propagation32,33 and the Wigner func- harmonic oscillator eigen-state oscillates and changes n times tion was obtained by calculating (3) numerically afterward. In its sign. Note that despite the fact that the Wigner function Fig. 4 the anharmonicity parameter α grows from left to right (31a) is time-independent there is a permanent circular steady starting with α = 0 (part a) and increasing to α = 1/4 (part b) Wigner functionflow in phase space as indicated by the arrows and α = 3/4 (part c). One clearly sees in Fig. 4 how the diver- in Fig. 3. gence between the direction of the Wigner function flow and In section 3 we demonstrated that for harmonic potentials the tangentials to the lines of constant Wigner function values quantum dynamical effects are absent. This means, the dy- grows with increasing α. namics of the Wigner function follows the classical Liouville equation because all quantum terms in the quantum Liouville equation vanish for harmonic potentials. Also in equation (23) quantum terms vanish, which has consequences for the geom- etry of the Wigner function flow of the eigen-states of the har- 4.3. Scattering by a potential barrier monic oscillator. Using equations(17) and (23a) one can show that j(x, p) w(x, p) = 0 and, therefore, the Wigner func- tion flow is always ∇ tangential to the lines of constant Wigner Scattering of a wave packed by a potential barrier is a well function values. This is a feature that the harmonic oscillator known textbook system to exemplify quantum effects. De- eigen-states share with all time-independent classical phase pending on the potential’s shape and the initial condition of space distributions. Also for classical distributions the flow the wave packet some part of the wave packet may be reflected (27) is tangential to the lines of constant phase space distribu- while the other part is transmitted; a process that has no anal- tion. ogon in classical physics of point-like particles. If the potential has non-quadratic terms then the quantum The quantum dynamical scattering dynamics is commonly terms in (23) will affect the structure of the Wigner eigen- visualized by the wave function in position space4,5 and some- functions and the Wigner function flow and the lines of con- times in momentum space6. Figure 5 illustrates scattering of stant Wigner function values are no longer tangential to each an initial Gaussian wave packet by a potential barrier in phase other. Thus, the divergence between the direction of the space at different points in time. The scattering potential bar- Wigner function flow and the tangentials to the lines of con- rier is located around x = 0 and its shape is given by stant Wigner function values may be interpreted as a measure how strong the non-classical terms in the quantum Liouville equation (14) affect the structure of the eigen-states. = x + x + V(x) V0 tanh ∆ 1 tanh ∆ 1 , (32) In Fig. 4 we show the Wigner function ground states of the − anharmonic oscillator potential V(x) = mω2 x2/2+αx4 for var- ious values of the anharmonicity parameter α. For this figure see also Fig. 5 d. For this figure the Schrödinger wave func- tions was propagated by a Fourier split operator method34 and the Wigner function was obtained by calculating (3) numeri- cally afterward. The system in Fig. 5 starts at time t = 0 (Fig. 5 a) with an asymptotically free Gaussian wave packet. Here we consider a thin potential with height V0 = 64 and characteristic width ∆ = 1/8. The wave packet’s initial kinetic energy equals Ekin = 32. Accordingly to the laws of classical physics the kinetic energy is too small to cross the barrier. As the wave packet moves towards the potential barrier it starts to interact with the barrier. Due to the non-classical terms of the quan- tum Liouville equation the Wigner function starts to develop regions with negative values such that at about t = 1/2 sig- nificant interference patterns have emerged, see Fig. 5 b. At about t = 1 the wave packet has split into two parts that are traveling into opposite directions in position space, a reflected part and a transmitted part that has tunneled trough the po- tential barrier. In phase space this corresponds to two isolated FIG. 3: (Color online) False color plot of the Wigner function of wave packets with approximately Gaussian shape moving into the harmonic oscillator eigen-function with energy E = 3 + 1/2 (di- opposite directions, see also Fig. 5 c. However, there is also mensionless units with ~ = 1, m = 1, and ω = 1). Gray solid lines a region with non-zero Wigner function between these two indicate levels of constant Wigner function values while black arrows packets that exhibits strong oscillations featuring positive and indicate the steady Wigner function flow (17). The arrows’ length is negative values reviling the quantum nature of the coherent proportional to the flow’s magnitude. superposition of the two wave packets14,17. 7
FIG. 4: (Color online) Wigner function ground state of the anharmonic oscillator potential V(x) = mω2 x2/2 + αx4 for various values of the anharmonicity parameter α with α = 0 (part a), α = 1/4 (part b), α = 3/4 (part c) (dimensionless units with ~ = 1, m = 1, and ω = 1/2).
5. Conclusions 10C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechan- ics in Phase Space: An Overview with Selected Papers (World Scientific, 2005). In this contribution we have explored ways to visualize quan- 11W. P. Schleich, Quantum optics in phase space (Wiley-VCH, 2001). tum mechanics in phase space by means of the Wigner func- 12S. Habib, K. Shizume, and W. H. Zurek, Physical Review Letters 80, 4361 tion and the Wigner function flow. The mathematical formal- (1998). 13D. T. Smithey, M. Beck, and M. G. R. A. Faridani, Physical Review Letters ism of quantum mechanics in phase space in terms of the 70, 1244 (1993). Wigner function resembles classical Hamiltonian mechanics 14C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997). of non-interacting classical particles. Thus, teaching quantum 15G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997). 16 mechanics in phase space may have some advantages when A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Physical Review Letters 87, 050402 (2001). moving from classical to quantum physics. Non-classical fea- 17S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, tures are highlighted in phase space, for example by negative and S. Haroche, Nature 445, 510 (2008). values of the phase space distribution or the divergence be- 18I. Bialynicki-Birula, P. Górnicki, and J. Rafelski, Physical Review D 44, tween the direction of the Wigner function flow and tangen- 1825 (1991). 19 tials to the lines of constant Wigner function values of eigen- G. R. Shin, I. Bialynicki-Birula, and J. Rafelski, Physical Review A 46, 645 (1992). states. 20S. Varró and J. Javanainen, Journal of Optics B: Quantum and Semiclassical Optics 5, S402 (2003). 21C. H. Keitel, Contemporary Physics 42, 353 (2001). 22O. T. Serimaa, J. Javanainen, and S. Varró, Physical Review A 33, 2913 (1986). Acknowledgments 23R. L. Hudson, Reports on Mathematical Physics 6, 249 (1974). 24J. E. Moyal, Mathematical Proceedings of the Cambridge Philosophical So- ciety 45, 99 (1949). N. R. I. would like to express her appreciation to the organiz- 25A. Peres, Quantum Theory: Concepts and Methods (Kluver Academic Pub- ers of the XVth International Summer Science School of the lishers, 1993). city of Heidelberg and to the Max-Planck-Institut für Kern- 26We do not distinguish in our notation between the Wigner function and the physik for its hospitality. classical probability distribution. It should be clear from the context when- ever w(x, p, t) denotes a Wigner function or a classical probability distribu- tion. 27 1B. Thaller, Visual Quantum Mechanics (Springer, 2000). P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge Non- 2B. Thaller, Advanced Visual Quantum Mechanics (Springer, 2000). linear Science Series, Vol. 9 (Cambridge University Press, 1998). 28 3R. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and F. Schwabl, Statistical mechanics, Advanced texts in physics (Springer, Visualized Examples, 2nd ed. (Oxford University Press, 2006). 2006). 29 4S. Brandt and H. D. Dahmen, The Picture Book of Quantum Mechanics, 3rd A. Kenfack and K. Zyczkowski,˙ Journal of Optics B: Quantum and Semi- ed. (Springer, 2001). classical Optics 6, 396 (2004). 30 5A. Goldberg, H. M. Schey, and J. L. Schwartz, American Journal of Physics M. Mahmoudi, Y. I. Salamin, and C. H. Keitel, Physical Review A 72, 35, 177 (1967). 033402 (2005). 31 6A. Goldberg, H. M. Schey, and J. L. Schwartz, American Journal of Physics D. Leibfried, T. Pfau, and C. Monroe, Physics Today 51, 22 (1998). 32 36, 454 (1968). A. Goldberg and J. L. Schwartz, Journal of Computational Physics 1, 433 7R. P. Feynman, QED: The Strange Theory of Light and Matter (Princeton (1967). 33 University Press, 1986). A. Goldberg and J. L. Schwartz, Journal of Computational Physics 1, 448 8E. Wigner, Physical Review 40, 749 (1932). (1967). 34 9M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Physics Re- M. D. Feit, J. A. Fleck, Jr., and A. Steiger, Journal of Computational ports 106, 121 (1984). Physics 47, 412 (1982). 8
FIG. 5: (Color online) Scattering of a wave packed by a potential barrier (32) in phase space. Potential parameters are V0 = 64 and ∆ = 1/8, the wave-packet’s initial mean momentum is 8, all values in dimensionless units with ~ = 1 and m = 1. The three upper panels show the Wigner function, the position space probability distribution and the momentum space probability distribution at different times; t = 0 (part a), t = 1/2 (part b), and t = 1 (part c). Part d illustrates the scattering potential (32).