The Contradiction Between the Measurement Theory of Quantum Mechanics and the Theory that the Velocity of Any Particle Can Not be Larger than the Velocity of Light
By the measurement theory of quantum mechanics and the method of Fourier transform,we proved that the wave function psi(x,y,z,t)= (8/((2(pi)(2L(exp (1/2)))(exp 3))(Phi(L,t,x)Phi(L,t,y)Phi(L,t,z)). According to the theory that the velocity of any particle can not be larger than the velocity of light and the Born interpretation, when absolute value of delta greater than (ct+ L),Phi(L,t,delta) = 0. But according to the calculation, we proved that for some delta, even if absolute value of delta is greater than (ct+L), Phi(L,t,delta) is not equal to 0.
By the measurement theory of quantum mechanics and the method of Fourier transform,we proved that the wave function psi(x,y,z,t)= (8/((2(pi)(2L(exp (1/2)))(exp 3))(Phi(L,t,x)Phi(L,t,y)Phi(L,t,z)). According to the theory that the velocity of any particle can not be larger than the velocity of light and the Born interpretation, when absolute value of delta greater than (ct+ L),Phi(L,t,delta) = 0. But according to the calculation, we proved that for some delta, even if absolute value of delta is greater than (ct+L), Phi(L,t,delta) is not equal to 0.
The propagation of the wave function of a particle is characterised by a group and a phase velocity. The group velocity is associated with the particle's classical velocity, which is always smaller than the speed of light, and the phase velocity is associated with the propagation speed of the wave function phase and is treated as being unphysical, since its value is always greater than the speed of light. Here we show, using Sciama's Machian formulation of rest mass energy, that this physical interpretation, for the group and the phase velocity of the wave function, is only valid if the weak equivalence principle strictly holds for the propagating particle, except for the photon. In case this constraint is released the phase velocity of the wave function could acquire a physical meaning in quantum condensates.
An arbitrary function in the eigenfunction space of some quantum-mechanical Hamiltonian may be thought to represent the initial configuration ψ(q,0) of a nonstationary state. The system develops in time according to ψ(q,t)=exp(−itH)ψ(q,0)= ∑ k=0∞(−it)kk!Hkψ(q,0).Defining F(t)≡〈ψ(q,0), ψ(q,t) 〉, and hk≡〈ψ(q,0), Hkψ(q,0) 〉, and taking the Fourier transformG(ω)= ∫ −∞∞dtexp(−iωt)F(t),we obtain G(ω)=2π ∑ k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctions ψ(q,0)= ∑ n=0∞cnφn(q)+ ∫ 0∞dωc(ω)φ(ω,q),the Fourier transform represents G(ω)=2π ∑ n=0∞|cn|2δ(ω+ωn)+2π|c(−ω)|2,which exhibits, in principle, the entire eigenvalue spectrum. In this paper, a direct method of calculating eigenvalue spectra, based on the foregoing principle, is proposed. Two modifications are required for computational practicability: (i) use of a finite representation for the delta function and truncation of the summation (a); (ii) replacement of the integrals hk by hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to be GN(τ,ω,q′)=2π ∑ k=0Nhk(q′)k!χτ(k)(ω),with χτ(ω)≡sinτω/πω. The sequence GN(τ,ω,q′) is shown to converge as N→∞ if in the Expansion (b) the coefficients cn and c(ω) decrease with ω as exp(—ω/λ) or faster. Assuming convergence, the spectral function represents a broadened eigenvalue spectrum.
In recent notes by Einstein, Podolsky and Rosen and by Bohr, attention has been called to the fact that certain results of quantum mechanics are not to be reconciled with the assumption that a system has independently real properties as soon as it is free from mechanical interference. We here investigate in general, and in abstract terms, the extent of this disagreement. When suitably formulated, such an assumption gives to certain types of questions the same answers as does quantum mechanics; this is true of the formulas usually given in discussions of the theory of measurement. There exists, however, a general class of cases in which contradictions occur. That such contradictions are not restricted to the abstract mathematical theory, but can be realized in the commonest physical terms, is shown by the working out of an example..
We learn from Quantum Mechanics that the observation of the microscopic world, the measurement (M) of a quantum object, i.e. a particle, inexorably modifies the physical system we wish to examine. What happens is that with the M it takes place a reduction of the state vectors, that is the ‘wave function collapse’ of the measured particle. Why does it happen? No one knows. The enigma of the so-called Measurement Paradox, in our opinion, could be solved if we considered that the light quantum(LQ), as suggested by the Principle of Equivalence Mass-Energy, carries out a dynamic-mass equivalent to its energy. The LQ is indispensable to carry out a M. No M can be carried out without using the quantum of light. Calculus show that a photon of the optic band hits an electron with a momentum bigger than the mass of the electron itself. This may explain why the M induces the implosion of the quantum object observed, together with the collapse of its wave function, giving rise to the Measurement Paradox.
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is (18.1) Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read ( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity ( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5) ( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = + B A SS 21 Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants 21 and SS 1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to ( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6) Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.
Abstract Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes <?CDATA ψ(k,t)=ψ(k)exp(−iωkt)?> ψ ( k , t ) = ψ ( k ) exp ( − i ω k t ) yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t) = (ℏ/2i)[ψ +( x , t)∂ μ ψ( x , t) − ∂ μ ψ +( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities <?CDATA H(x,t)=cP0(x,t)?> ( x , t ) = c 0 ( x , t ) and <?CDATA P(x,t)?> ( x , t ) differ from ℘ μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density <?CDATA H(x,t)?> ( x , t ) , the canonical charge density ϱ( x , t), the energy pseudo-density <?CDATA H~(x,t)=c℘0(x,t)?> ˜ ( x , t ) = c ℘ 0 ( x , t ) , and the Born density ∣ψ(