Meaning of wavefunction (English)

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1-5: The Meaning of the Wave Function

As learned on the page 1-3,

the fundamental equation

of quantum mechanics

is the Schroedinger equation.

It was introduced

as a wave equation

which did not contradict

the relation between

the energy and the momentum

of a particle

in classical mechanics.

As a result,

the wave function

became a complex number.

This looked quite strange,

because using a complex number

for a physical quantity

could not be accepted.

Then, what does

the wave function mean?

As for the interpretation

of the wave function,

various ideas were proposed.

Among them,

the probability interpretation

proposed by

M. Born

(Germany, UK, 1882 - 1970)

has been accepted

to be the most orthodox,

and all the theories

of quantum mechanics are constructed

on the basis

of this interpretation.

[The Probability Interpretation of the Wave Function]

Suppose a small volume

(box)

dV = dx dy dz

with three edges of

dx ,

dy ,

and

at a coordinate point

r = (x , y , z )

in a three-dimensional space.

(See the figure below.)

Let the probability

that a particle

will be found in this volume

at a time t

P (x,y,z,t )dxdydz.

The function

P (x,y,z,t )

is considered as the probability

in the unit volume

in the neighborhood

of the point

(x,y,z),

and is sometimes called

the probability density.

[N.B.]

In the above discussion,

the word, "probability"

was used.

This however does not mean

that the particle,

e.g. an electron,

exists in a form

of a "particle" or "corpuscle"

and this particle moves

in accordance with

the degree of probability.

An electron is

neither mere "particle"

nor mere "wave".

An electron exists

with particle-wave

duality.

The word "probability" used

above denotes the chance

that the particle

will be found

in the volume

when it is detected.

Born's probability

interpretation claims that

the probability density that a particle will be found is equal to the square of the absolute value of the wave function.

Namely, the probability

that the particle will be found

in the small volume

shown in the above figure

is considered to be

Although the wave function

is a complex number in general,

the square of the absolute value of it

is always positive (or zero),

and we have no difficulty.

[Normalization of the Wave Function]

At an arbitrary time,

the particle should exist

somewhere in the whole space.

The total probability

that the particle

will be found somewhere

must therefore be 100 %.

This total probability

is obtained by integrating

Eq. (1) over the whole space.

Accordingly,

the wave function

should satisfy

the normalization condition

A solution of the

Schroedinger equation,

does not always satisfy

this normalization condition

(2).

We therefore have to

normalize the wave function

so that it satisfies

the normalization condition.

Let

be a solution of the

Schroedinger equation.

A function obtained

by multiplying

an arbitrary non-zero constant

to the ,

i.e., ,

is also a wave function

satisfying the same

Schroedinger equation.

Therefore we can normalize

the wave function

by arranging the normalization constant

to make

satisfy the normalization condition

(2).

It is easy to see

that the normalization constant

is given by

On the one hand,

in the classical theory,

we can precisely

determine the

orbit (or trajectory)

and the momentum of a particle

all over the time

by using Newtonian

equation of motion.

On the other hand,

the Schroedinger equation

in quantum mechanics

determines the

wave function

from which we

can derive only

the probability

that a particle

will be found somewhere.

You may feel

that this is not

entirely satisfactory.

However, this is necessary

and sufficient

in order to describe

the microscopic world.

This is enough

from the point of view

of Heisenberg's uncertainty principle.

You will gradually

be able to understand this

through this Seminar.

A typical proof of

the correctness

of the probability interpretation

of the wave function

will be shown

in the following page.

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