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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 dz 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 be 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 dV 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 dV 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 C 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 C 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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