# What is eigenvalue and eigenfunction in quantum mechanics?

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?

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.