Rectangular Potential Barrier Boundary Conditions with E=V

In summary, the conversation discusses the calculation of transmission and reflection coefficients for a rectangular finite potential barrier for a particle with energy equal to the height of the barrier. The equations from the Wikipedia page are referenced, but there is confusion about the disappearance of an imaginary number from the exponentials attached to the Cl term. The conversation ends with an agreement that the term should not become real and a suggestion to start the barrier at x=0 instead of (-a,a) for easier boundary condition application.
  • #1
GoliathPSU
2
0

Homework Statement


I am trying to calculate the transmission and reflection coefficients for rectangular finite potential barrier between (-a, a) for a particle of mass m with energy equal to the height of the barrier (E = V0 > 0).

Homework Equations


http://en.wikipedia.org/wiki/Rectangular_potential_barrier#E_.3D_V0

The Attempt at a Solution


I understand the general solutions for the potential region are linear, and am trying to match the boundary conditions to evaluate the coefficients, but I am having trouble understanding why the Wikipedia page linked above has the imaginary number disappear from the exponentials attached to the Cl term. Instead there is just a + sign there. Is this just a mistake on the page? As far as I can tell, there is no reason the term should suddenly become real.
 
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  • #2
I agree with you. No one notices, because they let Cl=0 for T and R .

By the way, usually the barrier is from x=0 to x=a to make applying the boundary conditions a little easier.
 

1. What is a rectangular potential barrier boundary condition with E=V?

A rectangular potential barrier boundary condition with E=V is a scenario in quantum mechanics where a particle encounters a potential barrier with a constant energy (E) equal to the height of the potential barrier (V). In this scenario, the particle has a finite probability of crossing the barrier, even though its energy is less than the potential barrier's height.

2. How is the behavior of particles affected by rectangular potential barriers with E=V?

The behavior of particles encountering rectangular potential barriers with E=V is governed by the laws of quantum mechanics. According to these laws, particles have a finite probability of penetrating the barrier, even if their energy is less than the barrier's height. This is known as quantum tunneling and is a unique phenomenon that is only observed at the quantum scale.

3. What are some real-world applications of rectangular potential barrier boundary conditions with E=V?

Rectangular potential barrier boundary conditions with E=V have several real-world applications, including in the development of electronic devices like transistors and diodes. They are also used in nuclear physics to explain the behavior of particles in radioactive decay and in astrophysics to understand the behavior of particles in stars.

4. Why is the concept of rectangular potential barrier boundary conditions with E=V important in quantum mechanics?

The concept of rectangular potential barrier boundary conditions with E=V is essential in quantum mechanics because it helps explain the behavior of particles at the quantum level. It also plays a crucial role in understanding and developing various technologies that rely on quantum phenomena, such as quantum computing and communication.

5. How can the behavior of particles in rectangular potential barriers with E=V be mathematically described?

The behavior of particles in rectangular potential barriers with E=V can be mathematically described using the Schrödinger equation, which is a fundamental equation in quantum mechanics. This equation takes into account the particle's wave function, energy, and the potential barrier's height to predict the probability of the particle crossing the barrier.

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