|
Suppose that we have a system of particles that is described by the wave function Y. The Copenhagen interpretation tells us that |Y|2 is the probability distribution to find the particles at some specific positions, and, if they have spin, with some specific spin. As all particles must be somewhere and have some spin, the probability to find the particles somewhere and with some spin must be equal to 1. This can be written as
where t stands for all coordinates and spins of the particles and the integration is over all possible values of these coordinates and spins. This equation is called a normalization condition, and it is said that the wave function has to be normalized.
Very often a wave function is given in a form that is not normalized. The problem is then how to normalize it. There is a simple part to the solution of this problem and a possibly complicated part. The simple part is the method to solve the problem. Suppose that we have a wave function Y that is not normalized: i.e.,
where N differs from 1. We want to know a constant C so that CY is a normalized wave function.
|
ó
õ |
dt |CY|2 = |C|2 |
ó
õ |
dt |Y|2 = |C|2N = 1. |
|
The last equal sign indicates our objective. It is easy to see that C = 1/ÖN is an appropriate solution. The possibly complicated part is the determination of N, because the integral that defines N may be very difficult to calculate.
Because this method is so easy, non-normalized wave functions are very often employed, and equations are used that give correct results whether a wave function is normalized or not. For example, the expectation value of an operator A is defined as
when Y is normalized. If we define it as
however, we see that if we multiply by some constant the expectation value does not change. For normalized wave functions both definitions are equivalent, but the latter can also be used for non-normalized wave functions.
The method to normalize a wave function above has also other solutions. The general solution can be written as C = exp(ia)/ÖN with a a real number. The factor exp(ia) is called a phase factor, and it is characterized by |exp(ia)| = 1 for any real a. If we multiply a wave function that is normalized by a phase factor, we get another wave function that is also normalized. In fact, no property of a system at all changes when we multiply a system's wave function by a phase factor. (Remember that a system's wave function determines all its properties.) So if two wave functions differ only by a phase factor those wave functions are regarded as essentially the same.
|
|
