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In Sakurai's Modern Quantum Mechanics (2nd ed) p.99, he says

We know from the theory of partial differential equations that (time-independent Schrodinger equation) subject to boundary condition ($\psi(\mathbf{x}')\rightarrow 0$ as $|\mathbf{x}'|\rightarrow\infty$) allows nontrivial solutions only for a discrete set of values of $E$. It is in this sense that the time-independent Schrodinger equation yields the quantization of energy levels. Once the partial differential equation (time-independent Schrodinger equation) is written, the problem of finding the energy levels of microscopic physical systems is as straightforward as that of finding the characteristic frequencies of vibrating strings or membranes.

Although I've found a paper ( about this, the relationship is still not very explicit. So would you please explain briefly about the relationship between the Schrodinger equation and string/membranes? Many thanks.

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This question (v2) seems partly spurred by a conflation of non-relativistic material strings, such as, e.g., a guitar string, and relativistic string theory. – Qmechanic Aug 5 '14 at 12:16
up vote 9 down vote accepted

The relationship that Sakurai seems to be talking about is a mathematical one: both the Schr:odinger equation in wave mechanics and models of other physical systems, like small vibrations of a stretched elastic membrane, reduce to solving eigenvalue problems with elliptic differential operators.

In the quantum mechanical setting, the operator is the Hamiltonian, the eigenvectors are the stationary solutions, and the eigenvalues the energies.

In the vibrating membrane context, the operator is the Laplacian, the eigenvectors are the normal modes, and the eigenvalues are (minus the squares of) the corresponding frequencies of oscillation.

The physics is different, and Sakurai is not suggesting that the two things are physically related. One problem is 'as straightforward as' the other because they are mathematically closely related.

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