Why does this term always approach 4?

In summary, the conversation revolves around the derivation of the formula for tunnelling probability in a finite square potential well. The standard result is an approximation to a more complex result, and the term before the exponential is always approximately equal to 4 for any values of V and E. The reason for this is not clear, but it is often ignored by authors, leading to a factor of 4 error in the already approximate answer. There is some physical reason why V / E0 is usually close to 2, but it is not known. The conversation ends with a request for references that make the claim about the 4 approximation.
  • #1
houlahound
908
223
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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  • #2
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.
 
  • #3
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?
 
  • Like
Likes Demystifier
  • #4
not sure to be honest. will do a search.
 

FAQ: Why does this term always approach 4?

Why does this term always approach 4?

The term "approach 4" is often used in mathematics and physics to describe a phenomenon where a certain value or quantity gets closer and closer to the number 4 as it increases or decreases. This can be due to the specific properties or equations involved in the calculation, leading to a converging value towards 4.

What causes a term to approach 4?

There can be various factors that contribute to a term approaching 4. Some possible reasons include the specific mathematical or physical principles at play, the values of other variables in the equation, and the overall behavior of the system being studied.

Is there a specific reason why 4 is the value being approached?

In many cases, the value of 4 is used as a reference point or a target value in the equation or system being studied. This can be due to its significance in the context of the problem or because it is a convenient value to work with mathematically.

Can a term approach a value other than 4?

Yes, a term can approach any value depending on the specific equation or system being studied. The value being approached may also change depending on the initial conditions or other factors in the problem.

How is the rate of approach determined?

The rate at which a term approaches 4 is determined by various factors, including the equation or system being studied, the values of other variables, and the properties of the term itself. It can also be affected by external factors such as external forces or changes in the environment.

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