dorsal/arxiv
View SchemaClassical trajectories compatible with quantum mechanics
| Authors | B. Roy Frieden, A. Plastino |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0006012 |
| URL | https://arxiv.org/abs/quant-ph/0006012 |
| DOI | 10.1016/S0375-9601(01)00481-9 |
Abstract
Consider any stationary Schroedinger wave equation (SWE) solution $psi (x)$ for a particle. The corresponding PDF on position QTR{em}{x} of the particle is QTR{em}{p}$_{X}(x)=|psi (x)|^{2}$. There is a classical trajectory QTR{em}{x(t)} for the particle that is consistent with this PDF. The trajectory is unique to within an additive constant corresponding to an initial condition QTR{em}{x(0).} However the value of QTR{em}{x(0)} cannot be known. As an example, a free particle in its ground state in a box of length QTR{em}{L} obeys a classical trajectory QTR{em}{x/L - (1/2}$pi)sin (2pi x/L)+t_{0}=t.$ The constant QTR{em}{t}$_{0}$ is an unknowable time displacement. Momentum values, however, cannot be determined by merely differentiating QTR{em}{d/dt} the trajectory QTR{em}{x(t)} and, instead, follow the usual quantification rules of Heisenberg's. This permits position and momentum to remain complementary variables. Our approach is fundamentally different from that of D. Bohm.
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"abstract": "Consider any stationary Schroedinger wave equation (SWE) solution $psi (x)$\nfor a particle. The corresponding PDF on position QTR{em}{x} of the particle is\nQTR{em}{p}$_{X}(x)=|psi (x)|^{2}$. There is a classical trajectory\nQTR{em}{x(t)} for the particle that is consistent with this PDF. The trajectory\nis unique to within an additive constant corresponding to an initial condition\nQTR{em}{x(0).} However the value of QTR{em}{x(0)} cannot be known. As an\nexample, a free particle in its ground state in a box of length QTR{em}{L}\nobeys a classical trajectory QTR{em}{x/L - (1/2}$pi)sin (2pi x/L)+t_{0}=t.$ The\nconstant QTR{em}{t}$_{0}$ is an unknowable time displacement. Momentum values,\nhowever, cannot be determined by merely differentiating QTR{em}{d/dt} the\ntrajectory QTR{em}{x(t)} and, instead, follow the usual quantification rules of\nHeisenberg\u0027s. This permits position and momentum to remain complementary\nvariables. Our approach is fundamentally different from that of D. Bohm.",
"arxiv_id": "quant-ph/0006012",
"authors": [
"B. Roy Frieden",
"A. Plastino"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/S0375-9601(01)00481-9",
"title": "Classical trajectories compatible with quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0006012"
},
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