Schroedinger equation (English)

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1-3: The Schroedinger Equation

E. Schroedinger

(Austria, 1887 - 1961)

was interested

in de Broglie's matter wave

(de Broglie's wave).

He investigated

what type of equation

does this de Broglie's wave obey,

and he developed a

wave equation (1926).

This is

Schroedinger's wave equation.

Today it is simply called the

Schroedinger equation.

(The precise spelling

of Schroedinger is "".)

The Schroedinger equation

is nowadays

the fundamental equation

of quantum mechanics

which governs the

microscopic world.

It can be properly compared

with Newton's equation of motion

which is the fundamental equation

in classical mechanics.

By solving

the Schroedinger equation,

all the results

of Bohr's quantum theory

have completely been derived,

and the mysteries

in the microscopic world

have been resolved

one after another.

[The Schroedinger Equation for a Free Particle]

Let us consider

a one-dimensional space

for simplicity.

We treat the motion

of a free particle

on which no force is exerted

at all.

The free-particle motion

is the simplest one

moving with a constant velocity.

As already discussed,

particles

in the microscopic world

combine a particle nature

and a wave nature.

Then, how can

this double nature of

the free particles

be described

(or formulated)?

The momentum of a particle

of mass m moving

at a speed v

is p = mv.

The energy of this particle is

The relation between

E and p

is, of course,

On one hand,

these quantities,

the momentum

p and the energy E,

representing the motion

as that of a particle

characterize the

particle nature.

On the other hand,

the wavelength

and the frequency

of the de Broglie wave

characterize the wave nature

of the particle motion.

The relations between these quantities

characterizing the double nature

are just the Einstein-de Broglie's relations

The function that represents

the de Broglie wave

expressed as

which is called the

wave function.

Taking account of

the double nature mentioned above,

Schroedinger proposed

a wave equation

which the wave function

for a free particle

must obey;

i.e. the Schroedinger equation

for a free particle

in the one-dimensional space,

Here,

another expression

for the Planck constant, ,

defined by

is used.

We will meet

this again hereafter.

A somewhat detailed

explanation on

the underlying idea

of the above Schroedinger equation

is given on this other page,

1-3-A: The Underlying Idea of the Schroedinger Equation

but you may skip it,

if you feel it tedious.

[The Wave Function of a Free Particle]

The simplest solution

of the Schroedinger equation

(4) is given by

This is the wave function

of a free particle.

A solution of the

Schroedinger equation,

i.e. a wave function,

is in general

a complex function.

This result looks

somewhat strange.

The wave function has

originally been considered

as a realization

(or representation)

of the de Broglie wave.

Now, however, it

is complex.

This looks peculiar.

Then, what does the

complex wave function mean?

It is logically natural

that a complex wave function

comes out of

the Schroedinger equation,

because an imaginary number unit i

is contained in it.

Is the Schroedinger equation

wrong?

No, it isn't.

To tell the truth,

there is an unexpected

deep meaning

contained in the wave function.

As this will be

discussed later in detail,

let us overlook

it for a while.

[The Schroedinger Equation for a Particle Exerted a Force]

Comparing the relation

between the energy

and the momentum

of a particle,

Eq. (2),

with the Schroedinger equation

(4),

we realize the correspondences

Namely, if we

replace

and

in Eq. (2)

with the differential operators

Eq. (7)

and operate with

the result on a wave

function

then we obtain

the Schroedinger equation

(4).

In other words,

we can change

the energy-momentum relation

in the classical mechanics,

Eq. (2),

over the Schroedinger equation

in the quantum mechanics,

(4),

by the replacement

(7).

Hence we sometimes call

this procedure

"quantization".

Next, we discuss

the case in which a force

is exerted on a particle.

Let the potential of the force be

V (x ).

The relation between the energy

and the momentum

is given by the

energy conservation law,

Carrying out

the replacement

and operating with the result

on a wave function

we have a wave equation

This is the generalized form of Schroedinger equation

which is applicable

for a particle

being acted upon by a force.

[The Three-Dimensional Schroedinger Equation]

In order to treat

some realistic problems,

we have to extend

the Schroedinger equation

(9)

to be applicable

in the three-dimensional space.

In this case,

the force potential must be

a function of a three-dimensional

coordinate

(x,

z ).

The wave function is

therefore given by a function of

variables

(x, y, z, t ),

and

the three-dimensional Schroedinger equation

is written

[A Serious Problem Remained !]

It was clarified that

the Schroedinger equation

(9)

applied to various systems,

e.g. the hydrogen atom,

produced

very good results

that were beautifully

fit to experiment.

With this

Schroedinger equation,

various structures

in the atomic

and molecular world

were well explained.

Thereby, people considered that

the Schroedinger equation

is the most fundamental equation

which can describe

and rule the microscopic world.

However, they could not

understand

what the wave function means,

so that the serious problem

of "the interpretation of wave function"

has remained.

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