# Finite Square Well, Ψ[SUB]III[/SUB] const related too Ψ[SUB]II[/SUB]?

• The Floating Brain
In summary, the conversation is about solving for the probability distribution in a system with two ledges and a cavern using Griffith's Modern Physics 2nd edition chapter 5. The speaker is confused about why the region I distribution does not equal the region II distribution at 0 and is seeking clarification on how to solve for G and why the actions on one side affect the other. They mention using the equation -αGe-α(L) = (αC/Qs)sin(Q(L)) + C/cos(Q(L)) and using ΨI(0) = ΨII(0) to solve for C = QA/α.
The Floating Brain
Homework Statement
There are two ledges with of U potential energy, and a cavern in between them of 0 potential energy. Such that

U(x) = { x <= 0, U, x > 0, x >= L, U }

Find the probability distribution for all 3 regions
Relevant Equations
(ħ[SUP]2[/SUP]/2m)((d[SUP]2[/SUP]Ψ)/(dx[SUP]2[/SUP]) + U(x)v = EΨ

∫ Ψ * Ψ dx = 1
I'm following Griffith's Modern Physics 2nd edition chapter 5.

I got to the part where we make ΨI(0) = ΨII(0) I get that

αCeα(0) = QAsin(Q(0)) - QBsin(Q(0)) => C = QA/α

But when I try to graph it, the region I distribution doesn't seem to equal the region II distribution at 0.

The book goes on on to insert C into an equation to solve for G (-αGe-α(L) for the other side of the well), but I don't understand why it does this.

-αGe-α(L) = (αC/Qs)sin(Q(L)) + C/cos(Q(L))Why does what happens on one side affect what happens on the other?

Delta2
The Floating Brain said:
Homework Statement:: There are two ledges with of U potential energy, and a cavern in between them of 0 potential energy. Such that

U(x) = { x <= 0, U, x > 0, x >= L, U }

Find the probability distribution for all 3 regions
Relevant Equations::2/2m)((d2Ψ)/(dx2) + U(x)v = EΨ∫ Ψ * Ψ dx = 1

I'm following Griffith's Modern Physics 2nd edition chapter 5.

I got to the part where we make ΨI(0) = ΨII(0) I get that

αCeα(0) = QAsin(Q(0)) - QBsin(Q(0)) => C = QA/α

But when I try to graph it, the region I distribution doesn't seem to equal the region II distribution at 0.
Your graph doesn't correspond to your work above. Back up a bit and tell us what you're using for ##\psi_I## and ##\psi_{II}##. And how did you get rid of ##B## to solve for ##C## from one equation?

## 1. What is a finite square well potential?

A finite square well potential is a model used in quantum mechanics to describe the behavior of a particle in a confined region with a finite depth. It is represented by a potential energy function that is constant within a certain range and then abruptly drops to zero outside of that range.

## 2. How is Ψ[SUB]III[/SUB] const related to Ψ[SUB]II[/SUB] in a finite square well?

In a finite square well potential, Ψ[SUB]III[/SUB] const is the wave function of the particle in the region outside of the well, where the potential energy is zero. Ψ[SUB]II[/SUB] is the wave function inside the well, where the potential energy is constant. The two wave functions are related through the continuity of the wave function and its derivative at the boundaries of the well.

## 3. What is the significance of Ψ[SUB]III[/SUB] const in a finite square well?

Ψ[SUB]III[/SUB] const represents the probability amplitude of finding the particle outside of the finite square well. It is important in determining the overall behavior of the particle in the system and can give insight into its energy levels and allowed states.

## 4. How does the depth of the finite square well potential affect Ψ[SUB]III[/SUB] const?

The depth of the finite square well potential affects Ψ[SUB]III[/SUB] const by changing the overall behavior and energy levels of the particle. A deeper well will result in a larger potential energy barrier for the particle to overcome, leading to a different wave function and probability amplitude for Ψ[SUB]III[/SUB] const.

## 5. Can Ψ[SUB]III[/SUB] const ever equal Ψ[SUB]II[/SUB] in a finite square well?

No, Ψ[SUB]III[/SUB] const and Ψ[SUB]II[/SUB] cannot be equal in a finite square well potential. This is because Ψ[SUB]III[/SUB] const represents the wave function outside of the well, where the potential energy is zero, while Ψ[SUB]II[/SUB] represents the wave function inside the well, where the potential energy is constant. These two regions have different potential energy values and therefore, the wave functions cannot be equal.

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