Finding the Boundary Conditions of a Potential Well

In summary, the author is trying to solve for the boundary conditions of a potential well of V_{0} but is having trouble because he is missing two pieces of information, q and alpha. He is sure that once he understands these two pieces of information, the solution will be clear.
  • #1
jmtome2
69
0

Homework Statement



Show that the conditions for a bound state, Eqn1 and Eqn2, can be obtained by requiring the vanquishing of the denominators in Eqn3 at k=i[tex]\kappa[/tex]. Can you give the argument for why this is not an accident?


Homework Equations



Eqn1: [tex]\alpha[/tex]=q*tan(qa)
Eqn2: [tex]\alpha[/tex]=-q*cot(qa)

Eqn3:

R= i*e[tex]^{-2ika}[/tex][tex]\frac{(q^{2}-k^{2})*sin(2qa)}{2kq*cos(2qa)-i(q^{2}+k^{2})*sin2qa}[/tex]
T=e[tex]^{-2ika}[/tex][tex]\frac{2kq}{2kq*cos(2qa)-i(q^{2}+k^{2})*sin2qa}[/tex]

The problem concerns a potential well of V[tex]_{0}[/tex] from -a to a.
I believe that k[tex]^{2}[/tex]=[tex]\frac{2mE}{h(bar)^{2}}[/tex] and that [tex]\kappa[/tex][tex]^{2}[/tex]=-[tex]\frac{2m(E-V_{0})}{h(bar)^{2}}[/tex]

I also know that in order to reach these boundaries through another method in the book, they let -[tex]\alpha[/tex][tex]^{2}[/tex]=[tex]\frac{2mE}{h(bar)^{2}}[/tex]...


The Attempt at a Solution


I removed the demoninators from Eqn3 and tried to substitute k=[tex]\sqrt{\frac{2mE}{h(bar)^{2}}}[/tex] into Eqn3... but not only was this result messy but I couldn't get the boundary conditions, Eqn1 and Eqn2 to come out. Help?
 
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  • #2
I'm not 100% clear on your problem statement...First, what is [itex]q[/itex]? Second are these the equations you meant to write:

[tex]R= ie^{-2ika}\frac{(q^{2}-k^{2})\sin(2qa)}{2kq\cos(2qa)-i(q^{2}+k^{2})\sin(2qa)}[/tex]

[tex]T=e^{-2ika}\frac{2kq}{2kq\cos(2qa)-i(q^{2}+k^{2})\sin(2qa)}[/tex]

[tex]k=\left(\frac{2mE}{\hbar^2}\right)^{1/2}[/tex]

[tex]\kappa=\left(\frac{2m(V_0-E)}{\hbar^2}\right)^{1/2}[/tex]

[tex]\alpha=\left(-\frac{2mE}{\hbar^2}\right)^{1/2}[/tex]

?
 
  • #3
[tex]
q=\left(\frac{2m(E+V_0)}{\hbar^2}\right)^{1/2}
[/tex]
 
  • #4
Okay, let's look at your eqn 3..."the vanquishing of the denominators" is just a fancy way of saying that [itex]2kq\cos(2qa)-i(q^{2}+k^{2})\sin(2qa)=0[/itex]...so [itex]\tan(2qa)=[/itex]___?
 
  • #5
[tex]=\left(\frac{sin(2qa)}{cos(2qa)}\right)?[/tex]
 
  • #6
Well of course, but what is that equal to when [itex]2kq\cos(2qa)-i(q^{2}+k^{2})\sin(2qa)=0[/itex]?
 
  • #7
Alright, I'll take a closer look at this at a later time to make sure that I understand it but I think the missing link is that I initially overlooked q and [tex]\alpha[/tex]. I'm sure that the solution will be clear once I make it through the alegbra, thanks
 

1. What is a potential well?

A potential well is a concept in physics that refers to a localized region in space where the potential energy of a particle is lower compared to its surroundings. This results in the particle being confined to the well and having a stable equilibrium position.

2. How do you find the boundary conditions of a potential well?

The boundary conditions of a potential well can be found by solving the Schrödinger equation for the system. This involves setting up the appropriate potential function and applying the appropriate boundary conditions, such as continuity of the wavefunction and its derivative at the boundaries of the well.

3. What factors affect the boundary conditions of a potential well?

The boundary conditions of a potential well are affected by the shape and depth of the potential well, as well as the mass and energy of the particle. These factors determine the behavior and stability of the particle within the well.

4. How do boundary conditions impact the behavior of a particle in a potential well?

The boundary conditions of a potential well play a crucial role in determining the energy levels and allowed wavefunctions of a particle within the well. They also affect the probability of finding the particle in different regions of the well and its overall stability.

5. How are the boundary conditions of a potential well used in practical applications?

The concept of potential wells and their boundary conditions is used in various fields such as quantum mechanics, solid state physics, and nuclear physics. They help in understanding the behavior of particles in confined environments and are used in the design of devices such as transistors and quantum dots.

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