Probability densities

In the following, we will introduce some basic mathematical concepts that are required to formulate the probabilistic aspect of the wave function.

Figure: In this example, the random variable $x$ is the distance $x=r$ to the center.

Let $x\in [-\infty,\infty]$ be a random variable, e.g. the outcome of a measurement. The probability density $\rho(x)$ of $x$ is defined in the following way: $\rho(x)dx$ is the probability that $x$ lies in $[x,x+dx]$. Clearly, $\rho(x)$ has to be normalized, i.e. $\int_{-\infty}^{\infty}dx \rho(x)=1$. Alternatively, one often uses the following

Definition 1 (Probability Density)   Let $P(a\le x \le b)$ be the probablity for the random variable $x$ to be in the interval $[a,b]$. We define the probability density $\rho(x)$ by

$$P(a\le x \le b) = \int_{a}^{b}dx \rho(x).$$ (25)

Remarks:

• $\rho(x) \ge 0$.
• $\int_{\infty}^{\infty}dx \rho(x) \equiv P(-\infty \le x \le \infty) =1.$
• $P(-\infty\le x'\le x)$ is called distribution function.