Quantum mechanics of light (English)

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3-4: Quantum Mechanics of Light

The Schroedinger equation

learned so far

has been introduced

in order to describe

the wave nature

of the substance particles,

i.e.

the de Broglie wave

accompanying

the motion of the particles.

It has been clarified

that this

Schroedinger equation

can beautifully describe

the duality of the particle nature

and the wave nature.

On the other hand,

it has also been

emphasized that

light possesses

the wave-particle duality.

In the classical theory,

light was originally

an electromagnetic wave

which obeys Maxwellian equations.

It consists of

an oscillating electric field

and

a magnetic field

as shown in

Fig. (A).

How can this wave motion

get the particle nature?

Fig. (A): The wave nature of light

In the classical theory,

light consists of an oscillating electric field

and a magnetic field

B .

[The Wave Nature of Light]

In the classical theory, light consists of an oscillating electric field E and magnetic field B These two are perpendicular to each other, and have the same frequency and the same wave length Therefore the speed of light is The light speed in the vacuum is

[The Energy of Cavity Radiation]

According to Maxwell's

electromagnetism,

the energy U

of the electromagnetic waves

(or light or radiation)

in a cavity (vacuum)

is written

where the integration

should be over the whole cavity.

The electric field

and

the magnetic field

can be written

in the form of superpositions of plane waves.

Here, let the wave vector

of the plane wave be

and the wavelength be

Then there is a relation

For example,

the electric field

can be expanded as

where

the polarization vectors

and

are unit vectors

in the vertical

and horizontal directions,

respectively.

The magnetic field

is similar too.

Here it should

be noted that

the electric

and magnetic fields

are superpositions

of many types of waves with

various magnitudes

of wave numbers

(wavelengths).

Using this results,

we can write the energy

of the cavity radiation as

where

The equations (4)

are equivalent

to the equations of spring motion

(or harmonic oscillator)

with the mass 1.

Let

and

be the coordinate

and momentum of

the k th harmonic oscillator,

respectively.

The first equation of

Eqs. (4)

is the relation between

the velocity of the mass

and the momentum,

and

the second equation of

Eqs. (4)

is Newtonian equation of motion

for the harmonic oscillator.

Since it is somewhat tedious

to explain how to derive

these results

from Maxwellian electromagnetism,

we omit the details here.

You may be asked

to accept the following conclusions:

Looking at the two equations,

Eqs. (3) and (4),

we can say that

a cavity radiation is equivalent to a set of infinite number of harmonic oscillators.

[Quantization of Electromagnetic Wave (Light)]

As stated above,

we know that

light is a set of

infinite number of

harmonic oscillators.

As explained on the pages,

1-6: Tunnel Effect

3-1: Energy Eigenvalues, Eigenstates,

if we treat a harmonic oscillator

with quantum mechanics,

or if we quantize it,

then

the eigenvalues would be

discrete and

integral multiples

of a unit energy quantum,

which is given by

This is nothing else than

Planck's energy quanta.

Namely, the energy of the radiation in a cavity exists in the corpuscular (or particle-like) form.

Only if we deal

with the classical

electromagnetic waves

in the quantum mechanical way,

then the energy quanta

appear at once.

How splendid this is!

[Interaction between Light and Charged Particles]

Because light

(electromagnetic wave)

consists of

electric and magnetic fields,

when meeting such

a charged particle

as an electron,

it exerts a force

on the particle

and there occurs

an interaction between them.

In order to correctly treat

this interaction,

we have to combine

the Schroedinger equation

for the charged particle

and the quantum mechanics

of the electromagnetic waves.

The formulation

for this purpose is

the quantum electrodynamics.

Since its contents

are slightly advanced,

they are omitted here.

Using the theory

of the quantum electrodynamics,

we can completely solve

all the mysteries concerning

the interactions between

lights and substance particles,

e.g.

Einstein's theory

on the photoelectric effect

and Compton effect

discussed on the pages

of the previous Seminar,

3-6: The Hypothesis of Light Quanta and the Photoelectric Effect

and

3-7: Compton Effet.

Thus the duality

of the particle nature

and the

wave nature

has completely been

unified within

quantum mechanics

and

all the mysteries

in "the microscopic world"

have been solved perfectly.

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