As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen’s collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles.
The trajectory representation renders microstates of the Schrödinger’s wave function for the bound state problem. Each microstate by the equation
ψ = (2m)1/4cos(W/h ̄)/(W′)1/2[a − c2/(4b)]1/2
(aφ2 + bθ2 + cφθ)1/2/[a − c2/(4b)]1/2 cos[arctan(b(θ/φ) + c/2)/(ab − c2/4)1/2 = φ
is sufficient by itself to determine the Schrödinger’s wave function. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that the Schrödinger’s wave function be an exhaustive description of non-relativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and its corresponding Schrödinger wave function with sub-barrier energy that tunnels through the barrier with certainty. Hence, tunneling with certainty is a counter example refuting Born’s postulate of the Copenhagen interpretation that attributes a probability amplitude to the Schrödinger’s wave function. As the trajectory representation is deterministic and does not need ψ, much less to assign a probability amplitude to it, the trajectory representation does not need a wave packet to describe or localize a particle. The equation of motion,
t − τ = ∂W/∂E, where t is the trajectory time, relative to its constant coordinate τ, and given as a function of x;
for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet. Normalization, as previously noted herein, is determined by the nonlinearity of the generalized Hamilton-Jacobi equation for the trajectory representation and for the Copenhagen interpretation by the probability of finding the particle in space being unity. Though probability is not needed for tunneling through a barrier, the trajectory interpretation for tunneling is still consistent with the Schrödinger representation without the Copenhagen interpretation. The incident wave with compound spatial modulation of amplitude and phase for the trajectory representation,
has only two spectral components which are the incident and reflected unmodulated waves of the Schrödinger representation.
Trajectories differ with Feynman’s path integrals in three ways. First, trajectories employ a quantum Hamilton’s characteristic function while a path integral is based upon a classical Hamilton’s characteristic function. Second, the quantum Hamilton’s characteristic function is determined uniquely by the initial values of the quantum stationary Hamilton-Jacobi equation, while path integrals are democratic summing over all possible classical paths to determine Feynman’s amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept.
Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties.
Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties. Anonymous referees of the Copenhagen school have had reservations concerning the representation of the incident modulated wave as represented by the equation
before the barrier. They have reported that compoundly modulated wave represented by the above equation is only a clever superposition of the incident and reflected unmodulated plane waves. They have concluded that synthesizing a running wave with compound spatial modulation from its spectral components is nonphysical because it would spontaneously split. By the superposition principle of linear differential equations, the spectral components may be used to synthesize a new pair of independent solutions with compound modulations running in opposite directions. Likewise, an unmodulated plane wave running in one direction can be synthesized from two waves with compound modulation running in the opposite directions for mappings under the superposition principle are reversible.