How is a bound state defined in quantum mechanics for states which are not eigenstates of the Hamiltonian i.e. which do not have definite energies? Can a superposition state like $$\psi(x,t)=\frac{1}{\sqrt{2}}\phi_1(x,t)+\frac{1}{\sqrt{2}}\phi_2(x,t), $$ where $\phi_1$ and $\phi_2$ are energy eigenstates be a bound state? How to decide?
Answer
Bound states are usually understood to be square-integrable energy eigenstates; that is, wavefunctions $\psi(x)$ which satisfy $$ \int_{-\infty}^\infty|\psi(x)|^2\text dx<\infty \quad\text{and}\quad \hat H \psi=E\psi. $$
This is typically used in comparison to continuum states, which will (formally) obey the eigenvalue equation $\hat H\psi=E\psi$, but whose norm is infinite. Because their norm is infinite, these states do not lie inside the usual Hilbert space $\mathcal H$, typically taken to be $L_2(\mathbb R^3)$, which is why the eigenvalue equation is only formally true if taken naively - the states lie outside the domain of the operator. (Of course, it is possible to deal rigorously with continuum states, via a construct known as rigged Hilbert spaces, for which a good reference is this one.)
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