In quantum mechanics, the Born rule tells us how a wave function determines the probability of observing a particle. More precisely, the probability of detecting a particle, such as an electron, is proportional to the square of the magnitude of its wave function. This page explores whether the Born rule can be derived from deeper principles.
The Born rule is usually treated as one of the fundamental principles of quantum mechanics. A fundamental principle is an assumption that is not derived from something more basic. For that reason, most physicists do not try to derive the Born rule itself. So why look for a derivation?
I began by asking myself whether the Born rule really has to be fundamental. According to the rule, the probability of observing an electron is proportional to the square of the absolute value of its wave function. I believe, however, that probability should ultimately be understood in the spirit of Laplace's theory of probability. The next section briefly reviews Laplace's approach.
Laplace's theory of probability gives a simple way to calculate probabilities. The French mathematician Pierre-Simon Laplace introduced this approach in 1814. As an example, imagine a bag containing three red balls and two blue balls.
If you reach into the bag and draw one ball, what is the probability that it will be red? In Laplace's theory, we calculate this by counting events and elementary events. An elementary event is an outcome that cannot be subdivided any further. For example, the event R, "drawing a red ball," consists of the following three elementary events:
Each outcome, such as "drawing red ball r-1," is elementary because it cannot be broken down into smaller outcomes. In Laplace's theory, the probability of an event is the number of favorable elementary events divided by the total number of elementary events. In this example, the probability of event R is 3/5, because three of the five possible outcomes are red balls.
The Born rule calculates probability in a very different way. Why, in quantum mechanics, do we square the magnitude of the wave function instead of simply counting elementary events? I call this the probability problem of quantum mechanics. On this page, we explore this problem using the many-worlds interpretation and probability theory. The next section introduces the many-worlds interpretation from this point of view.
In the many-worlds interpretation, all of reality is described by a universal wave function. Different people picture this "wave function of the universe" in different ways. To build an intuitive picture, let us start with the wave function of a single particle and then extend the idea step by step.
To approach the probability problem, we consider the wave function of many particles.
If the position of the k-th elementary particle is \begin{align*} r_k = (x_k, y_k, z_k), \end{align*} then the multi-particle wave function can be written as follows:
\begin{align*}
\Psi = \Psi(r_1, r_2, r_3, \dots, r_n)
\end{align*}
This wave function describes the positions of n elementary particles. By varying those positions, we can represent the possible configurations of an entire universe.
Each point in the multi-particle wave function corresponds to a different world. The probability distribution P is given by:
\begin{align*}
P(r_1, r_2, r_3, \dots, r_n) = \bigl|\Psi(r_1, r_2, r_3, \dots, r_n)\bigr|^2
\end{align*}
The following paper presents a derivation of the Born rule using the many-worlds interpretation together with probability theory.
A central idea in this derivation is the hierarchical universe. For the full argument, please see the paper linked above.
Here is a brief overview of the hierarchical universe.
The hierarchical universe is a nested structure of universes.
Within our universe, each particle is associated with a wave function. I propose that there is a smaller universe whose radius is equal to the magnitude of that wave function.
Inside that smaller universe, there is another particle with its own wave function, which in turn gives rise to an even smaller nested universe whose radius is equal to the magnitude of that wave function.
Conversely, the wave function that determines the radius of our universe describes a particle that exists in an even larger universe.
In this way, our universe and the smaller universe are connected through the corresponding wave function.
© 2012-2015 xseek-qm.net
広告
