# Solving Potential Barrier Homework Problem

• Feldoh
Your name]In summary, the problem involves a particle with energy E = -\hbar ^2 q^2/2m, coming in from negative infinity with amplitude 1 and encountering a potential energy function V(x) that is not continuous at x = ±a. The problem is broken into two cases, depending on if E > V or E < V in region 2. The general solutions for the wave function in each region are ψ(x) = exp(iqx) + A exp(-iqx) and ψ(x) = B exp(ikx) + C exp(-ikx) respectively. Applying the continuity condition at x = -a and x = a, we can find the values of A,
Feldoh

## Homework Statement

Consider the diagram described by: http://filer.case.edu/pal25/well.jpg

If a particle with $$E = \frac{-\hbar^2 q^2}{2m}$$ comes in from negative infinity with amplitude 1, what is the wave function for negative x?

Oh and V(x) < -a and > a = 0

## The Attempt at a Solution

I think the problem needs to be broken into two cases, depending on if E > V or E < V in region 2.

If E > V then...

Region I:
$$\frac{\partial ^2 \psi}{\partial{x}^2} = -q^2 \psi$$

Which gives the solution:

$$\psi (x) = exp(iqx) + A exp(-iqx)$$

So we have the wave equation be we still need to determine the coefficient A, so we have to solve for the other two regions...

Region II:
$$\frac{\partial ^2 \psi}{\partial{x}^2} = -\frac{2m}{\hbar ^2}(E-V_0) \psi$$

Which gives the solution:

$$\psi (x) = B exp(ikx) + C exp(-ikx)$$ where $$k^2 = \frac{2m}{\hbar ^2}(E-V_0)$$

Region III:

$$\psi (x) = D exp(iqx)$$

-----------------------

So I guess I need to do continuity conditions for these equations? I have a feeling this is going to be really tedious and that there is a different way?

Last edited by a moderator:

You are correct in thinking that the problem needs to be broken into two cases, depending on if E > V or E < V in region 2. This is because the potential energy function is not continuous at x = ±a.

For the case of E > V, we can use the general solution for region I that you have correctly stated:

ψ(x) = exp(iqx) + A exp(-iqx)

In region II, we have the potential energy function V(x) = 0, so we can use the general solution for that region:

ψ(x) = B exp(ikx) + C exp(-ikx)

where k^2 = 2m(E-V_0)/\hbar^2.

Now, we need to apply the continuity condition at x = -a and x = a. This means that the wave function and its derivative must be continuous at these points. This gives us two equations:

ψ(-a) = exp(-iaq) + A exp(iaq) = B exp(-ika) + C exp(ika)

and

ψ'(-a) = iq exp(-iaq) - iqA exp(iaq) = ikB exp(-ika) - ikC exp(ika)

Similarly, at x = a we have:

ψ(a) = exp(iaq) + A exp(-iaq) = B exp(ika) + C exp(-ika)

and

ψ'(a) = iq exp(iaq) - iqA exp(-iaq) = ikB exp(ika) - ikC exp(-ika)

Now, we can solve these four equations to find the values of A, B, C, and k. Once we have these values, we can use them to write the wave function for negative x as:

ψ(x) = exp(iqx) + A exp(-iqx)

I hope this helps. Let me know if you have any further questions or if you need any clarification.

## 1. How do I identify a potential barrier in a homework problem?

To identify a potential barrier in a homework problem, you should look for a sudden change or discontinuity in the problem. This could be in the form of a sudden increase or decrease in a value, a sudden change in direction, or a sudden change in the behavior of a system. These are all indications of a potential barrier that needs to be solved.

## 2. What is the difference between a potential barrier and a potential well?

A potential barrier is an energy barrier that prevents particles from crossing it, while a potential well is a region where particles are confined and have lower energy. In other words, a potential barrier is a barrier that particles need to overcome to move from one side to the other, while a potential well is a region where particles are trapped and cannot escape.

## 3. How do I solve a potential barrier homework problem?

To solve a potential barrier homework problem, you will need to use mathematical equations and principles such as the Schrödinger equation, the wave function, and boundary conditions. You will also need to understand the physical properties of the barrier, such as its height and width, to determine how particles will behave when encountering it. It is important to carefully analyze the problem and use the appropriate equations and principles to solve it correctly.

## 4. What are the key factors that affect the behavior of particles in a potential barrier?

The key factors that affect the behavior of particles in a potential barrier are the height and width of the barrier, the energy of the particles, and the potential energy function of the barrier. These factors determine whether particles will be able to pass through the barrier or be reflected back, as well as the probability of particles being found in certain locations.

## 5. Can a potential barrier be overcome?

Yes, a potential barrier can be overcome if particles have enough energy to surpass it. This is known as tunneling, where particles have a small probability of crossing the barrier even if their energy is lower than the barrier's height. However, the probability of tunneling decreases as the barrier height increases, making it more difficult for particles to overcome the barrier.

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