Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 12
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When following the solutions starting from a fixed initial point and runningto all possible final points qi at a time t, the associated actions of these solutions forma function A(qi , t). Due to (1.18), this satisfies precisely the first of the equations(1.58):pi =∂A(qi , t).∂qi(1.62)Moreover, the function A(qi , t) has the time derivativedA(qi (t), t) = pi (t)q̇i (t) − H(pi (t), qi (t), t).dt(1.63)Together with (1.62) this implies∂t A(qi , t) = −H(pi , qi , t).(1.64)If the momenta pi on the right-hand side are replaced according to (1.62), A(qi , t)is indeed seen to be a solution of the Hamilton-Jacobi differential equation:∂t A(qi , t) = −H(∂qi A(qi , t), qi , t).1.2(1.65)Relativistic Mechanics in Curved SpacetimeThe classical action of a relativistic spinless point particle in a curved fourdimensional spacetime is usually written as an integralA = −Mc2Zdτ L(q, q̇) = −Mc2Zqdτ gµν q̇ µ (τ )q̇ ν (τ ),(1.66)H. Kleinert, PATH INTEGRALS111.3 Quantum Mechanicswhere τ is an arbitrary parameter of the trajectory.
It can be chosen in the finaltrajectory to make L(q, q̇) ≡ 1, in which case it coincides with the proper time ofthe particle. For arbitrary τ , the Euler-Lagrange equation (1.8) reads#"1d1gµν q̇ ν =∂µ gκλ q̇ κ q̇ λ .dt L(q, q̇)2L(q, q̇)(1.67)If τ is the proper time where L(q, q̇) ≡ 1, this simplifies tod 1gµν q̇ ν =∂µ gκλ q̇ κ q̇ λ ,dt2or1∂ g − ∂λ gµκ q̇ κ q̇ λ .gµν q̈ =2 µ κλAt this point one introduces the Christoffel symbolν1Γ̄λνµ ≡ (∂λ gνµ + ∂ν gλµ − ∂µ gλν ),2(1.68)(1.69)(1.70)and the Christoffel symbol of the second kind3Γ̄κν µ ≡ g µσ Γ̄κνσ .(1.71)q̈ µ + Γ̄κλ µ q̇ κ q̇ λ = 0.(1.72)Then (1.69) can be written asSince the solutions of this equation minimize the length of a curve in spacetime,they are called geodesics.1.3Quantum MechanicsHistorically, the extension of classical mechanics to quantum mechanics becamenecessary in order to understand the stability of atomic orbits and the discretenature of atomic spectra.
It soon became clear that these phenomena reflect thefact that at a sufficiently short length scale, small material particles such as electronsbehave like waves, called material waves. The fact that waves cannot be squeezedinto an arbitrarily small volume without increasing indefinitely their frequency andthus their energy, prevents the collapse of the electrons into the nucleus, whichwould take place in classical mechanics. The discreteness of the atomic states of anelectron are a manifestation of standing material waves in the atomic potential well,by analogy with the standing waves of electromagnetism in a cavity.3In many textbooks, for instance S.
Weinberg, Gravitation and Cosmology, Wiley, New York,1972, the upper index and the third index in (1.70) stand at the first position. Our notation follows J.A. Schouten, Ricci Calculus, Springer, Berlin, 1954. It will allow for a closeranalogy with gauge fields in the construction of the Riemann tensor as a covariant curl ofthe Christoffel symbol in Chapter 10. See H.
Kleinert, Gauge Fields in Condensed Matter ,Vol. II Stresses and Defects, World Scientific Publishing Co., Singapore 1989, pp. 744-1443(http://www.physik.fu-berlin.de/~kleinert/b2).121.3.11 FundamentalsBragg Reflections and InterferenceThe most direct manifestation of the wave nature of small particles is seen in diffraction experiments on periodic structures, for example of electrons diffracted by a crystal. If an electron beam of fixed momentum p passes through a crystal, it emergesalong sharply peaked angles.
These are the well-known Bragg reflections. Theylook very similar to the interference patterns of electromagnetic waves. In fact, itis possible to use the same mathematical framework to explain these patterns as inelectromagnetism. A free particle moving with momentump = (p1 , p2 , . . . , pD ).(1.73)through a D-dimensional Euclidean space spanned by the Cartesian coordinate vectorsx = (x1 , x2 , . . . , xD )(1.74)is associated with a plane wave, whose field strength or wave function has the formΨp (x, t) = eikx−iωt ,(1.75)where k is the wave vector pointing into the direction of p and ω is the wave frequency.
Each scattering center, say at x0 , becomes a source of a spherical wavewith the spatial behavior eikR /R (with R ≡ |x − x0 | and k ≡ |k|) and the wavelength λ = 2π/k. At the detector, all field strengths have to be added to the totalfield strength Ψ(x, t). The absolute square of the total field strength, |Ψ(x, t)|2, isproportional to the number of electrons arriving at the detector.The standard experiment where these rules can most simply be applied consistsof an electron beam impinging vertically upon a flat screen with two parallel slitsa distance d apart.
Behind these, one observes the number of particles arriving perunit time (see Fig. 1.1)dNdt211∝ eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ) eikxFigure 1.1 Probability distribution of particle behind double slit, being proportional tothe absolute square of the sum of the two complex field strengths.H. Kleinert, PATH INTEGRALS131.3 Quantum Mechanics2 111dN∝ |Ψ1 + Ψ2 |2 ≈ eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ) 2 ,(1.76)dtRwhere ϕ is the angle of deflection from the normal.Conventionally, the wave function Ψ(x, t) is normalized to describe a single particle. Its absolute square gives directly the probability density of the particle at theplace x in space, i.e., d3 x |Ψ(x, t)|2 is the probability of finding the particle in thevolume element d3 x around x.1.3.2Matter WavesFrom the experimentally observed relation between the momentum and the size ofthe angular deflection ϕ of the diffracted beam of the particles, one deduces therelation between momentum and wave vectorp = h̄k,(1.77)where h̄ is the universal Planck constant whose dimension is equal to that of anaction,hh̄ ≡= 1.0545919(80) × 10−27 erg sec(1.78)2π(the number in parentheses indicating the experimental uncertainty of the last twodigits before it).
A similar relation holds between the energy and the frequency ofthe wave Ψ(x, t). It may be determined by an absorption process in which a lightwave hits an electron (for example, by kicking it out of the surface of a metal, thewell-known photoeffect). From the threshold property of the photoeffect one learnsthat an electromagnetic wave oscillating in time as e−iωt can transfer to the electronthe energyE = h̄ω,(1.79)where the proportionality constant h̄ is the same as in (1.77). The reason for thislies in the properties of electromagnetic waves. On the one hand, their frequency ωand the wave vector k satisfy the relation ω/c = |k|, where c is the light velocitydefined to be c ≡ 299 792.458km/s. On the other hand, energy and momentumare related by E/c = |p|.
Thus, the quanta of electromagnetic waves, the photons,certainly satisfy (1.77) and the constant h̄ must be the same as in Eq. (1.79).With matter waves and photons sharing the same relations (1.77), it is suggestiveto postulate also the relation (1.79) between energy and frequency to be universal forthe waves of all particles, massive and massless ones. All free particle of momentump are described by a plane wave of wavelength λ = 2π/|k| = 2πh̄/|p|, with theexplicit formΨp (x, t) = N ei(px−Ep t)/h̄ ,(1.80)where N is some normalization constant.
In a finite volume, the wave functionis normalized to unity. In an infinite volume, this normalization makes the wavefunction vanish. To avoid this, the current density of the particle probabilityj(x, t) ≡ −i↔h̄ ∗ψ (x, t) ∇ ψ(x, t)2m(1.81)141 Fundamentals↔is normalized in some convenient way, where ∇ is a short notation for the differencebetween right- and left-derivatives↔→←ψ ∗ (x, t) ∇ ψ(x, t) ≡ ψ ∗ (x, t) ∇ ψ(x, t) − ψ ∗ (x, t) ∇ ψ(x, t)≡ ψ ∗ (x, t)∇ψ(x, t) − [∇ψ ∗ (x, t)] ψ(x, t).(1.82)The energy Ep depends on the momentum of the particle in the classical way,i.e., for nonrelativistic material particles of mass M it is Ep = p2 /2M, for relativisticqones Ep = c p2 + M 2 c2 , and for massless particles such as photons Ep = c|p|.
Thecommon relation Ep = h̄ω for photons and matter waves is necessary to guaranteeconservation of energy in quantum mechanics.In general, momentum and energy of a particle are not defined as well as in theplane-wave function (1.80). Usually, a particle wave is some superposition of planewaves (1.80)Zd3 pi(px−Ep t)/h̄.(1.83)Ψ(x, t) =3 f (p)e(2πh̄)By the Fourier inversion theorem, f (p) can be calculated via the integralf (p) =Zd3 x e−ipx/h̄ Ψ(x, 0).(1.84)With an appropriate choice of f (p) it is possible to prepare Ψ(x, t) in any desiredform at some initial time, say at t = 0.
For example, Ψ(x, 0) may be a functionsharply centered around a space point x̄. Then f (p) is approximately a pure phasef (p) ∼ e−ipx̄/h̄ , and the wave contains all momenta with equal probability. Conversely, if the particle amplitude is spread out in space, its momentum distributionis confined to a small region. The limiting f (p) is concentrated at a specific momentum p̄. The particle is found at each point in space with equal probability, withthe amplitude oscillating like Ψ(x, t) ∼ ei(p̄x−Ep̄ t)/h̄ .In general, the width of Ψ(x, 0) in space and of f (p) in momentum space areinversely proportional to each other:∆x ∆p ∼ h̄.(1.85)This is the content of Heisenberg’s principle of uncertainty.
If the wave is localizedin a finite region of space while having at the same time a fairly well-defined averagemomentum p̄, it is called a wave packet. The maximum in the associated probabilitydensity can be shown from (1.83) to move with a velocityv̄ = ∂Ep̄ /∂ p̄.(1.86)This coincides with the velocity of a classical particle of momentum p̄.H. Kleinert, PATH INTEGRALS151.3 Quantum Mechanics1.3.3Schrödinger EquationSuppose now that the particle is nonrelativistic and has a mass M.
The classicalHamiltonian, and thus the energy Ep , are given byH(p) = Ep =p2.2M(1.87)We may therefore derive the following identity for the wave field Ψp (x, t):Zhid3 pf(p)H(p)−Eei(px−Ep t)/h̄ = 0.p3(2πh̄)(1.88)The arguments inside the brackets can be removed from the integral by observingthat p and Ep inside the integral are equivalent to the differential operatorsp̂ = −ih̄∇,Ê = ih̄∂t(1.89)outside. Then, Eq. (1.88) may be written as the differential equation[H(−ih̄∇) − ih̄∂t )]Ψ(x, t) = 0.(1.90)This is the Schrödinger equation for the wave function of a free particle.
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