Heisenberg's Quantum Mechanics (English)

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1-4: Heisenberg's Way of Thinking

At almost the same time

as the above-mentioned

Schroedinger's wave mechanics

was proposed,

W. K. Heisenberg

(Germany, 1901 - 76)

published a new theory

which has later been called

the matrix mechanics.

Let us consider

a particle of mass

whose momentum

and coordinate (position)

are represented

and

q ,

respectively.

In the classical mechanics,

the coordinate

and the velocity

or the momentum

p = mv

are determined

as a function of time

by Newtonian equation of motion.

In other words,

the particle moves

along an orbit

described by a function

q (t ),

and

the momentum

p (t ),

is definite

at any point

on the orbit.

Accordingly, the orbit

and the state of motion

of a particle are

completely described

by the momentum

and the coordinate

in the classical mechanics.

[Heisenberg's Matrix Mechanics]

The above-mentioned variables

and

in classical mechanics

are ordinary variables.

When they are multiplied

by each other,

the order of multiplication

can be exchanged;

i.e.,

However,

in Heisenberg's theory,

the result of

multiplication changes

depending on its order.

This means that

the physical quantities

in Heisenberg's theory

are not ordinary variables;

they are matrices,

well-known

in mathematics.

Let us denote these

matrix-variables

by boldface letters as

q .

A matrix consists

of many elements as

As with ordinary variables

the four basic operations

of arithmetic

(addition, subtraction,

multiplication and division)

can be defined for matrices.

However, the results

of multiplication of

and

differ depending on the order as

Heisenberg assumed their difference to be

where

is the imaginary

number unit and

is a unit matrix.

Heisenberg replaced

the ordinary variables

in classical mechanics

with the matrix variables

mentioned above

and assumed

the condition (4).

Thereby, he obtained

the correct values of frequencies

and strengths

of the spectra of hydrogen.

These results are

exactly the same

as those obtained

by Schroedinger's wave mechanics.

This was surprising

because it appeared

that there might be

nothing common

between these two theories,

the matrix mechanics

and the wave mechanics.

Schroedinger has

however shown that

these two theories are

equivalent to each other

and one can be derived

from the other (1926).

Nowadays, these Heisenberg's

matrix mechanics

and Schroedinger's wave mechanics

are generically named

quantum mechanics.

[Stop Drawing the Orbit -- Abandon Common Sense !]

Looking back

at what we have

studied so far,

we are somewhat skeptical

whether the concept

of the orbit

of a particle

may be valid

or acceptable

in the microscopic world.

In our common sense,

a particle is considered to be

a point,

and the motion

of the particle

is represented

by the position

(coordinate) and

the velocity

(or momentum).

Namely, we assume that

the moving point particle

draws an orbit,

and the momentum

of that point particle

is specified at each point

or position on the orbit.

This is the common sense

in the macroscopic world.

Whereas, in the

Schroedinger wave mechanics,

a particle

does not only move

as a particle

but accompanies a wave.

In the Heisenberg

matrix mechanics,

the variables of position

and momentum

are no longer ordinary variables

but matrices.

Consequently, it seems that

position

and momentum might not

necessarily

be variables of unique values

as our common sense suggests.

Quantum mechanics seems to say,

"Abandon the common sense !"

Is it allowed

in the microscopic world

to abandon

the common understandings which

have long been accepted

since ancient time?

[Heisenberg's Uncertainty Principle]

As seen above,

Heisenberg asserted that

"the common sense

might be abandoned"

(1927).

The basis is the following

uncertainty principle.

In order to explain this,

let us borrow

an interesting illustration

used by

G. Gamow

(Russia, USA, 1904 - 68).

(See the following figure.)

This pleasant illustration

was represented

in Gamow's enlightening book,

"Thirty Years That Shock Physics"

(Anchor/Doubleday,

New York, 1966),

to easily explain the uncertainty

relation.

As for Gamow,

we will see him

in the next section.

Now let us consider

the thought (Gedanken)

experiment

shown in the figure below.

("Gedanken" is a German word.)

Gamow's Gedanken experiment to explain the uncertainty principle

In a completely vacuous room,

an electron ejected

horizontally from a gun

is going down

vertically, being acted on by

gravity.

When we detect the trajectory

by shining illuminating light

on the electron,

what happens?

Let us consider

the Gedanken

experiment

in the above figure.

In a completely vacuous room,

an electron ejected

horizontally from a gun

is going down vertically,

being acted on by

gravity.

First, let us analyze

this situation

with the classical theory.

According to Newtonian mechanics,

the electron must

fall down

along a

parabolic orbit .

To detect this orbit,

we turn on the lamp sometimes

and illuminate

the electron.

Since light applies

pressure to the electron

and distorts the orbit,

we should weaken

the strength of the

light very much

to minimize its impact.

It is possible

to weaken the strength

without bound,

because there is no lower

limit of the strength

of light

in the classical theory.

As a consequence,

the orbit measured in this way

would approach

a parabola ultimately.

This is just our common sense

that must be proved

by the measurement.

However, if we take

into account that

illuminating light

is emitted as photons

the above discussion

would be incorrect.

Let the frequency

and the wavelength

of the light

and , respectively.

The light collides

with the electron

as a particle with a given energy

and momentum

The electron is

given a momentum

at most.

As a result,

the orbit of the electron

is distorted.

In order to reduce

this distortion,

we have to make

the momentum transfer

as small as possible,

i.e., to make the wavelength

extremely large.

But we cannot identify

the position of the electron,

if the wavelength

is larger than the size of

the telescope

being used for observation.

Namely, the uncertainty

(or error) of the

position

of the electron

is at least about

the wavelength of the light,

Accordingly,

the uncertainty

(or error) of the

momentum of

the electron becomes

Consequently,

we have a relation

This relation

(6)

is called

Heisenberg's uncertainty principle.

This principle means that,

if we intend to reduce

the uncertainty of position,

then the uncertainty of momentum

becomes large,

and vice versa.

In other words,

it is impossible

to simultaneously specify

precisely the values of

both the coordinate position

and corresponding momentum

of a particle

in an experiment

because of the particle-wave

duality of light.

We referred to uncertainty

(or error)

in the above discussion,

but this does not imply

the measurement errors

like misreading of apparatuses

or errors caused

from poor experimental

settings.

Even if the apparatuses

have no defect

and the experiment is carried

out without any error,

we would have essential errors

(or uncertainties)

due to the particle-wave

duality of light.

[The Assertion of Heisenberg and Bohr]

After doing various

hypothetical (Gedanken)

experiments,

Heisenberg and Bohr

came to the conclusion

that

no experiment can

specify the values

of the coordinate position

and momentum of a particle

more precisely

than the limits given

by the uncertainty principle

(6).

Hence, they asserted that

we may abandon the classical common sense

that the coordinate

and the momentum of a particle

are ordinary variables

possessing precisely

specified values

and that

a particle moves

along a trajectory

drawn in a curved line.

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