# Particle in a well, can someone explain how this step works

• Theodore0101
In summary, we are trying to determine the energy for bounded states of a particle with mass m, under potential V(x) = ∞ when x<0, -V0 when 0<x<a, 0 when x>a. For part A, the relation for energy is given by tan(√(2ma^2(E+V0)/h^2)=-√(E+V0/-E). For part B, we need to find the values of V0 for which the particle has only one bounded state. This is achieved by choosing the cotangent function so that it only has one period in the corresponding interval -V0<E<0. This leads to the inequality pi^2*h^2/
Theodore0101
Homework Statement
A particle with mass m is effected by potential V(x) = ∞ when x<0, -V0 when 0<x<a, 0 when x>a
A) Set up a relation from which the energy for bounded states can de determined
B) For which values V0 does the particle have only one bounded state?
Relevant Equations
tan(√(2ma^2(E+V0)/h^2)=-√(E+V0/-E)
Homework Statement:: A particle with mass m is effected by potential V(x) = ∞ when x<0, -V0 when 0<x<a, 0 when x>a
A) Set up a relation from which the energy for bounded states can de determined
B) For which values V0 does the particle have only one bounded state?
Homework Equations:: tan(√(2ma^2(E+V0)/h^2)=-√(E+V0/-E)

Hi! I understand the most of this example problem, I have no problem with A), which the answer is the relation tan(√(2ma^2(E+V0)/h^2)=-√(E+V0/-E), and I mostly understand B), it is just one part in the end I can't follow. Choosing the tangens function so that it only has one period in the corresponding intervall -V0<E<0 and the equations only have one point that they meet in. Therefor we put that pi/2 < √(2ma^2(E+V0)/h^2) <3pi/2 since pi/2 to 3pi/2 is a period for a tangens function, but from there they say that the equation gives pi^2*h^2/8ma^2 < V0 < 9pi^2*h^2/8ma^2, but I don't follow this step. What happened to E, why are you allowed to take it away? Shouldn't it be pi^2*h^2/8ma^2 < V0 + E < 9pi^2*h^2/8ma^2 (instead of just V0)?

Thanks

Are you sure it is ##\tan## function only? Please check your solution. For that you don't need the restriction π/2 to 3π/2. The curve will always intersect the graph between 0 to π/2. See the graph below.
.

If you meant ##\cot## then the restriction applies. In that case simply note that since by defination of bound state ##-V_0 \leq E \leq 0##, ##E+V_0## is positive only if E=0 for then only inequalty is valid. See below.

In both the graphs I have changed variables for my convenience.

Last edited:

## 1. What is a particle in a well?

A particle in a well refers to a quantum mechanical system where a particle is confined within a potential well, meaning it is limited to a certain region of space.

## 2. How is the particle in a well system set up?

The particle in a well system is typically set up by placing a particle, such as an electron, in a potential well created by a physical barrier or a potential energy barrier.

## 3. What is the significance of studying particle in a well?

Studying the particle in a well system allows us to understand the behavior of particles at a quantum level and how they are affected by potential barriers. It also has practical applications in fields such as solid-state physics and quantum computing.

## 4. Can you explain how the particle in a well system works?

In the particle in a well system, the particle's motion is described by a wave function, which determines the probability of finding the particle at a specific location within the well. The potential barrier acts as a boundary for the particle, causing it to behave like a wave and exhibit unique quantum properties.

## 5. How does the particle in a well system relate to the Schrödinger equation?

The Schrödinger equation is used to describe the time evolution of a quantum system, including the particle in a well system. By solving the Schrödinger equation for the particle in a well, we can determine the allowed energy levels and corresponding wave functions for the particle.

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