Bound states in 1D potentials

• luisgml_2000
In summary: It turns out that this is precisely what is needed for an attractive potential.The theorem asserts that any attractive 1D potential has at least one bound state. The bound state is a minimum energy state, and is specific to the potential. Other potentials with the same energy may have different bound States. In summary, the theorem asserts that every attractive 1D potential has at least one bound state, and that the bound state is specific to the potential.

luisgml_2000

Hello:

There is a well known theorem which asserts that every attractive 1D potential has at least one bound state; in addition, this theorem does not hold for the 2D or 3D cases. I've been looking for a proof in my textbooks on qm but I've been unable to find it. Can you help me out?

Thanks!

It has to do with the variational method. You can always come up with a wavefunction that is "everywhere inside the well" and "piecewise flat", (think cross-section of a muffin tin) , so that $$\left\langle \psi | H | \psi \right\rangle < 0$$. This puts an upper-bound on the ground state of the system.

I never really understood how the proof breaks down in higher dimensions. I think it is because in 2D and 3D, the Hamiltonian has a centrifugal barrier. Well, other than that, explicit counter-examples are known. I hope someone else fills in the gaps.

I'm guessing that the bare minimum case for an attractive potential would have to be the delta function in any number of dimensions. I know what the bound state looks like in 1D; in 2d or 3d if you could solve for the delta potential and show there are no bound states, then I think that would be pretty much it.

The variational method does not require the wavefunction be a solution of the schrodinger equation.

luisgml_2000 said:
Hello:
There is a well known theorem which asserts that every attractive 1D potential has at least one bound state; in addition, this theorem does not hold for the 2D or 3D cases.!
In 1D, the wave function can be finite at the origin, and can always decrease monotonically to zero at infinity. In 2D or 3D, the "equivalent 1D" wave function is $$u=\sqrt{r}\psi$$
for 2D or $$u=r\psi$$ for 3D. In either case, u must equal zero at the origin.
This requires a strong enough attraction to make u turn over at some point.

What are bound states in 1D potentials?

Bound states in 1D potentials refer to the quantum mechanical phenomenon where a particle is confined in a one-dimensional potential well, such as a potential barrier or a harmonic oscillator potential. In this case, the particle is unable to escape and its motion is restricted to a certain region.

What is the significance of bound states in 1D potentials?

Bound states in 1D potentials have significant implications in various fields of physics, such as solid-state physics, atomic and molecular physics, and quantum mechanics. They play a crucial role in understanding the behavior of particles in confined systems, leading to developments in technology and advancements in fundamental physics.

How are bound states in 1D potentials calculated?

The bound states in 1D potentials are calculated using the Schrödinger equation, which describes the quantum mechanical behavior of a particle in a potential well. The solutions to this equation give the possible energy levels and corresponding wavefunctions for the confined particle.

What are the conditions for bound states in 1D potentials to exist?

For bound states to exist in 1D potentials, the potential well must be deep enough to confine the particle, but also have a finite width to allow for the particle's wavefunction to exist. Additionally, the potential must be symmetric to allow for symmetric and anti-symmetric wavefunctions, which correspond to even and odd energy levels, respectively.

Can bound states in 1D potentials have multiple particles?

Yes, bound states in 1D potentials can have multiple particles, as long as the potential well is able to confine all the particles. In this case, the bound state solutions will be dependent on the interactions between the particles, leading to more complex energy levels and wavefunctions.