Analogy between quantum and classical mechanics
--- syntax of mathematical language of quantum mechanics
In Section 21, Dirac defines the quantum Poisson bracket (PB),
which is the commutator ih[u, v] = uv - vu, and which is
the key to his symbolic version of quantum mechanics. The quantum PB is
supposed to be an analogy to classical PB, but I had had difficulty seeing the analogy,
because the quantum PB does not appear similar to the classical PB, except that in
classical PB, the two partial derivative operators d/dp and d/dq (here d means partial
derivative) also appear in the commutator form of ab-ba.
Dirac uses a general argument
to show that if [u, v] is to behave like classical PB, then we must have
[u1, v1](u2v2-v2u2) = (u1v1-v1u1)[u2, v2]. For classical variables, this is just
0 = 0. For quantum variables (matrices or operators) that do not commutate, [u, v] must be proportional to
uv - vu. This is a forceful argument, but still the formal analogy is not clear, in the
sense that the quantum and classical PBs do not appear in the same form.
Dirac's early papers solve this puzzle. In his 1925 paper "The Fundamental Equation of
Quantum Mechanics", he actually shows that for the simple periodical system, xy-yx indeed
corresponds to the classical PB, by corresponding differences and differentials for large
quantum numbers. This point is carefully explained in Heisenberg's book
"The Physical Principles of the Quantum Theory".
In his 1926 paper "Quantum Mechanics and a Preliminary
Investigation of the Hydrogen Atom", Dirac further points out that one can in general use the
differential formula of classical PB to calculate the quantum PB. So the
quantum PB and the classical PB can indeed be written in the same form.
The quantum PB and the classical PB actually share the same algebraic structure.
The formal analogy between quantum and classical mechanics was the motivation of
Heisenberg in his breakthrough paper. Heisenberg noticed that his tables (matrices) do not
commute, but he did not appreciate the importance of the commutator xy-yx. However, he
already obtained a preliminary matrix form of quantum condition by re-interpreting the old quantum
condition of Bohr-Sommerfeld, by corresponding differences and differentials
for large quantum numbers, a scheme that was later adopted by Dirac in finding the classical
correspondence of xy-yx.
The 1925 paper by Born and Jordan, which followed upon Heisenberg's paper and
appeared before Dirac 1925, worked out the analogy between quantum and classical mechanics in terms of the Hamilton equations
(instead of PB) and qp-pq = ih. The latter is just Dirac's quantum PB for p and q. It
should be called Born's quantum condition, because Born postulated it first.
It governs the quantum algebra, and leads
to Heisenberg's uncertainty principle. It is the root of quantum behavior, in that it leads to
a discrete set of eigen values of the Hamiltonian, as well as the creation and annihilation operators that
act on the discrete set of eigen states.
It is deeper and more mysterious than the
uncertainty principle, which is more concrete and intuitive.
In both Dirac 1925 and Born and Jordan 1925, quantum differentiation is defined.
This is needed if one wants to establish a formal analogy based on the Hamilton equations,
because one needs to define dH/dp and dH/dq (here d is partial derivative).
Dirac's 1925 paper shows that dx/dv = xa - ax = [x, a] for some a. I still cannot follow the whole argument
because it appears that one needs to assume that the linear coefficients alpha's do
not depend on x. The Born and Jordan's definition is much closer to regular
differentiation, and the differentiations like dH/dp and dH/dq are indeed in
the form of quantum PB. Quantum PB captures the abstract algebraic essence of
differentiation, and the regular differentiation is but a concrete manifestation of
this abstract algebra.
The correspondence between quantum and classical PBs for the simple periodic system and the
quantum differentiation do not appear in Dirac's book, perhaps because the former is just
a special case, and the latter, though quite general, is not needed in specifying the
fundamental laws.
Another related issue is that quantum PB should converge to classical PB as h goes
to 0. J. E. Moyal in the 1940s obtained an exact correspondence for finite h in his paper
"Quantum Mechanics as a Statistical Theory", where the sine of h/2 times the
classical PB appears in the transition probabilities on the phase space. Moyal
had a lengthy argument with Dirac over many letters.
Schroedinger obtained his equation also by analogy to classical physics, although
through a totally different path. He observed the analogy between the Hamilton's
least action principle in mechanics and Fermat's least time principle for geometric
optics, and established the correspondence between mechanical property in the former
and the speed of light ray in the latter. Since there is a wave theory for light,
Schrodinger then deduced a wave theory for mechanics. The Ehrenfest theorem connects
the Schrodinger equation to classical mechanics by taking expectation.
It is remarkable that by simply replacing p by -ih d/dx and E by ih d/dt in classical Hamiltonian, one gets
Schroedinger equation. In this aspect, it seems quantum (wave) mechanics is even
simpler than classical mechanics, because such a substitution scheme leads to the dynamics
directly, without the need for the Hamilton equations. In other words, this substitution
scheme encodes both the Born's quantum condition and the Hamilton equations.
The mathematical fact is that the energy-momentum state in the form of
the de Broglie waves are the eigen states
of the above differential operators, with classical momentum and energy as eigen values.
Penrose calls this fact a miracle in his book "The Road to Reality." As Veltman points out
in his diagrammatica book, the physical content is that a particle with well defined
momentum-energy is described by a de Broglie plane wave in space-time.
The Schroedinger equation tells us that one only needs to specify the Hamiltonian for the physical
system. The discrete eigen values of the Hamiltonian are the different engergies that the system
can take. Moreover, the Hamiltonian is the generator that drives the change of the state of the
system over time. Specifically, the energy is the time frequency of the change of phase of the state at
each energy level.
In Born's quantum condition qp - pq = ih (1), if q is diagonalized as in Schrodinger
representation, then p = -ih d/dq (2). Moreover, p and q form the Fourier pair,
i.e., for eigen kets |p> and |q>, < q|p> \propto e^{ipq/h} (3), so that for any ket
|X>, < p|X> and < q|X> are Fourier transforms of each other. See Section 22 of Dirac's book.
(1), (2) and (3) are
three manifestations of the quantum condition. (1) is fundamental, abstract and general.
(2) is convenient for calculation. (3) reveals the physical meaning of quantization.
(in this paragraph, in (1) p and q are operators, in (2) q is classical variable and
p is operator, and in (3), p and q are classical variables.)
The idea of canonical quantization is to promote the classical variables to operators, or more
intuitively, change the "nouns" to "verbs", while tying them by quantum condition or
Fourier transform. These "verbs" can then act on different things (eigen states) with
different consequences (eigen values). This immensely expands the expressive power of the
mathematical language without even altering the original sentences or increasing the number
of words in each sentence. It adds the notion of states, each of which is a superposition
of eigen states, whose coefficients (squared) in the linear superposition are the probabilities
of observing the corresponding eigen values. It leads to quantum phenomena in that the Hamiltanian
has a discrete set of eigen values, whose corresponding eigen states may be interpreted as
states of different energy levels or different numbers (kinds) of particles. The operators that
transform one eigen state to another are then said to have created or annihilated particles. It also
makes a lot of magical things possible that are originally inconceivable with scalars, such as
making the square root of (m^2 + p^2) linear in m and p, as in Dirac equation.
Meanwhile, these newly promoted "verbs" are then on the same footing as those "verbs" (differential
operators) in the original classical equations.
The general scheme of canonical quantization is to start from the Langrangian,
identify the canonically conjugate variables such as p and q, or phi and pi,
then obtain the Hamiltonian, and finally derive
the dynamics by quantum PB or commutator. Dirac's symbolic approach is most abstract
and fundamental. His transformation theory relates different concrete reprenstations such
as Heisenberg picture and Schrodinger picture. It is like seeing the same movement
from different viewpoints. Both the movement of the system and the movement of the
viewpoint are unitary transformations.
The correspondence between quantum and classical mechanics is most intuitive in
Feynman's path integral or sum over histories, where the probability amplitude of a path
is e^{iS/h}, where S is the action of the path.
The original idea already appeared in Dirac's 1933 paper. See also Section 32 of Dirac's book.
The motivation of path integral was again to seek a formal mathematical correspondence between
quantum and classical mechanics, this time through
the Langrangian formulation, instead of the previous Hamiltonian formulation.
Dirac gave an
explanation why the classical stationary path of the least action
emerges as h goes to 0, because those paths that are away from the stationary
path cancel each other out through destructive interference due to fast change in phase,
whereas those paths that are close to the stationary path reinforce each other through
constructive interference due to slow phase change because the dirivative at the stationary
path is 0. In fact, this is also the justification of Fermat's least time principle for geometric optics
from Huygens principle for the wave theory of light. So the connection between
geometric optics and wave optics, which was a useful analogy for Schrodinger's discovery of his
equation, is also a useful analogy for considering the connection between least action principle in
classical mechanics and path integral in quantum mechanics.
In path integral formulation, quantum mechanics is actually more natural than classical mechanics,
because superposition is more natural than minimization. The path integral formulation
also appears simpler than canonical quantization, because there is no need
to explicitly promote classical variables to non-commutating operators. The superposition
already takes care of it implicitly.
However, the physical content of path integral formulation can be less transparent than in the canonical
formation, because it deals with long time consequence directly rather than instaneous change. For instance,
the existence of discrete energy levels and the creation and annihilation of particles are not explicit
in path integral formulation. The discrete energy levels and states appear only in the spectral decomposition of
the transition kernel of the path integral.