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The energy eigenstates for a particle in an infinite 1d square potential well and the modes for displacement of a standing wave of a string (length L) between two rigid posts have a similar form:

$y(z)=A\sin(n\pi z/L) \text{ (Classical Standing wave)} \\ \psi(z)=B\sin(n\pi z/L) \text{ (Square Well Stationary State)}$

In the classical case (the standing wave of a string), is the coefficient A arbitrary? If so, what does it depend on? And in the quantum mechanical case, it seems to me like the coefficient is not arbitrary, because we need to integrate the modulus squared of wavefunction to get probability density, so we normalize the wavefunction such that the coefficient is B=sqrt(2/L). Is this correct, or is B arbitrary like A is as well?

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In the classical case, as in the quantum case, the coefficient A and B respectively denote the amplitude of the wave. The amplitude is basically how big of an oscillation you can have. In the classical case, A is the maximum displacement from 0 of any point along the string.

You are correct that B is generally normalized such that the probability density integrates to 1. This makes calculations easy and simple to understand. However, you could neglect to normalize B and simply normalize your probabilities later down the line as well (e.g. when you are actually making a physical prediction).

In the classical case, A is not constrained like B is. A generally is proportional to the square-root of the energy of the wave. The higher the energy of the wave, the higher A will be. However, there are limits to what A can be of course - at some energies the string will break.

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  • $\begingroup$ Thank you for the response! So, would it be fair to say that both A and B are, to some degree, arbitrary, since it's not a hard requirement that B be normalized? Also, can you explain why A is proportional to the square root of the energy of the wave? I had thought that A would just mathematically be the result of initial conditions imposed on the wave. $\endgroup$ – Mingzu Chen May 17 '18 at 15:14

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