Particle in a Box/Wave function decomposition.

In summary, the problem involves proving that the decomposition of the normalized wave function into three parts yields a specific cn value. The wave function is already proven to be normalized and the integral should be added for the three parts instead of being integrated from -infinity to infinity.
  • #1
spikethekitty
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Homework Statement


There is a particle in an infinite square well and the normalized wave function is divided into 3 parts. Psi = A between 0< x < L/3, B where L/3< x < 2L/3, and C from 2L/3 < x < L. I have to prove that the decomposition into psin yields a specific cn.


Homework Equations


cn= Integral psi*Psi dx.

psin= Sqrt(2 / L) Sin(n pi x / L)

The Attempt at a Solution



I have already proven that the wave function is normalized. The only thing i am confused about is what i do about the integral. Since Psi is in three parts, I am pretty sure that I integrate A times psin in terms of x from 0 to L/3, instead of -infinity to infinity, and likewise for B and C. What i don't know is what to do after that. Do i add the three answers together or multiply them? or am i on the complete wrong track already?
 
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  • #2
You must add the integrals, basic calculus:

(Integral from a to c) = (Integral from a to b) + (Integral from b to c)

If you didn't use this, how did you prove the function was normalized?
 

FAQ: Particle in a Box/Wave function decomposition.

What is a Particle in a Box?

A Particle in a Box refers to a theoretical model in quantum mechanics where a particle is confined to a finite space, typically a one-dimensional box. This model is used to understand the behavior and properties of particles in quantum systems.

What is the significance of this model in quantum mechanics?

The Particle in a Box model is significant because it helps us understand the quantization of energy in quantum systems. It also serves as a simplified system to study the behavior of particles in confined spaces.

What is the Wave function decomposition in this model?

The Wave function decomposition in the Particle in a Box model refers to the process of breaking down the wave function, which describes the probability of finding a particle in a certain location, into its individual components. This allows for a better understanding of the behavior of the particle in the box.

How is the wave function decomposed in the Particle in a Box model?

The wave function is typically decomposed into a series of sine or cosine functions, known as the Fourier series. These functions represent the various energy states that the particle can occupy in the box. The coefficients of these functions determine the probability of the particle being in a certain energy state.

What are some real-life applications of the Particle in a Box model?

The Particle in a Box model has applications in various fields, such as quantum computing, solid-state physics, and spectroscopy. It is also used to understand the behavior of electrons in atoms and molecules, which has implications in chemistry and materials science.

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