DOCSLIB.ORG

Problem: Evolution of a Free Gaussian Wavepacket Consider a Free Particle

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Problem: Evolution of a Free Gaussian Wavepacket Consider a Free Particle Problem: Evolution of a free Gaussian wavepacket Consider a free particle which is described at t = 0 by the normalized Gaussian wave function µ ¶1=4 2a 2 ª(x; 0) = e¡ax : ¼ The normalization factor is easy to obtain. We wish to ¯nd its time evolution. How does one do this ? Remember the recipe in quantum mechanics. We expand the given wave func- tion in terms of the energy eigenfunctions and we know how individual energy eigenfunctions evolve. We then reconstruct the wave function at a later time t by superposing the parts with appropriate phase factors. For a free particle H = p2=(2m) and therefore, momentum eigenfunctions are also energy eigenfunctions. So it is the mathematics of Fourier transforms! It is straightforward to ¯nd ©(k; 0) the Fourier transform of ª(x; 0): Z 1 dk ª(x; 0) = p ©(k; 0) eikx : ¡1 2¼ Recall that the Fourier transform of a Gaussian is a Gaussian. Z 1 dx Á(k) ´ ©(k; 0) = p ª(x; 0) e¡ikx (1.1) ¡1 2¼ µ ¶ µ ¶ 1=4 Z 1 1=4 2 1 2a ¡ikx ¡ax2 1 ¡ k = p dx e e = e 4a (1.2) 2¼ ¼ ¡1 2¼a What this says is that the Gaussian spatial wave function is a superposition of di®erent momenta with the probability of ¯nding the momentum between k1 and k1 + dk being pro- 2 portional to exp(¡k1=(2a)) dk. So the initial wave function is a superposition of di®erent plane waves with di®erent coef- ¯cients (usually called amplitudes). Note that exp ikx for each real k is also an eigenfunction ¹h2k2 of the Hamiltonian with eigenvalue Ek = 2m . Therefore, we know that exp(ikx) evolves in time with a time dependence given by e¡iEkt=¹h . So the time-evolved state, ª(x; t), is given by Z 1 dk ª(x; t) = p Á(k) eikx e¡iEkt=¹h (1.3) ¡1 2¼ µ ¶ 1=4 Z 1 2 1 dk ¡ k ¡iE t=¹h ikx = p e 4a e k e (1.4) 2¼a ¡1 2¼ where we know Ek and this integral has to be done. The exponential has an argument given by µ ¶ k2 2i¹hat bk2 1 2i¹hat ¡ [ 1 + ] = ¡ where b = 1 + : 4a m 2 2a m 1 Thus the integral to be done is µ ¶1=4 Z 1 1 dk 2 p e¡bk =2 eikx 2¼a ¡1 2¼ and this can be done in the usual way in spite of the fact that b is complex since the integral is well-de¯ned. We obtain, using r Z 1 2 ¡bk2=2 ikx 2¼ ¡ x dk e e = e 2b ; (1.5) ¡1 b µ ¶ r 1=4 2 1 1 ¡ x the result ª(x; t) = e 2b (1.6) 2¼a b µ ¶ 2a 1=4 1 ax2 = q exp[ ¡ ] : (1.7) ¼ 2i¹hat (1 + 2i¹hat ) 1 + m m The probability density is given by (after a wee bit of algebra) r 2 ¡ 2ax 2 2a 1 2¹hat 2 jª(x; t)j = q e 1+( m ) (1.8) ¼ 2¹hat 2 1 + ( m ) from which we can read o® " µ ¶ # 1 2¹hat 2 hx2i = 1 + : (1.9) 4a m 1 Note that the initial width 4a increases quadratically as a function of time. To understand this result it is useful to rewrite it. We note that since < x >= 0 we have 2 2 the initial < x > is 1=(4a). Let us call this σ0 = 1=(4a), in a great notational leap forward. 2 So the width at time t is given by (chasing the factors of a and replacing them by σ0=2) 2 2 2 2 2 ¹h t hx it ´ σx(t) = σ0 + 2 2 : (1.10) 4m σ0 Now we note that the uncertainty in the momentum can be read o® from the wave function in k-space. Since Á(k) is Gaussian we ¯nd 2 2 2 2 ¹h hk i = a and hence σp = ¹h a = 2 4σ0 1 where we have rewritten a in terms of σ0 and used the de Broglie relation. σ2t2 Therefore, σ2(t) = σ2 + p : (1.11) x 0 m2 1Why can the initial wave function be called a minimum uncertainty wave packet? How does the momentum uncertainty change as a function of time ? Why ? 2 So the initial uncertainty in the position increases with an extra part which can be thought of as follows: σpt=m is the distance traveled by a classical particle with momentum equal to the uncertainty σp in a time t. So for t << σ0=(σp=m) the width remains unchanged. Given an arbitrary initial wave function ª(x; 0) for a free particle we can ¯nd ª(x; t). First ¯nd its decomposition in terms of Á(k): 2 Z 1 dx ª(x; 0) = p Á(k) eikx : ¡1 2¼ Then each plane wave, an eigenfunction of the free particle of the Hamiltonian with eigenvalue ¡iE t=¹h Ek, evolves in time and at time t picks up a phase factor e k . Therefore we have Z 1 dk ª(x; t) = p Á(k) eikx e¡iEkt=¹h (1.12) ¡1 2¼ Z Z 1 dk 1 dy = p eikx e¡iEkt=¹h p e¡ikyª(y; 0) replacing Á(k) (1.13) ¡1 2¼ ¡1 2¼ Z ·Z ¸ 1 1 dk = dy eik(x¡y) e¡iEkt=¹h ª[y; 0] (1.14) ¡1 ¡1 2¼ Z 1 = dy G(x; t; y; 0) ª[y; 0] (1.15) ¡1 where we have de¯ned the corresponding Green's function for the SchrÄodingerequation: Z ·Z ¸ 1 1 dk G(x; t; y; 0) ´ dy eik(x¡y) e¡iEkt=¹h : ¡1 ¡1 2¼ Note that given the function ª(y; 0) (y is just a dummy variable), G tells you how to propagate the system in time and obtain the function ª(x; t) at time t. G is known as the free-particle propagator. It is also called the Feynman propagator for a non-relativistic free particle. Performing the integral we ¯nd the explicit expression r 2 m i m(x¡y) G(x; t; y; 0) = e 2¹ht 2¼i¹ht for t > 0. As t ! 0 this goes over to a delta function as one can check. Finding the Green's function for other problems such as the quantum harmonic oscillator is somewhat tedious. (See R. P. Feynman and A. R. Hibbs, \Quantum Mechanics and Path Integrals", unfortunately, poorly proof-read.) 2Á(k) can be found as usual by the inverse Fourier transform: Z 1 dy Á(k) = p ª(y; 0) e¡iky : ¡1 2¼ 3.
Load more

Recommended publications
  • Statistics of a Free Single Quantum Particle at a Finite
    STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • The S-Matrix Formulation of Quantum Statistical Mechanics, with Application to Cold Quantum Gas
    THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pye Ton How August 2011 c 2011 Pye Ton How ALL RIGHTS RESERVED THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS Pye Ton How, Ph.D. Cornell University 2011 A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation (“two-body approximation”) corresponds to the ex- act resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applica- tions of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at T 2πn/[m log(2π/mg)], where n c ≈ is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of µ/T ) and computed from the numerical solutions.
    [Show full text]
  • Class 6: the Free Particle
    Class 6: The free particle A free particle is one on which no forces act, i.e. the potential energy can be taken to be zero everywhere. The time independent Schrödinger equation for the free particle is ℏ2d 2 ψ − = Eψ . (6.1) 2m dx 2 Since the potential is zero everywhere, physically meaningful solutions exist only for E ≥ 0. For each value of E > 0, there are two linearly independent solutions ψ ( x) = e ±ikx , (6.2) where 2mE k = . (6.3) ℏ Putting in the time dependence, the solution for energy E is Ψ( x, t) = Aei( kx−ω t) + Be i( − kx − ω t ) , (6.4) where A and B are complex constants, and Eℏ k 2 ω = = . (6.5) ℏ 2m The first term on the right hand side of equation (6.4) describes a wave moving in the direction of x increasing, and the second term describes a wave moving in the opposite direction. By applying the momentum operator to each of the travelling wave functions, we see that they have momenta p= ℏ k and p= − ℏ k , which is consistent with the de Broglie relation. So far we have taken k to be positive. Both waves can be encompassed by the same expression if we allow k to take negative values, Ψ( x, t) = Ae i( kx−ω t ) . (6.6) From the dispersion relation in equation (6.5), we see that the phase velocity of the waves is ω ℏk v = = , (6.7) p k2 m which differs from the classical particle velocity, 1 pℏ k v = = , (6.8) c m m by a factor of 2.
    [Show full text]
  • Chapter 3 Wave Properties of Particles
    Chapter 3 Wave Properties of Particles Overview of Chapter 3 Einstein introduced us to the particle properties of waves in 1905 (photoelectric effect). Compton scattering of x-rays by electrons (which we skipped in Chapter 2) confirmed Einstein's theories. You ought to ask "Is there a converse?" Do particles have wave properties? De Broglie postulated wave properties of particles in his thesis in 1924, based partly on the idea that if waves can behave like particles, then particles should be able to behave like waves. Werner Heisenberg and a little later Erwin Schrödinger developed theories based on the wave properties of particles. In 1927, Davisson and Germer confirmed the wave properties of particles by diffracting electrons from a nickel single crystal. 3.1 de Broglie Waves Recall that a photon has energy E=hf, momentum p=hf/c=h/, and a wavelength =h/p. De Broglie postulated that these equations also apply to particles. In particular, a particle of mass m moving with velocity v has a de Broglie wavelength of h λ = . mv where m is the relativistic mass m m = 0 . 1-v22/ c In other words, it may be necessary to use the relativistic momentum in =h/mv=h/p. In order for us to observe a particle's wave properties, the de Broglie wavelength must be comparable to something the particle interacts with; e.g. the spacing of a slit or a double slit, or the spacing between periodic arrays of atoms in crystals. The example on page 92 shows how it is "appropriate" to describe an electron in an atom by its wavelength, but not a golf ball in flight.
    [Show full text]
  • Informal Introduction to QM: Free Particle
    Chapter 2 Informal Introduction to QM: Free Particle Remember that in case of light, the probability of nding a photon at a location is given by the square of the square of electric eld at that point. And if there are no sources present in the region, the components of the electric eld are governed by the wave equation (1D case only) ∂2u 1 ∂2u − =0 (2.1) ∂x2 c2 ∂t2 Note the features of the solutions of this dierential equation: 1. The simplest solutions are harmonic, that is u ∼ exp [i (kx − ωt)] where ω = c |k|. This function represents the probability amplitude of photons with energy ω and momentum k. 2. Superposition principle holds, that is if u1 = exp [i (k1x − ω1t)] and u2 = exp [i (k2x − ω2t)] are two solutions of equation 2.1 then c1u1 + c2u2 is also a solution of the equation 2.1. 3. A general solution of the equation 2.1 is given by ˆ ∞ u = A(k) exp [i (kx − ωt)] dk. −∞ Now, by analogy, the rules for matter particles may be found. The functions representing matter waves will be called wave functions. £ First, the wave function ψ(x, t)=A exp [i(px − Et)/] 8 represents a particle with momentum p and energy E = p2/2m. Then, the probability density function P (x, t) for nding the particle at x at time t is given by P (x, t)=|ψ(x, t)|2 = |A|2 . Note that the probability distribution function is independent of both x and t. £ Assume that superposition of the waves hold.
    [Show full text]
  • Zitterbewegung in Relativistic Quantum Mechanics
    Zitterbewegung in relativistic quantum mechanics Alexander J. Silenko∗ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China, Research Institute for Nuclear Problems, Belarusian State University, Minsk 220030, Belarus Abstract Zitterbewegung of massive and massless scalar bosons and a massive Proca (spin-1) boson is analyzed. The results previously obtained for a Dirac fermion are considered. The equations describing the evolution of the velocity and position of the scalar boson in the generalized Feshbach- Villars representation and the corresponding equations for the massive Proca particle in the Sakata- Taketani representation are equivalent to each other and to the well-known equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy-Wouthuysen representation. Since the position and velocity operators in the Foldy-Wouthuysen representation and their transforms to other representations are the quantum-mechanical counterparts of the corresponding classical variables, Zitterbewegung is not observable. PACS numbers: 03.65.-w, 03.65.Ta Keywords: quantum mechanics; Zitterbewegung ∗ [email protected] 1 I. INTRODUCTION Zitterbewegung belongs to the most important and widely discussed problems of quantum mechanics (QM). It is a well-known effect consisting in a superfast trembling motion of a free particle. This effect has been first described by Schr¨odinger[1] and is also known for a scalar particle [2, 3]. There are plenty of works devoted to Zitterbewegung. In our study, only papers presenting a correct analysis of the observability of this effect are discussed. The correct conclusions about the origin and observability of this effect have been made in Refs.
    [Show full text]
  • Lecture 22 Relativistic Quantum Mechanics Background
    Lecture 22 Relativistic Quantum Mechanics Background Why study relativistic quantum mechanics? 1 Many experimental phenomena cannot be understood within purely non-relativistic domain. e.g. quantum mechanical spin, emergence of new sub-atomic particles, etc. 2 New phenomena appear at relativistic velocities. e.g. particle production, antiparticles, etc. 3 Aesthetically and intellectually it would be profoundly unsatisfactory if relativity and quantum mechanics could not be united. Background When is a particle relativistic? 1 When velocity approaches speed of light c or, more intrinsically, when energy is large compared to rest mass energy, mc2. e.g. protons at CERN are accelerated to energies of ca. 300GeV (1GeV= 109eV) much larger than rest mass energy, 0.94 GeV. 2 Photons have zero rest mass and always travel at the speed of light – they are never non-relativistic! Background What new phenomena occur? 1 Particle production e.g. electron-positron pairs by energetic γ-rays in matter. 2 Vacuum instability: If binding energy of electron 2 4 Z e m 2 Ebind = > 2mc 2!2 a nucleus with initially no electrons is instantly screened by creation of electron/positron pairs from vacuum 3 Spin: emerges naturally from relativistic formulation Background When does relativity intrude on QM? 1 When E mc2, i.e. p mc kin ∼ ∼ 2 From uncertainty relation, ∆x∆p > h, this translates to a length h ∆x > = λ mc c the Compton wavelength. 3 for massless particles, λc = , i.e. relativity always important for, e.g., photons. ∞ Relativistic quantum mechanics: outline
    [Show full text]
  • Relativistic Quantum Mechanics
    Relativistic Quantum Mechanics Dipankar Chakrabarti Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India (Dated: August 6, 2020) 1 I. INTRODUCTION Till now we have dealt with non-relativistic quantum mechanics. A free particle satisfying ~p2 Schrodinger equation has the non-relatistic energy E = 2m . Non-relativistc QM is applicable for particles with velocity much smaller than the velocity of light(v << c). But for relativis- tic particles, i.e. particles with velocity comparable to the velocity of light(e.g., electrons in atomic orbits), we need to use relativistic QM. For relativistic QM, we need to formulate a wave equation which is consistent with relativistic transformations(Lorentz transformations) of special theory of relativity. A characteristic feature of relativistic wave equations is that the spin of the particle is built into the theory from the beginning and cannot be added afterwards. (Schrodinger equation does not have any spin information, we need to separately add spin wave function.) it makes a particular relativistic equation applicable to a particular kind of particle (with a specific spin) i.e, a relativistic equation which describes scalar particle(spin=0) cannot be applied for a fermion(spin=1/2) or vector particle(spin=1). Before discussion relativistic QM, let us briefly summarise some features of special theory of relativity here. Specification of an instant of time t and a point ~r = (x, y, z) of ordinary space defines a point in the space-time. We’ll use the notation xµ = (x0, x1, x2, x3) ⇒ xµ = (x0, xi), x0 = ct, µ = 0, 1, 2, 3 and i = 1, 2, 3 x ≡ xµ is called a 4-vector, whereas ~r ≡ xi is a 3-vector(for 4-vector we don’t put the vector sign(→) on top of x.
    [Show full text]
  • Module IV. QUANTUM MECHANICS Planck's Law
    Module IV. QUANTUM MECHANICS Blackbody Radiation: A Blackbody is one which absorbs all the light which is incident on it and re-radiates all those absorbed radiations when it is heated to incandescence (red hot). Blackbody Radiation spectrum: A Blackbody is one which absorbs the entire radiation incident on it and emits all the absorbed radiation when it is hotter. A true blackbody does not exist practically. Many Scientists tried to device an ideal blackbody, however, a blackbody designed by Wien has features very close to the true blackbody. A blackbody at a particular temperature found to emit a radiation of all possible wavelengths. The radiations emitted are analysed and is called blackbody radiation spectrum. Nature of Blackbody radiation Spectrum: The radiations emitted is analysed and the intensity (or the energy density) of emitted radiation is plotted against the wavelength as shown in fig. (i)At a constant temperature, radiations of different wavelengths are emitted out in different quantities. It is a continuous spectrum starting from certain minimum wavelength to maximum wavelength. There exist a particular wavelength (Called휆푚) for which intensity of emitted radiation is maximum. (ii)There are different curves for different temperatures. The intensity emitted radiation increases with the increase of the temperature. (iii) As the temperature of the body increases, the wavelength corresponding to maximum intensity (휆푚) shifts towards lower wavelength side. (iv) The area under the curve gives the total radiant energy. The area under the curve is proportional to the forth power of the absolute temperature. This is called Stefan’s 4th power law [E 훼 푇4].
    [Show full text]
  • Klein-Gordon Equation and Wave Function for Free Particle in Rindler Space-Time
    International Journal of Advanced Research in Physical Science (IJARPS) Volume 7, Issue 9, 2020, PP 10-12 ISSN No. (Online) 2349-7882 www.arcjournals.org Klein-Gordon Equation and Wave Function for Free Particle in Rindler Space-Time Sangwha-Yi* Department of Math , Taejon University 300-716 , South Korea *Corresponding Author: Sangwha-Yi, Department of Math, Taejon University 300-716, South Korea Abstract: Klein-Gordon equation is a relativistic wave equation. It treats spinless particle. The wave function cannot use as a probability amplitude. We made Klein-Gordon equation in Rindler space-time. In this paper, we make free particle’s wave function as the solution of Klein-Gordon equation in Rindler space-time. Keywords: Wave Function; Free Particle; Klein-Gordon equation Rindler Space-time; PACS Number: 03.30.+p,03.65 1. INTRODUCTION At first, Klein-Gordon equation is for free particle field in inertial frame. mc 2 21 2 2 0 2ct 2 2 m is free particle’s mass (1) If we write wave function as solution of Klein-Gordon equation for free particle,[3] A0 exp i ( t k x ) A0 is amplitude, is angular frequency, kk is wave number (2) Energy and momentum is in inertial frame,[3] E , pk (3) Hence, energy-momentum relation is[3] E2 22 p 22 c m 24 c 222 k c m 24 c (4) Or angular frequency- wave number relation is 2mc 2 2 k2 (5) c22 2. KLEIN-GORDON EQUATION AND WAVE FUNCTION FOR FREE PARTICLE FIELD IN RINDLER- SPACE-TIME Rindler coordinates are c2 a cc22a ct (10 )sinh(0 ) , x ( 10 )cosh(0 ) ac0 a00 c a yz23, (6) We know Klein-Gordon equation in Rindler space-time.
    [Show full text]
  • Chapter 10 Relativistic Wave Equations
    Chapter 10 Relativistic Wave Equations 10.1 The Relativistic Wave Equation In the previous chapter we have recalled the basic notions of non-relativistic quan- tum mechanics. We have seen that, in the Schroedinger representation, the physical state of a free particle of mass m is described by a wave function ψ(x, t) which is itself a classical field having a probabilistic interpretation. For a single free particle this function is solution to the Schroedinger equation (9.79). A system of N inter- acting particles will be described by a wave function ψ(x1, x2,...,xN t) whose squared modulus represents the probability density of finding the particles; at the points x1, x2,...,xN at the time t. In this description the number N of particles is always constant that is it cannot vary during the interaction. Note that the conser- vation of the number of particles is related to the conservation of mass in a non- relativistic theory: The sum of the rest masses of the particles cannot change during the interaction. A change in this number would imply a variation in the sum of the corresponding rest masses. Strictly related to this property of the Schroedinger equation is the fact that the total probability is conserved in time. Let us recall the argument in the case of a single particle. The normalization of ψ(x, t) is fixed by requiring that the probability of finding the particle anywhere in space at any time t be one: d3x ψ(x, t) 2 1, | | = V! where V R3 representing the whole space.
    [Show full text]