# Wave Function for Delta Function Barrier with E<0

• Seriously
In summary, the conversation discusses the behavior of a particle incident from the left on a delta function barrier located at x=0 with potential V(x) = +a * delta(x). It is determined that there are no bound states and the wave function will exhibit a tunnelling effect, resulting in a lower energy wave after passing through the barrier. The question of what the wave function will look like with E<0 is raised, but it is uncertain due to the unusual nature of the potential step. The possibility of an error in the question is also considered.
Seriously
Given a delta function barrier located at x=0: V(x) = +a * delta(x)
If you have a particle incident from the left with E<0, what does the wave function look like??

I have trouble with this because I thought the particle energy needed to be greater than the minimum potential (E > Vmin) for you to get a solution.

I figure there are no bound states (only scattered states). But what does that mean for the wave function with E<0?

If E < V, you will get a tunelling effect (one of those things I will have to learn myself), but I think basically when you come out from the other end of the barrier, the energy of the wave is lower than the original wave. I'm not sure what the wave function $\Psi$ will look like though...

But here, Vmin=0!

In a usual potential step or potential barrier problem, E>Vmin, and the wave function inside the barier or walls is an atenuation of where E>V. Here however, E<V everywhere! So my intuition tells me that psi would be 0 everywhere in this situation.

Could it be that there is an error in the question? That it E>0 was meant rather than E<0?

Exactly! Vmin=0... and therefore E<0 always gives E<Vmin. So I don't understand.

I double checked the question. It specifically states E<0. It then asks if there are any bound states (answer=no), and wants me to draw the wave function.

I was told that E<0 just means that the energy is less than the minimum potential energy which can be drawn anywhere. But this makes no sense to me.

## 1. What is the significance of a "delta function barrier" in the context of wave functions?

A delta function barrier refers to a mathematical representation of a potential barrier that is infinitely narrow and infinitely high. It is often used in quantum mechanics to model a sudden change in the potential energy of a system.

## 2. How is the wave function affected by a delta function barrier when the energy is less than zero?

When the energy of a system is less than zero, the wave function experiences a discontinuity at the location of the delta function barrier. This means that the wave function abruptly changes from a non-zero value to zero at the barrier.

## 3. Can a particle with energy less than zero pass through a delta function barrier?

No, a particle with energy less than zero cannot pass through a delta function barrier. This is because the probability of finding the particle in the region of the barrier is zero, as the wave function is zero at that point.

## 4. How does the transmission coefficient for a delta function barrier with E<0 compare to that of a barrier with E>0?

The transmission coefficient for a delta function barrier with E<0 is zero, meaning that there is no possibility of the particle passing through the barrier. In contrast, the transmission coefficient for a barrier with E>0 is non-zero, indicating that there is a chance for the particle to pass through the barrier.

## 5. Can the wave function for a delta function barrier with E<0 be normalized?

Yes, the wave function for a delta function barrier with E<0 can be normalized. However, this normalization only applies to the regions where the energy of the particle is greater than zero, as the wave function is zero at the location of the barrier.

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