Find Range of V0 for Particle in Potential

In summary, the problem involves a particle in a potential with three different regions and the task is to determine the range of values for V0 in terms of a and m for which there is only one bound state. The solution involves using a general wave function and applying continuity conditions at the boundaries. One can then determine the number of solutions by analyzing the equation q=-kctg(ka) and considering the limiting cases for a and m.
  • #1
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Homework Statement


A particle is in the following potential:
V(x)=infinity for x<0; -V0 for 0<x<a; and 0 for x>a
Given that there's only one bound state I am asked to determine the range of values for V0 in terms of the width a and the particle's mass m.[/B]

Homework Equations

The Attempt at a Solution


For -V0<E<0 I chose the following general solution for the wave function:
ψ(x)=Asin(kx) 0<x<a; Bexp(-qx) x>0
where k=√(2m(E+V0)/ħ and q=√(2m|E|)/ħ
By demanding continuity at x=a for both wave functions and their derivatives I obtained the following solution:
q=-kctg(ka)
How may I proceed? I'd appreciate some guidance.
[/B]
 
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  • #2
How do you get the number of solutions from your last equation?
The borders for "a" and "m" are exactly the limiting cases for 1 and 2 solutions.
 

What is the "Find Range of V0 for Particle in Potential" problem?

The "Find Range of V0 for Particle in Potential" problem is a physics problem that involves finding the range of initial potential energy (V0) values for a particle to reach a certain point in a potential energy field. It is commonly encountered in quantum mechanics and classical mechanics.

What is a potential energy field?

A potential energy field is a region of space where a particle can experience a force due to its position. It is a concept used in physics to describe the energy that a particle possesses due to its position in a given field.

How is the range of V0 determined?

The range of V0 is determined by using the equations of motion and the principle of conservation of energy. By setting the initial and final positions and solving for the initial potential energy (V0), we can determine the range of values for V0 that will allow the particle to reach the desired final position.

What factors can affect the range of V0?

The range of V0 can be affected by several factors, including the mass of the particle, the shape of the potential energy field, and any external forces acting on the particle. Additionally, the chosen final position can also affect the range of V0.

Why is finding the range of V0 important?

Finding the range of V0 is important because it allows us to determine the initial potential energy required for a particle to reach a desired position in a potential energy field. This information is crucial in understanding the behavior and motion of particles in various physical systems.

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