Let us explain first,
what is canonical quantization, in general.
Any system of which we have
a classical description can be quantized
following a process known us canonical quantization,
which was elaborated by Dirac in his thesis in 1925.
The idea is the following: to give a description of a classical system,
we use a set of dynamic variables whose value is at time
t allow us to calculate at that time any quantity of the system,
such as its energy, linear momentum etc.
Moreover, these dynamic variables obey first order differential equations of time
so that their value at any short time t note
allows us to derive the future evolution after t note.
Now, among all the possible sets of dynamic variables some will play a particular role.
They can by pairs of canonically conjugate variables, q_j,
p_j, that are called respectively the coordinate and its canonically conjugate momentum.
The process of canonical quantization consists of
replacing each pair of a coordinate with
its conjugate momentum by a pair of operators q_j_^ and p_j_^ which do not commute.
The commutator is equal to iħ. As you roll, ħ = h/(2π).
If now, we can see that two operators
associated with different parts of canonically conjugate variables, they commute.
Their commutator is equal to zero.
This is indicated by the second equation where the canonical symbol,
delta j_k is null if j is different from k and takes the value
1 if j equals k. As soon as we
have replaced all pairs of
canonically conjugate variables by operators whose commutator equal iħ,
we can describe the behavior of
the quantized system using
the quantized Hamiltonian associated with classical Hamiltonian.
In classical physics the Hamiltonian of
a system is its energy expressed as a function of canonically conjugate variables.
When you replace a classical conjugate variables by
the corresponding quantum operators you obtain the quantum Hamiltonian of the system,
which is its quantum expression of the energy.
You can now elaborate the quantum description of
your system since you remember frame of course in quantum mechanics,
that once you know the Hamiltonian of a system you can obtain all its quantum properties.
You know, for instance,
that the quantized energies E_n of the system are
the eigenvalues of the Hamiltonian and that the corresponding eigenstates phi_n,
constitute a convenient basis of the space of the states of the system.
This equation is sometimes called the time independent Schrodinger Equation.
You also know that the time evolution of
the system is given by your first order differential equation,
also based on the quantized Hamiltonian iħ_d/dth at applied to the psi,
that describes a state of the system at time t. This is a Schrodinger equation.
It allows you to answer any question provided that you can solve it.
This is another story, not a simple one,
but let us come back to our canonical quantization.
At this point a question immediately arises.
How can we recognize parts of canonical and conjugate variables in a classical system?
There is a general procedure for answering that question,
starting from a quantity named the Lagrangian of the system.
Here we will use a simpler and more pragmatic approach based on
the classical Hamilton equations which describe the dynamics of classical systems.
The Hamilton equations are based on the classical Hamiltonian expression of the energy.
They link the evolution of the two canonically conjugate variables of the same pair.
They express the first order time derivative of one variable as
a function of a partial derivative of the Hamiltonian relative to the other variable.
They are almost the same except for the sign.
I must admit that this is quite abstract especially if you see it for the first time.
So let us take a simple example to find how it works.
Let us take the example of a material particle of mass m evolving in a potential U(x).
We restrict ourselves to a one dimensional problem.
Let us try the hypothesis that the canonical conjugate variable are
the position x and the momentum p equal m_dx/dt.
The expression of the energy is then E of x and p
equals the sum of the potential energy plus the kinetic energy.
And we can take it as Hamiltonian.
We can then write the Hamilton equations.
The first one gives p/m equals dx/dt. The second yields dp/dt equals minus du/dx.
That is a false.
We recognize Newton equations of motion.
That is to say the correct dynamics of the system and we can safely
conclude that x and p are canonically conjugate variables.
We thus quantize by taking operators x_^ and p_^ that do not commute and write
the quantum Hamiltonian as
a sum of two terms associated with the potential and the kinetic energy.
We can now write the Schrodinger equation and
if we take the form relevant to the language of
wave functions we obtain the standard Schrodinger equation that you have already used,
I am sure, to study the behavior of a particle in
a potential wave or even to calculate the energy levels of the hydrogen atom.
We could have made another conjecture, for instance,
that the canonically conjugate variables are position and velocity x and v call dx/dt.
Would it work? To test that hypothesis we now
assume that the Hamiltonian has the following form.
Here we consider V as a canonically conjugate of x.
If now we write the Hamiltonian equation for that hypothetic Hamiltonian we
do not recover Newton equation as you
can check yourself calculating the partial derivatives.
The dynamic variable x and V are not canonically conjugate variables.
You may feel unsatisfied by
these empiric way of recognizing canonically conjugate variables and regret
that I do not teach you the more general method starting from Lagrangian.
In my opinion the latter is not more rigorous since you must first guess
a specific form for the Lagrangian and then you must
verify that this expression leads to known dynamics of the system,
for instance, Newton equation in the previous case.
So the reasoning is basically the same as the one we do with the Hamiltonian formalism.
I do not mean to imply that there is no interest in using
the Lagrangian formalism but for this course it is enough to start
from the Hamiltonian formalism where we guess which of
the canonical conjugate variables and
verify that they are indeed the canonically conjugate variables.
We thus use the following criterion.
If the Hamilton equation associated to the energy of
the system yield the known dynamics of the system we
can conclude that the energy was expressed as a function of
canonically conjugate variables and proceed with quantization.
We will apply this criterion to the electromagnetic field to recognize
dynamic variables that will become operators obeying the canonical commutation relations.
Before effecting the quantization of
the electromagnetic field I want first to show you how
to do it in the particular case of a material harmonic oscillator.
The reason is that the electromagnetic field
behaves as a set of harmonic oscillators and we will learn soon.
So if you want to learn quantum orbits you better know
the formalism of the quantum harmonic oscillator in the form developed by Dirac.
You have already learned it in your course of
quantum mechanics but you must in a way watch carefully the next sequence in
order to refresh your memory and check that you are
comfortable with a formalism that will be used all along that course.
Before starting that new sequence you may want to search the following quiz.
If you think that understanding canonical quantization is not your priority,
you can skip the quiz.
Alain AspectProfessor
Michel BruneProfessor
