Solving Infinite Square Well: Eigen Functions & Solutions

In summary, the conversation discusses the derivation of eigen functions for an infinite square well, with one person finding them to be in agreement with Wikipedia while another person's course notes present solutions for a well between -a and +a. The confusion is cleared up by explaining that the solutions on [-L/2, L/2] can be obtained by replacing x with x + L/2. This results in cosine solutions for n=1,3,5... and sine solutions for n=2,4,6... The conversation also mentions graphing the ground state solution and how the two cases are identical except for the position of the origin in relation to the walls of the well.
  • #1
mr_whisk
7
0
Hi all,

Sorry if this question is not very challenging, but I am driving myself to confusion...

I happily derived the eigen functions for an infinite square well spanning from 0 to L and found them to be:

20276c56b7727f9948f8a7cacfc05d52.png


...in agreement with wikipedia.

However my course notes derive solutions for an inf sq well between -a and +a.

Surely L=2a, no?

In the notes n=1,3,5... produces cosine solutions and n=2,4,6... produce sine solutions.

Why is there now cosine solutions?

Please help, I'm going crazy :)
 
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  • #2
You can get the solutions on [-L/2 , L/2] from the solutions on [0, L] by replacing x by x + L/2 (i.e. moving the solutions to the left by L/2). For example, sin(πx/L) becomes sin((π/L)(x + L/2)) = sin(πx/L + π/2) = cos(πx/L).
 
  • #3
Thankyou dx,

That's brilliant, it all makes sense :)
 
  • #4
Draw a graph of the ground state (n = 1) solution for both cases. You'll see that the two graphs are identical except for the origin (x = 0) being in different places relative to the walls of the well. Likewise for n = 2, etc.
 

1. What is an infinite square well?

An infinite square well is a hypothetical potential energy function used in quantum mechanics to model the behavior of a particle confined to a certain region. It is a potential well with infinitely high walls and a finite depth, meaning that the particle cannot escape from the well.

2. What are eigenfunctions in the context of the infinite square well?

Eigenfunctions, or stationary states, are solutions to the Schrödinger equation that describe the allowed energy states of a particle in the infinite square well. They are characterized by a discrete set of energy levels and correspond to specific wavelengths and frequencies.

3. How are eigenfunctions related to the probability density function?

The probability density function, represented by the wave function, describes the likelihood of finding a particle in a certain location in the infinite square well. The square of the eigenfunctions give the probability density function and represent the probability of finding the particle in a specific energy state.

4. What is the significance of the boundary conditions in solving the infinite square well?

The boundary conditions, which state that the wave function must be continuous and have a finite slope at the boundaries of the well, help to determine the allowed energy levels and corresponding eigenfunctions of the particle. These conditions ensure that the wave function remains finite and physically meaningful within the well.

5. How are the energy levels and eigenfunctions of the infinite square well affected by the width and depth of the well?

The energy levels and eigenfunctions are directly related to the width and depth of the well. As the well becomes wider or deeper, the energy levels become more closely spaced and the eigenfunctions become more complex. Changing the dimensions of the well can also result in the appearance of degenerate energy levels, where multiple eigenfunctions correspond to the same energy level.

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