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Canonical Quantisation

K. Kong Wan

Published Aug 27, 2020• 0 citations
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Abstract

Q27(1) Show that Hamilton’s equations of motion in Eq. (27.8) for a classical harmonic oscillator whose Hamiltonian is given by Eq. (27.11) are equivalent to Newton’s equation of motion.
DOI
10.1201/9780429296475-27

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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=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[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. 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.
In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.