### Abstract

Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x = x(τ) and t = t(τ) instead of x = x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃ = p_{t}+H, where H is the usual Hamiltonian and p_{t} is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property Ĥψ= 0, which is the time-dependent Schrödinger equation, ih∂_{t}ψ=Ĥψ. We discuss the quantum mechanics of a relativistic particle as an example.

Original language | English |
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Pages (from-to) | 882-885 |

Number of pages | 4 |

Journal | American Journal of Physics |

Volume | 79 |

Issue number | 8 |

DOIs | |

Publication status | Published - 26 Jul 2011 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*American Journal of Physics*,

*79*(8), 882-885. https://doi.org/10.1119/1.3593270

**Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation.** / Deriglazov, A.; Rizzuti, B. F.

Research output: Contribution to journal › Article

*American Journal of Physics*, vol. 79, no. 8, pp. 882-885. https://doi.org/10.1119/1.3593270

}

TY - JOUR

T1 - Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation

AU - Deriglazov, A.

AU - Rizzuti, B. F.

PY - 2011/7/26

Y1 - 2011/7/26

N2 - Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x = x(τ) and t = t(τ) instead of x = x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃ = pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property Ĥψ= 0, which is the time-dependent Schrödinger equation, ih∂tψ=Ĥψ. We discuss the quantum mechanics of a relativistic particle as an example.

AB - Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x = x(τ) and t = t(τ) instead of x = x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃ = pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property Ĥψ= 0, which is the time-dependent Schrödinger equation, ih∂tψ=Ĥψ. We discuss the quantum mechanics of a relativistic particle as an example.

UR - http://www.scopus.com/inward/record.url?scp=79960823741&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960823741&partnerID=8YFLogxK

U2 - 10.1119/1.3593270

DO - 10.1119/1.3593270

M3 - Article

VL - 79

SP - 882

EP - 885

JO - American Journal of Physics

JF - American Journal of Physics

SN - 0002-9505

IS - 8

ER -