Tunnel effect (English)

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1-6: Tunnel Effect

We studied

on the preceding page

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

An experimental evidence

of the validity

of the above interpretation

was given

by Gamow's explanation

on the

alpha decay

of atomic nuclei.

The alpha decay of atomic nuclei

could be explained

by a strange phenomenon

called

tunnel effect.

This was an effect

inconceivable in the classical theory.

Let us learn it below.

["State" in Quantum Mechanics]

Before getting onto

the main point

of the tunnel effect,

let us explain the concept

of "state" in quantum mechanics.

It is a basic concept

by which all the informations

on the dynamical system

under consideration,

e.g. the energy and momentum

distributions of the system,

the probability density of

the particles

in the system,

and so on,

can be obtained.

All of these informations

about the system

under consideration

must be contained

in the wave function.

So that the wave function

is sometimes called

the state function.

[Stationary State]

Let us consider

a one-dimensional space

for simplicity.

The wave function of a free particle

is shown by

Eq. (6) on the page

"1-3: The Schroedinger Equation" as

This is a state

with a constant energy E .

Thereby we can consider that

a wave function

whose time factor is

is a state of

a constant energy E .

Let us assume

the wave function to be

As discussed above,

this state is

of a constant energy E .

Putting this into

the Schroedinger equation

we have a differential equation

independent of time t

which determines the

spatial part

of the wave function.

The above equation (4)

is named the

"time-independent Schroedinger equation" or sometimes

called the

"Schroedinger equation"

simply.

In the state represented

by the wave function

of the form of

Eq. (2),

the probability density

of the particle

is written

Because this probability

is independent of time t ,

this state is called

a stationary state,

which is just

what we have met in

Bohr's old quantum theory.

The ground state

and the excited states

of hydrogen atom

are of this type of states.

[Range of Motion of a Particle -- Classical Mechanics]

Let us see

the vibration

of a spring

or a harmonic oscillator

as shown in Fig. (A) below.

Suppose that a mass m

is attached to the

spring and neglect

the mass of the spring

itself.

Suppose that an x axis

is arranged in the direction of

the motion of the oscillator.

Let the position

of the natural length

of the spring be x = 0.

When an energy E

is given to the mass m ,

it oscillates

within the range

between the lower bound A

and the upper bound B.

The energies of this oscillator

are graphed in the following

Fig. (B).

When the spring is extended

by x

from the equilibrium position

(natural length),

then the force exerted

on the mass is

- kx ,

where k

is the spring constant.

At this point,

the potential energy

of the oscillator is

The total energy of the spring,

E ,

is the sum of the kinetic energy

and the potential energy,

i.e.,

For a given energy E ,

the possible range

of the oscillation

is as follows:

This is the range

between A and B

on the x axis in

Fig. (B).

In Fig. (B),

the

yellow-colored part

denotes

the potential wall.

The particle motion

is confined to the inner part

of the potential

(the outside of the wall).

The particle with energy

E is reflected

by the potential wall,

and repeats a back-and-forth

motion between A and B

on the x

axis.

[Energy Eigenvalues, Eigenstates]

Let us treat the same

oscillator motion

as discussed above

in quantum mechanics.

The quantum mechanical

state of motion

is determined

by the solutions of

the Schroedinger equation

(4).

In the present case,

it becomes

the following differential equation

The square of the absolute value

of the wave function, ,

denotes the probability density

where the particle will be detected.

So that

must be finite

over all the region of

In order that

the wave function

satisfies this condition,

the corresponding energy

should be restricted

to some specialized values.

While the energy

can take

a continuous value

in the classical mechanics,

it cannot be so

in quantum mechanics.

The allowed energy

in quantum mechanics

is not continuous

but discrete

(step-like).

Such discrete energies

are called

the eigenvalues

and the corresponding states

are called

the eigenstates.

We should pay attention

to the reason

why such eigenstates

with discrete energy

eigenvalues occur.

The reason is

that the particle motion

accompanies the wave function,

i.e.,

just due to

the particle-wave duality.

The requirement

of finiteness and continuity

of the wave function brings about

the discrete energy eigenvalues.

This property

is never obtained

in the classical mechanics

and

it is a surprising result.

Namely, the stationary states

in Bohr's old quantum theory

can be derived just from

the double nature

of electrons.

The energy eigenvalues of a harmonic oscillator

is given by

The allowed n

is 0 or positive integers.

These results are shown

in the form of energy levels

in the following

Fig. (C).

The eigenstates of a harmonic oscillator

The eigenvalues obtained

by solving

the Schroedinger equation

(9)

which are denoted

by the horizontal lines

(levels).

The thick solid curves

show the corresponding

wave functions.

In this figure,

the ground state,

n = 0,

is denoted by the symbol (0),

and the excited states,

n = 1, 2, 3, ...

(1), (2), (3), ... , respectively.

In the case of

a harmonic oscillator,

the energy levels

(horizontal lines in Fig. (C))

are at equal intervals.

This is a distinctive feature of

a harmonic oscillator.

The thick solid curves

in Fig. (C)

show the

wave function

corresponding to each

energy eigenvalue.

These results have been

obtained by solving

the Schroedinger equation

(9).

It can be solved

analytically

but it is somewhat tedious

to explain how to solve it,

so that we omit here.

If you want to study the details,

refer to some other

textbooks on quantum mechanics.

[Range of Motion of a Particle -- Quantum Mechanics]

In the classical mechanics,

the particle motion

is confined to the inner part

of the potential

(the outside of

the potential wall)

as shown in Fig. (B).

The particle cannot penetrate

into the potential wall

in the classical mechanics;

i.e.,

the range of the particle

motion in the classical mechanics

is just from A to B

in Fig. (B).

However, the situation

in quantum mechanics

is different.

In Fig. (C),

the wave function

corresponding to each eigenstate

goes beyond the limits

in the classical mechanics

and it penetrates

into the potential wall.

This means that

there is a probability that the particle will be found inside the potential wall.

This is a surprising fact

that we cannot imagine

in the classical mechanics.

[What's Tunnel Effect?]

As discussed above,

a particle

can penetrate into

a potential wall.

We then suppose

that a particle might

pass through

a potential wall

(or barrier).

That is true.

This strange phenomenon is called

"tunnel effect".

The wave function

of a free particle

written as

is a wave going

to the positive direction

on x

axis

with energy E

(right-going wave).

And the wave function

is a left-going wave

with the same energy.

Let us consider

a potential barrier

as shown

in Fig. (D) or Fig. (E).

Suppose that an incident

particle with

energy E

coming from the left

collides the potential barrier.

The energy E

is assumed to be lower

than the top of the barrier.

In the classical mechanics,

the particle is

completely reflected

by the potential barrier.

In quantum mechanics,

some probability

will be reflected,

but the other penetrates

the barrier

and pass through

into the right-hand side region

to proceed to the far right.

This is the tunnel effect.

[Penetration Coefficient]

In order to estimate

the rate of the probability

that penetrates the

potential barrier,

let us assume a simple

square potential

as shown in the above

Fig. (E).

Here, let the width

of the square potential be

and the height V₀.

The wave function

in the region

on the left-hand side of

the barrier

is a superposition

of the incident wave

(11)

and the reflected wave

(12);

i.e.,

Contrarily, the wave function

on the right-hand side

of the barrier

consists only of the

the penetrated wave as

Solving the Schroedinger equation,

we can determine the constants

A , B

and C .

Let us explain that

on the following other page,

1-6-A: Calculation of the tunnel effect in a simple case.

This page is not so difficult

but, if you feel it tedious,

you may pass it.

Then you should

accept the results.

As seen in the calculation

presented on the page 1-6-A,

a relation

holds.

The quantity

denotes

the intensity of the incident wave.

And

and

denote

the intensities of

the reflected

and transmitted waves, respectively.

The above relation (15)

means therefore that

the sum of the probabilities

of the reflected particles

and the transmitted particles

is equal to 100 %.

This is nothing but

the conservation of the probability.

The rate of the particles

penetrating the potential barrier

out of the incident particles,

i.e.

the penetration coefficient

(or transmission coefficient)

is given by

where

The function sinh

defined by

is called the

hyperbolic sine function.

An example of the

experimental evidences of

the tunnel effect

discussed in the present section

is the alpha decay of atomic nuclei

which will be explained

on the next page.

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