What is the use of eigenvalue and eigenfunction in quantum mechanics specially Schrodinger equation? What is the physical meaning of having an eigenvalue and eigenfunction in Schrodinger equation?


1 Answer 1


These concepts are absolutely central in quantum physics, so no short answer can do justice to the situation. In the case of Schrodinger's equation, the eigenvalues are the possible energies that the system can have if it is in a state of well-defined energy. Each eigenfunction (of the Hamiltonian) is the state of the system when its energy is equal to the associated eigenvalue. This can be compared to the study of a vibrating string in classical mechanics---a useful thing to know is the frequency of the fundamental mode and all the other modes, and the shape of the string for each case.

The energy eigenfunctions are especially important because they provide a very convenient way to express the evolution over time of the system. When the Hamiltonian is time-independent, each energy eigenstate evolves as $\exp(-i E t/\hbar)$, i.e. a very simple evolution. So a good way to proceed is to write the initial state of the system as a superposition of energy eigenstates: $$ \psi(x,0) = \sum_n a_n \phi_n(x) $$ when $\phi_n$ is the $n$'th energy eigenstate and the coefficients $a_i$ have to be worked out in each case from whatever the initial conditions are. Then, as long as the Hamiltonian is time-independent, you can immediately write down the state at any later time: $$ \psi(x,t) = \sum_n a_n \phi_n(x) e^{-i E_n t/\hbar} . $$ You should pause to appreciate how powerful a result that is! We just solved the whole problem of the evolution over time all in one go. Really, what it says is that the investment of effort to find the eigenvalues $E_n$ and eigenfunctions $\phi_n$ is well worth it.

The above method is very much like the method of normal modes when analysing a set of coupled oscillators in classical mechanics.

Finally, I should mention that every physical quantity that one could measure has its associated operator, and the eigenvalues of the operator give the possible values which that physical quantity can have, when the system is in a state in which the quantity has a precisely defined value.

As I say, the full answer to your question would have the length of a small textbook, but I hope this is helpful.


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