We cannot predict the exact trajectory of an atomic particle such as an electron. The best we can do is to predict the probability that a particle will be found in some region of space. The probability is determined by the particle’s wave function $\psi \left(x\right)$.

• $\psi \left(x\right)$ is a continuous, wave-like (i.e., oscillatory) function.

• The probability that a particle will be found in the narrow interval $\delta x$ at position $x$ is

  $$\text{Prob(in }\delta x\text{at }x)=\left|\psi \left(x\right)\right|{}^2\delta x$$

• $\left|\psi \left(x\right)\right|{}^2$ is the probability density $P\left(x\right)$.

• For the probability interpretation of $\psi \left(x\right)$ to make sense, the wave function must satisfy the normalization condition:

  $${\displaystyle {\int }_{-\infty }^{\infty }P\left(x\right)dx}={\displaystyle {\int }_{-\infty }^{\infty }{\left|\psi \left(x\right)\right|}^{2}dx=1}$$

  That is, it is certain that the particle is somewhere on the $x$-axis.

• For an extended interval

  $$\text{Prob}(x_{\text{L}}\le x\le x_{\text{R}})={\displaystyle {\int }_{x_{\text{L}}}^{x_{\text{R}}}{\left|\psi \left(x\right)\right|}^{2}dx=\text{area under the curve}}$$

A particle with wave-like characteristics does not have a precise value of position $x$ or a precise value of momentum $p_x$. Both are uncertain. The position uncertainty $\text{Δ}x$ and momentum uncertainty $\text{Δ}{p}_x$ are related by $\text{Δ}x\text{Δ}{p}_x\ge h\text{/}2$. The more you try to pin down the value of one, the less precisely the other can be known.