We present an operator derivation of the continuity equation in quantum mechanics. We first identify the (local) divergence of the current density operator, then show how it immediately implies the continuity equation. We first consider the case of a free particle, then extend the treatment to the case of applied vector and scalar potentials.
We present an operator derivation of the continuity equation in quantum mechanics. We first identify the (local) divergence of the current density operator, then show how it immediately implies the continuity equation. We first consider the case of a free particle, then extend the treatment to the case of applied vector and scalar potentials.
In (1), it is argued that the Schrodinger (nonrelativistic) quantum equation may be developed from steady state type classical equations linked to the continuity equation and Newton’s second law. Given the nonrelativistic treatment of (1), F=dp/dt = d/dt (mv) = m dv/dt partial + mv dv/dx. In (1) an equation (III.2) for many particles is obtained from the equation for a single particle: Md (dv/dt partial + v dot grad v) = dF + hbarhbar/4m grad dot (grad grad d - 1/d (grad d) (grad d) ). In (1), it is further assumed that v = grad (phi) for conservative forces, where phi is a function of x,t. One may then see that for d=1 case, for a single particle, that phi= -Et+px and that one may write a Schrodinger equation which is essentially a conservation of energy equation. Here, we consider starting with dp/dt=Force and using a special relativistic treatment. We note that for a conservative force, F = -grad V and suggest trying to write the LHS in terms of a gradient for a free particle so the gradient may be “pulled out” from both sides of dp/dt=F. From a physical point of view, F is a pull or push, but we argue that V (potential) is also physical and that given that p is almost like a force, that there should be meaning to p= grad phi(x,t). This is the basis premise. In the case, F=0, we show that one may obtain -EE + p dot p = constant via this approach which involves writing p = d/dx (-Et+px) (one dimension). We note that associating dp/dt = F with a Lorentz invariant -Et+px shows the close association of Newton’s second law with special relativity because dp/dt=Force is relativistic as long as p=mov/sqrt(1-vv/cc). We suggest that this approach associates operators id/dt partial and -i grad partial (hbar=1) with E and p and allows one to write a free particle equation (Klein-Gordon) in terms of exp(-iEt+ipx), which is a quantum form. We suggest that one way in which to see free particle quantum mechanics emerge (albeit in the simplest case-that of a free relativistic particle) is to try to consider dp/dt in terms of a gradient suggesting that there may be x, t features even for E,p constant. In Newtonian mechanics, a constant E and p immediately implies that derivatives are 0. This need not be the case if one considers p = d/dx (-Et+px). The legitimacy of considering p (which is closely linked to a force) is linked to the legitimacy of considering F = -grad V. In other words, V is a potential which is another way of considering the features of a force and p = d/dx (-Et+px) may mean that -Et+px (and exp(-iEt+ipx) may be another way of considering p,E at a deeper level than simply stating that one has a constant E,p for a free particle. We argue that one does not need to consider a continuity equation as done in (1) and that the full approach follows from analysis of a single free particle (constant E,p) in the special relativistic case, which then yields also the nonrelativistic one.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The 1 -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.
Local properties of a quantum system are defined as the expectation values of its observables in a microstate of some complete set of commuting observables. An equation for the time evolution of local properties is obtained for any system whose statistical operator (density matrix) evolves by unitary transformation in accordance with the von Neumann equation. The formalism is applied to the example of a system of one particle. In this case the local properties are fields, the time-evolution equation is an equation of continuity with source terms. For constants of motion the source terms vanish, giving equations of continuity for the fields. For each scalar field a flux vector for its transport current is defined. For momentum, a stress tensor is obtained. The effects on local properties of realization of a latent ensemble of the statistical operator (an entropy-increasing mechanism recently proposed to explain approach to equilibrium) are also considered; a non-negative local entropy production is identified, as well as a discontinuous redistribution of local properties among microstates.