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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 of x , y 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, n and l . Such numbers (integers) as these n and l 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), l 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 l expresses the speed of the revolution of the electron, i.e. the magnitude of the angular momentum of the electron, and m represents the orientation (direction) of angular momentum vector. The fact that these quantum numbers l and m 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 n and they are expressed as

The state with n = 1 is the lowest energy state (the ground state) and those with n = 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 r 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 at

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.
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