Mode of vibration comparing Classical and Quantum waves

Question

I'm now studying Quantum Mechanics, and I took a course on Vibration and Waves last year. I have been trying to make an analogy between classical and the quantum waves. Is it true that both the modes of a quantum and classical wave can not be excited individually?

Whenever you pluck a classical string, its motion can be described as a superposition of many different modes , n=1,2,3... you can never excite a single mode.

In analogy, can we say that whenever you try to measure the wave function, the measured value will be a composition of different eigenvalues?

Any ideas or contributions much appreciated.

Answer 1

No, that is a bad analogy. In fact, you've got things precisely the wrong way around. Whenever you perform a measurement that's in a state that is a superposition of basis states (eigenstates of the Hamiltonian, for instance) $$\psi=\sum\limits_n c_n \psi_n$$ where the $c_n$ are complex constants (arbitrary up to normalization) then this measurement will always return a value that is exactly the eigenvalue of one of these basis states. For instance, if we have $$\psi=\frac{1}{\sqrt{6}}(\psi_1+\psi_2+2\psi_3)$$ then a measurement will always return the eigenvalue corresponding to either $\psi_1, \psi_2$ (each with probability $\frac{1}{6}$) or $\psi_3$ (probability $\frac{2}{3}$).

Answer 2

Whenever you pluck a classical string, its motion can be described as a superposition of many different modes , n=1,2,3... you can never excite a single mode.

I don't think that's strictly true. You can have an object undergo low-amplitude oscillations in a single eigenmode; that's what makes cymatic sand patterns look the way they do. When they spontaneously change shape when you change the frequency, that's a manifestation of hopping between modes.

But one thing to take away from the classical picture is the idea that a vibrating string can either be thought of as a vibrating string, or as a discrete vector of coefficients describing the amount of each mode which is excited. If you think about it, both contain exactly the same information about the state of the system. Similarly, in quantum mechanics, you can either think of a wavefunction as a literal function on space (or whatever domain it is defined on), or as a vector in Hilbert space using an eigenstate basis for the representation.