Top of Part 1
Last page
Next page
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 is 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 E and the momentum p of a particle, Eq. (2), with the Schroedinger equation (4), we realize the correspondences

Namely, if we replace E and p in Eq. (2) with the differential operators in 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 E and the momentum p 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, y, 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.
Go back to the top page of Part 1.
Go back to the last page. Go to the next page.