# Simple question: half-infinite square well

• hootna
In summary, the conversation is about considering a square well with finite boundaries on one side and infinite boundaries on the other. The wave function inside the well is described by three equations with four variables, k and l being known. The issue at hand is the boundary conditions, where there are 3 equations to solve for 4 variables. The context involves calculating the probability of particles being reflected back from the left region of the well. The suggestion is to put the infinite wall at x = 0 and consider the wave function inside the well by making a mirror image and drawing the first few wave functions.
hootna
Hey, I'm considering a square well which is finite on one side (left) and infinite on the other (right).

So the wave function is:

Left-most region: Ae^(ikx) + Be^(-ikx)
Inside the well: Csin(lx) + D(cos(lx))
Right-most region: 0

where k and l are known.

The problem is with boundary conditions: On the left, we have both (continuity and continuity of derivative), but on the right we only have one (continuity). So we have 3 equations to solve for 4 variables... what gives?

If the context helps, I'm trying to figure out the probability that particles entering from the left in region 1 will be reflected back (i.e., the ratio of the probability flux in region 1 traveling to the left to the incident flux in region 1 traveling to the right).

Any help would be greatly appreciated.

Thanks,
Thomas.

Put the infinite wall at x = 0. Now, what can you say about the wave function inside the well?

Think about this, make a mirror image of the well and think about the finite well. Draw the first few wave functions and really think hard about what you are drawing and the answer will pop out at you.

## 1. What is a half-infinite square well?

A half-infinite square well is a theoretical model in quantum mechanics that describes a potential energy barrier that extends infinitely in one direction and is finite in the other direction. It is often used to study the behavior of a particle confined to a limited space.

## 2. What is the significance of a half-infinite square well?

The half-infinite square well is a simple and useful model for understanding the quantum behavior of particles in confined spaces. It helps us understand the concept of energy levels and the wave-like nature of particles.

## 3. How is the half-infinite square well different from a finite square well?

The main difference between the two is that a finite square well has both boundaries finite, while a half-infinite square well has one boundary extending to infinity. This leads to some differences in the energy levels and wavefunctions of the two systems.

## 4. How do energy levels change in a half-infinite square well?

The energy levels in a half-infinite square well are quantized, meaning they can only take on certain discrete values. As the depth of the well increases, the number of energy levels decreases and their spacing increases.

## 5. What is the physical significance of a half-infinite square well?

A half-infinite square well can represent various physical systems, such as a particle trapped in a potential well or a particle moving in a confined space. It allows us to study the behavior of particles in these systems and gain a better understanding of quantum mechanics.

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