5.B: Nodes and DeBroglie's principle
We are still examining the stationary states, or eigenstates, for a particle bouncing between two walls. Notice that the wave functions of successively higher-energy eigenstates get ``wavy-er'' - that is, they oscillate more rapidly as a function of position. The wave length, the spatial length of the oscillations decreases with increasing n. This is a consequence of de Broglie's principle which states that the energy of a particle described by a particular wave function increases with shorter wave length. In other words, ``more waviness means more energy''.
Each point where the wave function is zero as it changes sign from positive to negative, or negative to positive, is called a node. For example, the two nodes of the n = 3 state are indicated in the Figure. Since each node counts an oscillation of the wave function, the number of nodes is a measure of the waviness of the wave function and hence a measure of the of energy of a particle described by that wave function: More nodes means shorter wave length and higher energy.