Relation to classical theory (English)

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2-3: Relation to the Classical Theory

[Newtonian Equation of Motion]

Newtonian equation of motion

in a one-dimensional space

is written

where F is

a force exerted

on a particle of mass m,

and V ( x ) is

the potential energy.

The momentum of the particle, p,

is given by

So that the equation

of motion is rewritten

As discussed

previously,

the fundamental equation

of quantum mechanics,

the Schroedinger equation,

was promulgated

as the wave equation for

the de Broglie wave

accompanying

a substance particle motion.

Hence

the Schroedinger equation

must be closely

related to Newtonian

equation of motion.

What is the relation?

In order to study this,

let us start with the concept of

mean value

in quantum mechanics.

[The Mean Value of Position and Its Dispersion]

In quantum mechanics,

the position of a particle

is determined only probabilistically

(according to the probability).

Therefore, the result

would varies

whenever it is measured,

even if the measurement

is done under the same conditions.

The mean value

of these results, ,

is obtained by

carrying out the integration

where x

is the operator

denoting the position coordinate.

Of course, x is as well

a usual variable

representing the position

of the particle,

but it is also

considered as an operator

in quantum mechanics.

And

in Eq. (4)

the probability that the particle

will be found at x.

The wave function

should, of course, be

normalized.

Accordingly,

in the state

represented by the wave function , the mean value of the position of a particle, ,

is given by

Eq. (4).

There are many kinds

of physical quantities

like a position coordinate

x, or the

momentum of a particle

p, or the

energy of a particle

E, etc.

Let us represent these

physical quantities

by an operator

Thus, the mean value

of a physical quantity

can be obtained by integrating

over the whole space as

If we measure the position

of a particle many times

under the same condition,

the data would be

distributed around the mean value.

The dispersion

of the distribution, ,

must be given by

[The Operator of Momentum]

In quantum mechanics,

the position operator

is the usual variable x .

As explained on the page,

1-3: The Schroedinger Equation

the momentum operator p

is defined by the

differential operator,

i.e.,

In general, a physical quantity

in quantum mechanics

is expressed by an operator.

(We should think

that the usual variable

for the position

of a particle, x ,

is an operator as well.)

Hence, a product

of two different physical quantities

is not equal to the result

obtained by interchanging

the order of product

of these two quantities.

Thereby the usual common sense

in classical theory

is not valid

in quantum mechanics.

Accordingly, the mean value

of momentum, ,

in a state ,

is written

[Ehrenfest's Theorem]

Let us study the time-variation

of the mean value of momentum,

The derivative of

with time t

can be written

Namely, we can say that

"the time-variation of the mean value of the momentum of a particle is equal to the mean value of the force exerted on the particle".

This is Ehrenfest's theorem.

You should notice

that this theorem

is of the same form

as Newtonian equation of motion

in the classical mechanics.

In this sense,

Newtonian equation of motion

is involved in quantum mechanics.

It is not so difficult

to prove Ehrenfest's theorem

Eq. (9) as shown on the page,

2-3-A: Proof of Ehrenfest's Theorem.

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