For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
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(
The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.