Probability current

It is a real vector that changes with space and time.The probability current is invariant under gauge transformation.The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in Brownian motion and the Fokker–Planck equation.In non-relativistic quantum mechanics, the probability current j of the wave function Ψ of a particle of mass m in one dimension is defined as[2]where Note that the probability current is proportional to a WronskianThis can be simplified in terms of the kinetic momentum operator,The above definition should be modified for a system in an external electromagnetic field.In SI units, a charged particle of mass m and electric charge q includes a term due to the interaction with the electromagnetic field;[3]If the particle has spin, it has a corresponding magnetic moment, so an extra term needs to be added incorporating the spin interaction with the electromagnetic field.According to Landau-Lifschitz's Course of Theoretical Physics the electric current density is in Gaussian units:[4]Hence the probability current (density) is in SI units:[citation needed] The neutron has zero charge but non-zero magnetic moment, soFor composite particles with a non-zero charge – like the proton which has spin quantum number s=1/2 and μS= 2.7927·μN or the deuteron (H-2 nucleus) which has s=1 and μS=0.8574·μN [5] – it is mathematically possible but doubtful.The wave function can also be written in the complex exponential (polar) form:where R, S are real functions of r and t. Written this way, the probability density isFinally, combining and cancelling the constants, and replacing R2 with ρ,If we take the familiar formula for the mass flux in hydrodynamics:which is the same as equating ∇S with the classical momentum p = mv however, it does not represent a physical velocity or momentum at a point since simultaneous measurement of position and velocity violates uncertainty principle.in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function.The de Broglie-Bohm theory equates the velocity withwhere V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.and the integral equation can also be restated using the divergence theorem as:In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation, sans time derivative, is the probability of obtaining a value within V when the position of the particle is measured.The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the probability of the particle being measured in V is equal to the rate at which probability flows into V. By taking the limit of volume integral to include all regions of space, a well-behaved wavefunction that goes to zero at infinities in the surface integral term implies that the time derivative of total probability is zero ie.[7] This result is in agreement with the unitary nature of time evolution operators which preserve length of the vector by definition.where jinc, jref, jtrans are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors.In terms of a unit vector n normal to the barrier, these are equivalently:(that is, plane waves are stationary states) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed;illustrating that the particle may be in motion even if its spatial probability density has no explicit time dependence.For a particle in a box, in one spatial dimension and of length L, confined to the region
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