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Q & A: Airy functions in physics

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Most recent answer: 02/22/2013
Q:
How to solve the airy differential equation for a triangular well potential? also: How to find the matrix element for airy differential equation?
- sara medhet (age 23)
ISLAMABAD
A:
Sara- I've combined your two questions.

There's a nice discussion of these functions on Wikipedia:

and another on mathworld:
.

I'm not sure what you want us to add. It is interesting that in a formal treatment of the thermal distribution of an ideal atmosphere in a uniform gravitational field, the states of definite energy are Airy functions, That makes a formal solution somewhat messier than our typical informal arguments about states with rather well-defined heights, L, having Boltzmann factors of e-mgL/kT in their probabilities. (Here m is particle mass,  and kT is the thermal energy scale. So long as kT is large compared to the quantization energy scale for the gravitationally bound states, roughly m1/3g1/3h2/3, the informal Boltzmann arguments are ok.

I'm not sure what you mean by matrix elements here.

Mike W.

(published on 02/21/2013)

Follow-Up #1: properties of Airy function

Q:
you dont understand my question i am asking that while solving the triangular well potential we encounter airy differential equation...i m asking how to solve this equation here is the link http://mathworld.wolfram.com/AiryDifferentialEquation.html i have solved that uptill equation (26),i m not getting how to convert them into Ai form i,e: the airy function??can you please help me to simlify those that?? secondly, by matrix element i mean that inoder to find the uncertainity relation for airy function we need to know the variance and for that we need to have matrix elements of x,x^2 and similarly P and P^2...so how to compute them out??
- sara (age 23)
Islamabad Pakistan
A:
Sara- If I understand,  you have followed the derivation through equation 26, which is a simple re-expression of equation 25. () Equation 27 simply introduces a parameter k that's equivalent to changing the strength of the potential, or the system of units. It's really not hard to see that the k -dependence of the functions that follow is the right form.  So what you're concerned with, I think, is equation 29, which introduces the two independent Airy functions. One, Ai, is clearly picked to be the one that could correspond to physical solutions for a bound state, since it goes rapidly enough to zero for large x. The other, Bi, blows up.  I think the key is that you need exactly the right ratio of the coefficients of the two independent terms in eq. 26 to get the special behavior of Ai, but I'm not sure and haven't solved for that ratio. Sorry not to be of more help.

I think there's a shortcut to help with the second question.

You know that in any stationary state <p>=0 since the particle  isn't going anywhere.
You know from the virial theorem that <p2/2m>=En/3 and <mgx>=2En/3 for any particle stationary state with energy En in this linear potential. Here I'm assuming that it's a gravitational problem, but for some other type you just substitute the relevant field for mg.

So there are two problems remaining:
1. Although the En of the Airy function (or equivalently its zeros) are known, I'm not sure if there's a closed-form expression for them. A simple expression, which you no doubt have seen, gives an excellent approximation.
2. I haven't given how to get <x2>. A colleague has made a tentative suggestion for a technique, but I haven't gotten it to work yet.

Meanwhile, at least for states well above the ground state, we can use the time-average over classical trajectories to calculate <x2>, since the interference terms all drop out in the time average. This gives (you can do the math too) <x2>= (9/5)<x>2.

Maybe we should post this now just so you know we're trying.

Mike W.

(published on 02/22/2013)

Follow-up on this answer.