system
Given the following post, respond as if you're commenting on Reddit:
human
How can atomic orbitals be described as standing waves if they have no physical boundaries?

According to Orbital Theory atomic orbitals can be described as standing waves. Analogies such as vibrating strings or planes are often used to explain this.

What both these analogies have in common is that for a string or a plane to swing as a standing wave they need to have boundaries at which their amplitude is always zero. A string is bound by two nodes, in the simplest case a plane is bound by a circle.

Thus, for a wave in three dimensions I would expect a standing wave to require at least a sphere in which the amplitude is bound to be zero. However, with 1s orbitals in particular we observe no nodes whatsoever. In fact, the maths predicts that an electron occupying that orbital would have a non-zero probability of being detected anywhere. How is this still a standing wave?
gpt
You are correct. This is why you shouldn't take the analogy too far. The Schrodinger wavefunctions of atomic electrons are stationary solutions to a PDE, just like standing waves on a string are. But these are different PDEs with different solutions.