Why does this term always approach 4?

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Discussion Overview

The discussion centers on the algebraic reasoning behind why a specific term in the formula for tunneling probability in a finite square potential well approaches the value of 4. Participants explore the implications of this approximation without focusing on numerical verification.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that the term before the exponential in the tunneling probability formula approximates to 4, seeking an algebraic explanation for this observation.
  • Another participant challenges the assertion that the term approaches 4 for "any" values, stating it is exactly 4 when V = 2 * E0 and suggesting a possible physical reason for V/E0 being close to 2.
  • A request for references supporting the claim that the term approaches 4 is made, indicating a desire for further clarification or validation of this assertion.
  • A later reply expresses uncertainty about finding references to support the claim.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the universality of the approximation to 4, and the discussion remains unresolved.

Contextual Notes

The discussion highlights the dependence on specific values of V and E, and the potential limitations in understanding the conditions under which the approximation holds.

houlahound
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I have followed several derivations to obtain the formula for tunnelling probability in a finite square potential well.

the standard result that is actually an approximation to a more complex result is;

upload_2016-11-23_10-1-27.png


I get the derivation of this result shown in many standard texts.

what I do not get is the term before the exponential always approximates to the number 4.

I am not interested in verifying this with actual values for V, E.

I want to know algebraically why the term is always approximately equal to 4 for any values of V, E.

it is supposed to be obvious and probably is, eludes me atm.

authors often make a further approximation and ignore the entire term so they are always out by a factor of 4 on an already approximate answer.

when dealing with such small numbers is leaving the 4 out that pedagogically sound - that's a seprate question.
 
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It certainly isn't close to 4 for "any" values. It's exactly 4 when V = 2 * E0. So I suppose there's some physical reason why V / E0 is usually close to 2. But I don't know why that would be.
 
houlahound said:
what I do not get is the term before the exponential always approximates to the number 4.

Can you give some references that make this claim?
 
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not sure to be honest. will do a search.
 

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