The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)
The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)
In (1), we noted that a Gaussian wavefunction may be a solution to the time-independent Schrodinger equation (e.g. the ground state solution of the quantum oscillator) and simultaneously a solution of maximizing spatial Shannon’s entropy - density(x) ln(density(x)) subject to the constraint V(x) xx constant, where density(x) =W*(x)W(x) with W(x) being the wavefunction. In (2), which deals strictly with classical probabilities and the Maxwell-Boltzmann factor n(ei) = exp(-ei/T), we argued that this function is equivalent to: ln(n(ei)) = -ei/T. This in turn is equivalent to the maximization of ln(n(ei)!) with the constraint ei n(ei) which is part of: ln( N!/ Product over i n(ei)!) with the full constraint Sum over i ei n(ei)=Eave. Here n(ei)=N p(ei) where p(ei) is a probability and N a very large number. Thus a function of exp( something) may indicate that “something” * density(x) is a constraint with -density ln(density) being Shannon’s entropy. Quantum mechanics for a bound system yields a real wavefunction W(x) such that density(x)=W(x)W(x)=probability(x). Thus maximization of Shannon’s entropy - density(x) ln (density) subject to a constraint: density(x)B(x) should correspond to a solution for density of exp( B(x)). B(x) as part of the constraint, however, should represent a physical constraint. Thus if there is a solution to the Schrodinger equation of the form exp(.5B(x)) and B(x)density(x) represents the physical constraint in the system, classical maximization of probability solves the quantum problem in such a case. The above ideas may be applied to the quantum oscillator ground state and essentially repeat what is done in (1). We, however, try to generalize them to see if they may apply to a free quantum particle. In this case the probability used is exp(ipx) which is complex. It corresponds to P(x)=exp(ipx)exp(-ipx)=1 which is a uniform distribution which has maximum lack of information.
Experiments involving single or few elementary particles are completely described by Quantum Mechanics. Notwithstanding the success of that quantitative description, various aspects of observations, as nonlocality and the statistical randomness of results, remain as mysterious properties apart from the quantum theory, and they are attributed to the strangeness of the microscopic world. Here we restart from the fundamental relations of uncertainty to reformulate the probability law of Born including the temporal variable. Considering that both the spatial and the temporal variables play a symmetric role in the wave-function Ψ(x,t) , a temporal wavepacket is built and analysed. The probability density is written as p(x,t) = | Ψ(x,t) |^2, where the probabilistic interpretation for the temporal wavepacket is equivalent to Born's law for the spatial variable, x. For the convenience of the discussion of the role of the temporal variable, we write p(x_0,t) = | Ψ(x_0,t) |^2 for a free particle, expressing only the temporal wavepacket, then we discuss its spread. In the light of the evolution of this temporal wavepacket we analyse basic processes of matter-wave interaction, involving single and entangled entities. Nonlocality appears then as a consequence of the spread of the temporal wavepacket; and the position of each detected event in two-slits interferometry as due to the independent phases of the spatial and temporal wavepackets.
The solution is essentially a complex function which accounts for the matter wave spreading illustrated in Figure 28.3. As the wave function itself cannot be identified with a physical property of the free particle, an indirect interpretation was suggested by Born. He pointed out that its modulus ( tx,Ψ ) ( ),x tΨ is large where the particle becomes localized and small elsewhere, and hence ( ) ( ) 2, ,P x t dx x t dx= Ψ (29.1) is the probability of finding the particle described by ( )tx,Ψ at a point between x and at a time t. Therefore the matter wave spreading is associated with a decrease in the probability density . As the total probability of finding the particle somewhere on the x-axis must be unity, the wave function must be normalized to unity, that is dxx + ( txP , ) ( ) ( ) ( ), , ,P x t dx x t x t dx∞ ∞ ∗ −∞ −∞ = Ψ Ψ∫ ∫ = (29.2) The behaviour of the wave function ( )tx,Ψ is determined by Eq.(28.38), which may be rewritten as ( ) ( ) ( ) t x tx m itxttx ∆∂ Ψ∂+Ψ=∆+Ψ 2 2 , 2 ,, = (29.3) where the expression ( ) ( )/ , ,t x t t x t⎡∂Ψ ∂ ≅ Ψ + ∆ −Ψ ∆⎣ / t⎤⎦ ) , which holds for a small , has been used. In view of Eq.(29.3) the values of the wave function at any time can be derived from the initial value that describes a reference state of the particle.
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
Carrying over the idea of a wave function to relativistic quantum mechanics naturally leads to a multi-time wave function of the form ψ(t 1 , x 1 , ..., t N , x N ).It depends on one time coordinate t k and one space coordinate x k for each particle k = 1, ..., N .As already proposed by Dirac in 1932, the dynamics of a multi-time wave function can be described by a system of N equations of motion, which determine the evolution along the N time coordinates.The existence of a common solution to such a multi-time system is associated with an integrability condition, the so-called consistency condition, that substantially restricts the possibilities to introduce an interaction between the particles.The aim of this doctoral thesis is to treat several physically relevant possibilities for interacting dynamics in a mathematically rigorous fashion.After a physical introduction to multi-time wave functions and their meaning as probability amplitudes similarly as in non-relativistic quantum mechanics, the mathematical solution theory of systems of equations for a multi-time wave function is discussed.The consistency conditions in Dirac equations with added interaction potentials are the topic of the first main part of the thesis.A generalization of the no-go result by Petrat and Tumulka (2014) about non-existence of solutions in multi-time systems with interaction potentials is proven.It entails that no Poincaré invariant interacting potentials are compatible with the consistency condition.This possibility for the formulation of relativistic interacting quantum dynamics is hence ruled out.Dirac's suggestion from 1932 to circumvent the problem with interaction potentials is to employ second quantized potentials, which is taken on in the subsequent chapter.Together with Fock and Podolsky, he devised a model of quantum electrodynamics that describes the interaction of N Dirac fermions through an electromagnetic field.In the second main part of this thesis, that model is formulated mathematically, using a scalar field for simplicity and a cut-off to bypass the well-known ultraviolet divergences.First, the single-time equations are treated by proving essential self-adjointness of the Hamiltonians and also invariance of certain regular functions under the respective time-evolutions.Then, a consistent multitime evolution is constructed by glueing together the single-time evolutions.Besides the results on existence and uniqueness of solutions, it is also demonstrated that the model is indeed interacting.In the subsequent part, a well-defined interaction is constructed in another way, namely by particle creation and annihilation via interior-boundary conditions (IBCs).This proposed solution to the ultraviolet problem of quantum field theory is treated in a relativistic setting for the first time.In a simplified model in 1 + 1 dimensions where Dirac fermions create and annihilate each other when they meet, the class of IBCs that lead to probability conservation is identified.The theorem on existence and uniqueness in this model is the first rigorous result in a multi-time system with a variable number of particles.As final results, general theorems on linear systems of arbitrary order in the time derivatives are presented.Those show how the solution theory of multi-time equations changes if time derivatives of higher order are present.v ZUSAMMENFASSUNG Die Übertragung des Konzepts einer Wellenfunktion auf die relativistische Quantenmechanik führt auf natürliche Weise zu einer Vielzeiten-Wellenfunktion, welche die Form ψ(t 1 , x 1 , ..., t N , x N ) besitzt.Sie hängt von einer Zeitkoordinate t k und einer Ortskoordinate x k für jedes Teilchen k = 1, ..., N ab.Wie schon von Dirac 1932 vorgeschlagen, kann die Dynamik einer Vielzeiten-Wellenfunktion durch ein System aus N Bewegungsgleichungen beschrieben werden, welche die Entwicklung entlang der N Zeitkoordinaten bestimmen.Die Existenz einer gemeinsamen Lösung zu solch einem Vielzeiten-System ist hierbei mit einer Integrabilitätsbedingung verbunden, der so genannten Konsistenzbedingung, die die Möglichkeiten, eine Wechselwirkung zwischen den Teilchen einzuführen, erheblich einschränkt.Das Ziel dieser Doktorarbeit ist es, verschiedene physikalisch relevante Möglichkeiten für wechselwirkende Dynamiken auf mathematisch rigorose Weise zu behandeln.Nach einer physikalischen Einleitung über Vielzeiten-Wellenfunktionen und deren Bedeutung als Wahrscheinlichkeitsamplitude, ähnlich wie in der nicht-relativistischen Quantenmechanik, wird die mathematische Lösungstheorie von Gleichungssystemen für eine Vielzeiten-Wellenfunktion besprochen.Die Konsistenzbedingung in Dirac-Gleichungen mit addierten Wechselwirkungspotentialen ist Gegenstand des ersten Hauptteils der Arbeit.Es wird eine Verallgemeinerung des Unmöglichkeits-Resultats von Petrat und Tumulka (2014) über die Nichtexistenz von Lösungen in Vielzeiten-Systemen mit Wechselwirkungspotentialen bewiesen.Es beinhaltet, dass keine Poincaré-invarianten wechselwirkenden Potentiale mit der Konsistenzbedingung konform gehen.Diese Möglichkeit für die Formulierung von relativistischer, wechselwirkender Quantendynamik ist damit ausgeschlossen.Diracs Vorschlag aus dem Jahr 1932, das Problem mit Wechselwirkungspotentialen zu umgehen, ist durch Benutzung zweitquantisierter Potentiale, was im folgenden Kapitel angegangen wird.Zusammen mit Fock und Podolsky ersann er ein Modell der Quantenelektrodynamik, das die Wechselwirkung von N Dirac-Fermionen durch ein elektromagnetisches Feld beschreibt.Im zweiten Hauptteil der Arbeit wird das Modell mathematisch formuliert, wobei der Einfachheit halber ein skalares Feld und zur Verhinderung der wohlbekannten Ultraviolett-Divergenzen ein Cut-Off benutzt wird.Zuerst werden die Ein-Zeit-Gleichungen behandelt, indem die wesentliche Selbstadjungiertheit der Hamiltonians und auch die Invarianz gewisser regulärer Funktionen unter den entsprechenden Zeitentwicklungen gezeigt wird.Sodann wird eine konsistente Vielzeiten-Entwicklung konstruiert, indem man die Ein-Zeit-Entwicklungen zusammenfügt.Neben den Resultaten über Existenz und Eindeutigkeit von Lösungen wird auch aufgezeigt, dass das Modell tatsächlich wechselwirkend ist.Im darauf folgenden Abschnitt wird eine wohldefinierte Wechselwirkung mittels einer anderen Methode konstruiert, und zwar durch Teilchenerzeugung und -vernichtung via interiorboundary conditions (IBCs).Diese vorgeschlagene Lösung des Ultraviolettproblems in der Quantenfeldtheorie wird zum ersten Mal in einem relativistischen Rahmen behandelt.In einem vereinfachten Modell in 1 + 1 Dimensionen, in welchem Dirac-Fermionen sich bei Kontakt gegenseitig erzeugen und vernichten können, werden jene IBCs klassifiziert, die vii zu Wahrscheinlichkeitserhaltung führen.Das Existenz-und Eindeutigkeitstheorem in diesem Modell ist das erste rigorose Resultat in einem Vielzeiten-System mit einer variablen Teilchenzahl.Als letzte Resultate werden allgemeine Theoreme über lineare Systeme beliebiger Ordnung in den Zeitableitungen präsentiert.Diese zeigen auf, wie sich die Lösungstheorie von Vielzeiten-Gleichungen verändert, wenn höhere Zeitableitungen vorkommen.viii