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 a first order differential equation,

also based on the quantized Hamiltonian: iħ d(psi) over dt equals H hat applied to the ket 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 pairs of canonically 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 the 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 force.

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 hat and p hat

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 well 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 equal 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 are

the canonically 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 the quantization.

We will apply this criterion to the electromagnetic field to recognize the

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, as we will learn soon.

So if you want to learn quantum optics 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 anyway 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.