# Bound states as a solution of free particles?

• pellman
In summary, the conversation discusses the possibility of representing any well-behaved state as a superposition of free-particle momentum eigenstates by taking the Fourier transform. This has theoretical significance and provides a momentum space wave function, which is time-dependent in the case of a velocity dependent potential. It is also possible to use this method for perturbations to a free Hamiltonian in the Interaction Picture, which is useful for solving scattering problems.
pellman
It came to me just now that because we can always take the Fourier transform of a well-behaved function, this means we can think of any such state as a superposition of free-particle momentum eigenstates. E.g., the Hermite polynomial eigenfunctions of the harmonic oscillator. They have a Fourier transform (whatever it is) and can therefore be thought of as superpositions of functions e^(ipx) . These are free momentum eigenstates, even though individually they are not solutions to the harmonic oscillator Schrodinger equation.

From a math point of view, this is trivial. But does it have any theoretical significance?

Sure, this gives you the momentum space wave function.

pellman and mfb
Right. Of course. Thank you! The Fourier transform (momentum space wave function) will be time-dependent in this case. Unlike the free particle case. But it has the same physical meaning.

But does it still work if you have a velocity dependent potential? Classically, that changes the form of the canonical momentum. Mathematically you can still take the Fourier transform. I'm just wondering if the result in that case would have the same physical meaning: momentum space wave function.

In the case of small perturbations to a free Hamiltonian (even velocity-dependent ones), it makes sense to cast the perturbation term into this momentum basis, and compute the matrix elements of it between various free states. This is known as the Interaction Picture, and is a convenient way to set up a lot of scattering problems.

pellman

## 1. What are bound states and how are they related to free particles?

Bound states are a type of quantum state in which a particle is confined within a potential well. They are related to free particles in that they are also described by the Schrödinger equation, but with the addition of a potential term that causes the particle to be confined within a certain region.

## 2. How do bound states differ from unbound states?

Bound states have discrete energy levels and are confined within a potential well, while unbound states have continuous energy spectra and are not confined. Bound states also have a non-zero probability of being found within the potential well, while unbound states have a zero probability of being found within the potential well.

## 3. Can bound states exist in all types of potentials?

Yes, bound states can exist in all types of potentials, as long as the potential is finite and allows for confinement of the particle. This includes potentials such as the harmonic oscillator potential, the Coulomb potential, and the infinite square well potential.

## 4. How do bound states affect the behavior of particles?

Bound states have a significant impact on the behavior of particles, as they determine the energy levels and stability of the system. They also affect the probability of finding the particle in a certain region, as the wave function of a bound state is typically non-zero within the potential well and decays outside of the well.

## 5. What are some real-world applications of bound states?

Bound states have many practical applications, such as in the design of electronic devices, where the confinement of electrons in potential wells is utilized for various purposes. They are also important in understanding the properties of atoms and molecules, and in the study of nuclear physics and quantum computing.

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