Structure of a hydrogen atom (English)

Top of Part 3

Last page

3-2: The Structure of Hydrogen Atom

As mentioned before,

it is considered that

there is a heavy

proton at the center of

a hydrogen atom

and a light electron

revolves around it.

The proton may be thought

to be approximately

at rest at the origin

of the coordinate

(the center of the

hydrogen atom)

because proton is

about 1800 times heavier

than electron.

The Coulomb attractive force

works between the proton

and the electron.

Its potential is written

where r

is the distance between

the proton and the electron.

The Schroedinger equation

describing the motion

of the electron is

given by

Needless to say,

the wave function

is a function

x ,

and

z .

[The Eigenstates of Hydrogen Atom]

In the present case,

the polar coordinate

shown in the following

Fig. (A)

is more convenient than

the Cartesian coordinate .

We solve

the Schroedinger equation

(1) represented

in the polar coordinate

setting the boundary condition

that the wave function

should be smoothly continuous

at every point of

the coordinate space

and should converge to 0

at the infinitely long distance .

Then we have

a set of discrete energy eigenvalues

and the corresponding eigenstates.

The details of

the method to solve it is

omitted here.

If you want to study them,

please refer to some

other textbooks

of quantum mechanics.

The wave functions

of the eigenstates

is expressed as

Here the part

is called the

radial wave function

which is specified

by a set of integers,

and

l .

Such numbers (integers) as these

and

are sometimes called

quantum numbers,

which characterize

the eigenstates.

In the present case, they are

The part

denotes the angular wave function

which is specified

by a set of quantum

numbers (integers),

and

m , and

they are given by

The angular wave function

describes the revolving state

of the electron around

the coordinate origin (proton).

Namely,

the quantum number

expresses the speed

of the revolution of

the electron,

i.e. the magnitude of

the angular momentum

of the electron, and

represents

the orientation (direction)

of angular momentum vector.

The fact that these quantum numbers

and

are integers

means that

both the magnitude

and

the orientation

of the angular momentum

are step-like and discrete.

This result implies that

not only energy but also

angular momentum

and its orientation

are quantized

in quantum mechanics.

This was confirmed

by the Stern-Gerlach experiment

(1922).

Needless to say,

this also originates

from the particle-wave

duality of electrons.

And this can never understood

by the classical theory.

[The Energy Eigenvalues of Hydrogen Atom]

The energy eigenvalues

of hydrogen atom

are determined

only by the quantum number

and they are

expressed as

The state with n

= 1

is the lowest energy state

(the ground state)

and those with

= 2, 3, ...

are the excited states.

Thus the ground-state energy

is written

The energy eigenvalues

(energy levels)

are shown in the following

Fig. (B),

in which

the heights of the

horizontal lines (levels)

show the energy

eigenvalues obtained

by solving the Schroedinger equation

(1).

Fig. (B): The energy eigenvalues of hydrogen atom

The abscissa denotes

the position coordinate

of the electron

(the distance between

the proton and electron),

r ,

in units of the

Bohr radius ,

where

[The Probability that Electron will be Found]

It is very interesting

to see how much

the probability

for a electron to be found

at a point in the space is.

The probability density

at a position of the distance

from the center

is shown in

Fig. (C).

Integrating this probability density

over the whole space, we have

As shown in this integration,

the present wave functions

are so normalized

that the total

probability would be 1

(= 100 %).

The above Fig. (C) shows

the probability distribution

of electron in hydrogen atom.

The first figure

shows the ground state,

in which

almost all probability

concentrated

We can therefore consider

that the true radius

of a hydrogen atom is

almost equal to

the Bohr radius .

However, in the excited states,

the probability is

widely extended

at the distant area,

so that the excited

hydrogen atom

would be rather swollen.

[Conclusion of This Page]

We have studied

how the solutions

of the Schroedinger equation

for the case of hydrogen atom are.

All of Bohr's quantum theory

are completely included

in the solutions.

This means that

all the structure

of hydrogen atom

can perfectly be derived

just from the single equation,

the Schroedinger equation.

Thus we can understand

how splendid and how fundamental

the theory of Quantum Mechanics is.

Top
	Go back to the top page of Part 3. Go back to the last page. Go to the next page.