Wavevector in infinite square well

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Discussion Overview

The discussion revolves around the wavevector in the context of an infinite square well, specifically addressing the conditions under which different expressions for the wavevector (k) are used. Participants explore the implications of boundary conditions and the definitions of the well's width.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant notes that the wavevector k can be expressed as k = nπ / L, while another mentions k = 2nπ / L, questioning when to use each expression.
  • Another participant suggests that the correct k is determined by the boundary conditions of the wavefunction, which must vanish at the boundaries of the well.
  • A different viewpoint proposes that the discrepancy arises from varying definitions of the well's width, with some sources defining it as L and others as 2L.
  • One participant expresses uncertainty about whether they have the lengths defined correctly, indicating potential confusion in the definitions used.
  • Another participant recalls seeing the different expressions in the context of deriving the Density of States from the Free Electron Model, suggesting a practical application of the differing definitions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions of the well's width or the appropriate conditions for using each expression for k. Multiple competing views remain regarding the interpretation of boundary conditions and the definitions of the square well.

Contextual Notes

There are unresolved assumptions regarding the definitions of the well's width and the specific boundary conditions applied in different contexts, which may affect the interpretation of the wavevector expressions.

Master J
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Right guys, I want to get this one straight...

We have all seen the simple infinite square well a million times. From it, we can get the condition for the k-vector of the electron that

k = n.pi / L

Now, I also come across all the time that k = 2n.pi / L

When do we use which boundary condition? They both seem to come from the same situation, but I cannot see when one is used?

It's a simple situation that's been bugging me a while! Hope someone can clear this up.

Cheers!
 
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The wave-function needs to disappear at the 2 boundaries, so whichever k produces the disappearing wave-function at the 2 boundaries is the k that you should use. Usually the problem is defined so that L is the total length of the square well, so you should use the first one. The second one would be used if the total length of the square well was, for some reason, 1/2L.
 
It would help if you could give us specific references to sites/books/whatever that do it each way, otherwise we have to guess.

My guess is that it's because some sources define the well as having width L, with either 0 < x < L or -L/2 < x < +L/2; and some define the well as having width 2L, with -L < x < L.
 
Did I get the lengths backwards...? Should it be L and 2L? My apologies if I did, I just tried to work it out in my head.
 
I can't actually think of any right now, I just know I've seen it come up. For instance, in deriving the Density of States from the Free Electron Model.

So you think it's just from the fact that one can define the width of the well as L or 2L?
 

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