Understanding Potential Wells: Finite vs. Unbounded

[Feynmanfan]

I'm confused when I study potential wells. I understand that when E<V0 there are eigenstates and only certain values of energy. I can express this condition with an equation and eigenfunctions.

However, if E>V0, it's not bounded and all values of energy are possible. This, I don't understand. It is true that the sprectrum can be discrete or continuous but how would you explain this with a example, say the finite potential well.

Thanks.

 

[dextercioby]

The spectral problem admits solutions in any circumstances.It's just that some are nicer and require only Hilbert spaces and bounded operators and others rigged Hilbert spaces and unbounded operators.

This problem (of potential wells) is dealt with in a very mathematical way in the first volume of Galindo & Pascual:"Quantum Mechanics",Springer Verlag,1990.

Daniel.

 

[R0man]

Potential wells are an important concept in quantum mechanics, as they help us understand the behavior of particles in confined spaces. A potential well is essentially a region in space where the potential energy is lower than the surrounding areas. This creates a "well" where particles can be trapped and their energy levels are restricted.

There are two types of potential wells: finite and unbounded. In a finite potential well, the energy of the particle is limited by the potential barrier, represented by V0. This means that the particle can only have certain discrete energy levels, known as eigenstates. These eigenstates are represented by specific energy values and corresponding wave functions, or eigenfunctions.

On the other hand, in an unbounded potential well, the particle is not limited by a potential barrier and can have any energy level. This is because the potential energy is not restricted, allowing the particle to have continuous energy values. In this case, the energy spectrum is continuous rather than discrete.

To better understand this concept, let's consider a finite potential well, such as a particle trapped in a box. The particle's energy is limited by the potential barrier, and it can only exist in certain energy states, represented by the eigenstates. These energy states are like steps on a staircase, with each step representing a specific energy level. The particle can only exist on one of these steps at a time, and it cannot have energy levels in between them.

On the other hand, in an unbounded potential well, such as a particle in free space, there is no potential barrier limiting the particle's energy. This means that the particle can have any energy level, represented by a continuous spectrum. In this case, the energy levels are not restricted and the particle can move freely, without being confined to specific energy states.

In summary, the difference between finite and unbounded potential wells lies in the restriction of energy levels. In a finite potential well, the energy levels are discrete and limited by a potential barrier, while in an unbounded potential well, the energy levels are continuous and not restricted by a potential barrier. I hope this helps clarify the concept of potential wells for you.