# Potential barrier. Schroedinger equation.

• LagrangeEuler
In summary, the conversation discusses the application of the Schroedinger equation for a potential barrier with the condition that the barrier height is equal to the particle's energy. The first and third regions have free particles with wave functions of ##\psi_1(x)=Ae^{ikx}+Be^{-ikx}## and ##\psi_3(x)=Ce^{ikx}## respectively. In the second region, the wave function is constant and the boundary conditions are used to determine the coefficients. The discussion also touches on the concept of bond states, which have negative energy and result in exponentially decaying wave functions that do not propagate to infinity.
LagrangeEuler

## Homework Statement

Schroedinger equation for potential barrier.
What if ##V_0=E##
First region. Particles are free.
##\psi_1(x)=Ae^{ikx}+Be^{-ikx}##
In third region
##\psi_3(x)=Ce^{ikx}##

## Homework Equations

##\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(V_0-E)\psi=0##
where ##V_0## is height of barrier.
For region II

## The Attempt at a Solution

In second region
##\frac{d^2 \psi}{dx^2}=0##
from that
##\frac{d\psi}{dx}=C_1##
##\psi(x)=C_1x+C_2##
Boundary condition
##A+B=C_2##
##C_1a+C_2=Ce^{ika}##
##ikA-ikB=C_1##
##C_1=ikCe^{ika}##
System 4x4
Is this correct?
Could you tell me in this case do I have bond state?

Last edited:
Yes, it seems correct. Bond states have negative energy.

Bound states have negative energy? Can you explain me this. In case of this problem.

The oscillatory solutions of regions 1 and 3 happen because energy is positive and the wave is free to propagate to infinity. (unbounded particle). If the energy is negative you get a exponentially decaying wave function and the wave does not propagate to infinity (bounded particle).

I would like to clarify a few things about the content provided and offer a response to the question posed.

Firstly, the content provided does not specify the potential barrier, so it is difficult to accurately discuss the solution to the Schroedinger equation. However, I will assume that the potential barrier is a step function with height V_0 and extends from x=0 to x=a.

In this case, the solution to the Schroedinger equation in the first region (x<0) is given by ##\psi_1(x)=Ae^{ikx}+Be^{-ikx}##, as provided in the content. This solution represents a plane wave traveling in the positive and negative x-directions. However, the content does not specify the energy of the particles in this region, so I cannot comment on whether they are truly free or not.

In the third region (x>a), the solution provided is ##\psi_3(x)=Ce^{ikx}##. This represents a plane wave traveling in the positive x-direction.

Now, the content asks what happens if V_0=E, which implies that the energy of the particles is equal to the height of the potential barrier. In this case, the second region (0<x<a) becomes a potential well, and the solution to the Schroedinger equation in this region is given by ##\psi_2(x)=C_1\cos(kx)+C_2\sin(kx)##. This solution represents a standing wave with nodes at x=0 and x=a.

The boundary conditions provided in the content are not sufficient to determine the coefficients A, B, C, C_1, and C_2. However, assuming that we have a particle with energy E>V_0, we can use the continuity of the wavefunction at x=0 and x=a to determine the coefficients. This results in four equations and four unknowns, which can be solved to find the values of A, B, C, and C_2.

In this case, we do not have a bound state, as the particle can tunnel through the potential barrier if it has enough energy. The solution in the second region is not a standing wave, but rather a combination of incoming and outgoing waves.

In conclusion, the provided solution is not entirely correct and more information is needed to accurately solve the Schroedinger equation for a potential barrier.

## 1. What is a potential barrier?

A potential barrier is a region in which the potential energy of a particle or system of particles is higher than the surrounding areas. This can prevent the particles from moving freely through the barrier.

## 2. What is the significance of the potential barrier in the context of the Schrödinger equation?

The potential barrier plays a crucial role in the Schrödinger equation, which is a mathematical equation used to describe the behavior of quantum particles. In the equation, the potential barrier is represented by a potential energy function, and it affects the probability of a particle being found in a certain region.

## 3. How does the potential barrier influence particle behavior?

The potential barrier can act as an energy barrier that particles must overcome in order to move through it. This can lead to interesting phenomena such as tunneling, where particles can pass through the barrier even though they do not have enough energy to overcome it.

## 4. Can the potential barrier be manipulated?

Yes, the potential barrier can be manipulated by changing the potential energy function in the Schrödinger equation. This can be done, for example, by applying an external electric field to the system.

## 5. How does the potential barrier relate to the concept of energy levels in quantum mechanics?

In quantum mechanics, energy levels refer to the discrete values of energy that a particle can have. The potential barrier can affect these energy levels by either raising or lowering the energy of the particles as they interact with the barrier. This can have significant implications for the behavior of the particles in the system.

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