Can the Time Independent Schroedinger Equation Be Used to Find Unbound States?

[A. Neumaier]

I'm still not sure whether they (wavepackets, that is) remain normalizable

The norm is preserved by the dynamics, so normalizable at one time implies it at all times.

I was calling "bound" the space spanned by all bound states.

But this is not an interesting space as once you perturb the Hamiltonian you will get contributions from the scattering part even in your perturbed bound states.

The notion of a bound state is useful only for eigenstates, and this is why it is traditionally applied only to this case.

 

[TeethWhitener]

But this is not an interesting space as once you perturb the Hamiltonian you will get contributions from the scattering part even in your perturbed bound states.

Isn't this true for all states (even eigenstates)? I don't see how this statement makes the notion of a bound state more useful for eigenstates than for superpositions of eigenstates.

EDIT: I'm content to let things lie here. I'll simply change my usage to match standard usage and stop complaining.

 

[A. Neumaier]

Isn't this true for all states (even eigenstates)? I don't see how this statement makes the notion of a bound state more useful for eigenstates than for superpositions of eigenstates.

The point is that there is virtually no use for the class of superpositions of bound states, hence there is no point in giving these a special name. on the other hand, the discrete eigenstates are very important for many reasons and hence deserve having a short name for them, to facilitate communication.

 

[A. Neumaier]

Isn't this the definition of stationary states, not bound states?

These two concepts just differ by a phase; see the explanation in post #24.

 

[George Jones]

Here is my impression of common usage for "bound state" in standard (non-rigourous) quantum mechanics texts.

The energy spectrum for a Hamiltonian can have two parts, a continuous part, and a discrete part.

Bound states are stationary states that correspond to energies in the discrete part of the spectrum, and are normalizable.

Scattering states are stationary states that correspond to energies in the continuous part of the spectrum, and are not normalizable (or are delta function normalizable).

The stationary states of a harmonic oscillator are all bound states. The finite square well has both scattering and bound stationary states.

 

[George Jones]

To back up my previous post, I have attached a couple of pages from the third edition of "Quantum Mechanics: Foundations and Applications" by Arno Bohm, which is an advanced, somewhat rigourous text (e.g., it treats rigged Hilbert spaces).

From the second of these pages:

But the spectrum $\left\{ E_\alpha \right\}$ of $H$ is the combination of a continuous spectrum ... and a discrete spectrum ... Physically the continuous spectrum corresponds to scattering states and the the discrete spectrum to bound states

I have several other standard quantum texts that say similar things.

 

[vanhees71]

I admit, I am not familiar with the definition of bound states as eigenstates and only eigenstates. If that is the case, then you're right: superpositions of eigenstates are not, in general, eigenstates. I always considered a state to be "bound" if it had a vanishing chance of escaping to infinity (in a more precise sense, this would mean a bound state is normalizable as $x,t\to\pm\infty$--this properly excludes free wavepackets). But I could very well be using an incorrect term.

Indeed the common terminology is that "bound states" are the normalizable eigenstates of the Hamiltonian of the system under consideration. Since they are normalizable (or, in the position representation, the corresponding wave function is square-integrable) they represent true pure states of the system and since they are eigenstates of the Hamiltonian the time-dependence is just a phase factor $\exp(-\mathrm{i} E t/\hbar)$, where $E$ is the eigenstate. The eigenvalues of the bound states are in the discrete part of the Hamiltonian's spectrum. There can also be "scattering states", which are generalized eigenstates (distributions) of the Hamiltonian with the spectral value in the continuous part of the spectrum. These you can "normalize only to a $\delta$ distribution". The corresponding wave functions behave like oscillating exponential functions for large distances, and thus describe scattering states of particles, i.e., far away from the scattering potential the particles behave nearly like free particles ("asymptotic free states").