The linear potential propagator via wave function expansion

Nassar, Antônio B.; Bassalo, José Maria Filardo; Alencar, Paulo de Tarso Santos; Lopes, José Luiz Magalhães; Oliveira, José I. F. de; Cattani, M.

doi:10.1590/s0103-97332002000400024

Cited by 4 publications

(4 citation statements)

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“…where the coefficient out front is set by the same delta function boundary condition. Again, this matches the known propagator [10,11] up to a phase factor that is constant over all space and the result is achieved in a very simple fashion, since X (t ) is easily solvable for a constant force.…”

Section: The Constant Force Propagatorsupporting

confidence: 62%

Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i

Palenik¹

2014

Eur. J. Phys.

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Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics.Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F = dP dt , P = m dV dt , and [X, P ] = i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. We then show that for a general force F(X), by Taylor expanding X(t) in time, we can use this methodology to reproduce the Feynman path integral formula for the propagator. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

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Section: The Constant Force Propagatorsupporting

confidence: 62%

Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i

Palenik¹

2014

Eur. J. Phys.

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“…It is remarkable that from this the de Broglie-Bohm formulation it is possible to get the propagator with a constant mass by developing the quantum action, the exterior potential and the Bohm potential around classical path. Thus, let us take quickly what has been demonstrated in [9,10]. We suppose that…”

Section: Propagator In the De Broglie-bohm Theorymentioning

confidence: 99%

“…Then, replacing these changes in (3), a direct calculation gives equation (10). Now, let us see what becomes of the evolution equation ( 4).…”

Section: Space-time Transformation In the De Broglie-bohm Theorymentioning

confidence: 99%

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The Kanai–Caldirola propagator in the de Broglie–Bohm theory

et al. 2007

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A calculation for the Kanai–Caldirola propagator in the de Broglie–Bohm theory is proposed. The technique of space-time transformations used in the Feynman path integral is adapted and then the problem is converted to that of the harmonic oscillator. The quantum propagator is viewed as an expansion of the guiding wave function over the velocity space. The construction is re-examined by replacing the integration over the initial velocity of the classic path by its final extremity. This gives a certain affinity with the Feynman path integral approach.

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