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The missing ingredient is precisely the equation (4.32) relating velocity and
momentum, or rather this equation rewritten so that it relates velocity to the
momentum derivative of the energy. Differentiating H with respect to p yields
dH dK d p2 p
= = = .
dp dp dp2m m
But this is precisely the velocity v =dx/dt, so we get
dH dx
= . (4.35)
dp dt
Thesetwoequations,(4.34)and(4.35),arethefundamentaldynamicalequa-
tions of classical mechanics. This reformulation of Newtonian mechanics was
performedduringthe18th and19th centuriesbyEuler,Lagrange,Hamiltonand
Poisson. It is quite general and it is the formulation of classical mechanics in
which the translation to quantum mechanics is most easily performed. In the
general theory, where there might be more than one particle, one considers a
dynamical system described by a set of dynamical variables (x ,p ) .6 The
i i
{ }
energy, or Hamiltonian is function of these variables. Then the Hamiltonian
equations of motion become
dx ∂H
i
= (4.36)
dt ∂p
i
dp ∂H
i
= . (4.37)
dt −∂x
i
6Thesevariablescanbemoregeneralthanordinarypositionandmomentum,includingfor
exampleangularvariables.
79
Note how beautiful and symmetrical these equations are! However, they
can still be rewritten in an even more compact way in which the transition to
quantum mechanics is most easily performed. In order to do that, consider a
dynamical system with coordinates and momenta (x ,p ) where the indices
i i
{ }
ranges from 1 to n, n being the number of particles of the system. Let A and
B be two dynamical variables, both differentiable functions of the xs and ps.
′ ′
The Poisson bracket is defined by
n
∂A ∂B ∂B ∂A
A,B = . (4.38)
{ } X1 (cid:16)∂x i∂p i − ∂x i∂p i(cid:17)
As an example, calculate the brackets x,H p,H for a one-dimensional
{ } { }
system, i.e n=1.
∂x∂H ∂H ∂x ∂H
x,H = =
{ } ∂x ∂p − ∂x ∂p ∂p
∂p∂H ∂H ∂p ∂H
p,H = =
{ } ∂x ∂p − ∂x ∂p −∂x
Combiningtheseequationswith(4.36),(4.37)andgeneralizingtonparticles,
the Hamiltonian equations of motion can be written compactly
dx
i
= x ,H (4.39)
i
dt { }
dp
i
= p ,H . (4.40)
i
dt { }
In these equations,coordinates (positions) andmomenta appear completely
symmetrically.
Thereadermightfeelthatanedifice(ageneraltheoryofdynamics)hasbeen
erected on a very tiny base (the harmonic oscillator). That is indeed true. But
everything in this section can be developed quite rigorously from fundamental
principles of physics. A good accessible reference is [33]. As a last point, it
can be shown that the equation of motion for a function F of the dynamical
variables can be written
dF ∂F
= F,H + . (4.41)
dt { } ∂t
The second term ∂F/∂t, vanishes if there is no explicit time dependence for
F. This is often the case.
4.2.5 Quantization
Hamilton’s equations for classicaldynamics offers a very natural starting point
for quantization. The rules are
80
1. Replace classical dynamical variables A with the corresponding quantum
operators Aˆ
A Aˆ
−→
2. Replace the Poisson brackets , with commutators [, ]
{· ·} · ·
1
A,B [Aˆ,Bˆ].
{ }−→ i¯h
Upon these replacements,the equations of motion for the operatorsbecome
dFˆ 1
= [Fˆ,Hˆ]. (4.42)
dt i¯h
ThisformofquantumdynamicsbearsnoobviousrelationtotheSchr¨odinger