# Physical intepretation of nodes in quantum mechanics

I am taking my second course in QM, and my head is starting to spin as it probably should. But I would very much like to clear up my head about a few details regarding the wave function. As I know it is impossible to predict where particles are and one can only give a probability of where it should be.

The simplest case is a "frictionless" particle "bouncing" back and forth inside a infinite square well. E.g. a particle in the following potential

$$V(x) = \left( \begin{array}{cc} 0 \ , & \text{for} \ 0 \leq x \leq a \\ \infty \ , & \text{elsewhere} \end{array} \right)$$ Which gives rise to the following normalized solution $$\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left( \frac{\pi n}{a}x \right)$$ My problem is what the nodes in the square function represents. If I draw $$|\Psi_2(x,0)|^2=|\psi_2(x)|^2$$ I obtain a graph similar to the one below. What is the physical explanation that finding the particle around a small region around $$a/2$$ is close to zero? Or why is it so much less likely to find it near $$a/2$$ than $$a/4$$? E.g. why is $$P(a/2 -\varepsilon \leq X \leq a/2+\varepsilon) = \int_{a/2-\varepsilon}^{a/2+\varepsilon} \left| \psi_2(x) \right|^2 \,\mathrm{d}x \sim 0$$ for small $$\varepsilon$$

• An analogy is manifest with the modes of a vibrating string with fixed extremities. You have different harmonics, and for each harmonic, you have nodes. Aug 25, 2013 at 14:28
• I like this question and would like to see a good answer. In principle the node is "caused" by the assumptions of QM and these in turn are justified by a large body of evidence. This is counter-intuitive when coming from a macroscopic perspective but inherent in QM. Aug 25, 2013 at 22:35

Recall that the functions $\psi_n$ are energy eigenstates; these states are very special. A generic quantum state of the system is simply some continuous, square-integrable function $\psi$ on $[0,a]$ that vanishes at the endpoints of the interval; within these requirements, it can have any shape. Moreover, any such function can be written as a linear combination of the energy eigenfunctions $\psi_n$.