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 = 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.
| Original language | English |
|---|---|
| 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 |
Fingerprint
ASJC Scopus subject areas
- Physics and Astronomy(all)
Cite this
Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation. / Deriglazov, A.; Rizzuti, B. F.
In: American Journal of Physics, Vol. 79, No. 8, 26.07.2011, p. 882-885.Research output: Contribution to journal › Article
}
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
AN - SCOPUS:79960823741
VL - 79
SP - 882
EP - 885
JO - American Journal of Physics
JF - American Journal of Physics
SN - 0002-9505
IS - 8
ER -