Scattering and bound states

In summary, bound states (E < 0) always result in a solution of the Schrödinger equation with a discrete energy spectrum, while scattering states (E > 0) always result in a solution with a continuous energy spectrum. Bound states have sinusoidal time-independent wave functions, while scattering states have Gaussian-like wave functions. Additionally, bound states are always normalizable, while scattering states may not be. The reason for this is due to the limitation of possible frequencies/wavelengths for bound waves, resulting in a discrete energy spectrum, while unbound waves have a continuous spectrum due to the ability to go to infinity.
  • #1
Niles
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In all the possible potentials I have encountered so far, it seems that the bound states (i.e. E < [V(-infinity) and V(infinity)]) always results in a discrete spectrum of energies, whereas the scattering states (E > [V(-infinity) and V(infinity)]) always results in a continuous spectrum of energies.

I can't seem to find a logical explanation for this. If we use the anove defintion of bound and scattering states: The potential at plus/minus infinity of the harmonic oscillator is infinite, but so is the energy (for infinite n). But the harmonic oscillator has a bound spectrum.

I can't quite see this. I've taken the above from Griffith's, and sadly he never mentions whether "bound states = discrete spectrum" or not.
 
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  • #2
Ok, I can see that my question is poorly formulated. I'll refraise it:

First of all, a bound states is defined as the energy E < 0 and a scattering state is defined as E > 0. My questions on this topic are the following:

1) Will a bound state (i.e. E < 0) always result in a solution of the Schrödinger equation with a discrete energy spectrum?

2) Will a scattering state (i.e. E > 0) always result in a solution of the Schrödinger equation with a continuous energy spectrum?

In all the potentials I've encountered so far (harmonic oscillator, free particle, infinite square well and finite square well) it is so. But no where in the book (Griffiths) is an explanation of why. Can you guys enlighten me?
 
  • #3
1) Will a bound state (i.e. E < 0) always result in a solution of the Schrödinger equation with a discrete energy spectrum?

Yes. I think the easy way to see this is to note that every bound wave, classical or quantum, is limited to a discrete set of possible frequencies/ wave lengths. So if you span a cord, fix it on both ends, then pluck it, you will see you can not make it wave in every possible frequency.

The discrete wave spectrum translates in a discrete energy spectrum.

2) Will a scattering state (i.e. E > 0) always result in a solution of the Schrödinger equation with a continuous energy spectrum?

If the scattered particle is allowed to go to infinity, is not bound, then yes.
 
  • #4
Will a scattered particle always have an oscillating wave function? And similarly, will a bound particle always have an exponential wave function?

Again, these things I am taking from the examples in Griffith's QM book.

EDIT: And are bound states always normalizable, whereas scattering states are not?
 
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  • #5
bound states will always have sinusoidal time independent wave functions i think. and likewise I'm going to take a guess and say that an unbound scattering state will always have a guassian like time independent wave function.
 
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1. What is scattering in physics?

Scattering in physics refers to the phenomenon where particles interact with each other or with a medium, causing a change in their direction or energy. This can occur in various forms such as elastic scattering, inelastic scattering, and Raman scattering.

2. What are bound states in quantum mechanics?

Bound states in quantum mechanics are energy states of a particle that are confined to a specific region due to the presence of a potential barrier. This means that the particle is unable to escape the region and is considered to be "bound". Bound states are important in understanding the behavior of atoms, molecules, and other quantum systems.

3. How do scattering and bound states relate to each other?

Scattering and bound states are closely related as they both involve the interaction of particles with a potential barrier. In scattering, particles are scattered off the barrier, while in bound states, particles are confined within the barrier. Both phenomena can be described using similar mathematical models and principles in quantum mechanics.

4. What factors affect the scattering and bound states of particles?

The scattering and bound states of particles can be affected by various factors such as the strength and shape of the potential barrier, the energy and velocity of the particles, and the distance between the particles. Other factors such as the spin and charge of the particles can also play a role in determining their scattering behavior and bound states.

5. What are some real-life applications of scattering and bound states?

Scattering and bound states have many real-life applications, including in the fields of nuclear physics, material science, and medical imaging. For example, understanding the scattering behavior of particles in a material can help in developing new materials with specific properties. Bound states are also important in medical imaging techniques such as MRI, which uses the concept of nuclear bound states to produce images of the human body.

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