B Why does this term always approach 4?

[houlahound]

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;

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.

 

[secur]

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.

 

[PeterDonis]

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?

 

[houlahound]

not sure to be honest. will do a search.