Graduate How to calculate the parabolic cylinder function D

[BiGyElLoWhAt]

I have a computational project to do for quantum, and I am kind of tired of doing basic barriers, so I decided to try a quadratic barrier. Well, you get garbage. However, since I'm working with numerical approximations anyways, I figured I might try it, but I am not sure how to proceed. Here is what wolfram gives me:

http://www.wolframalpha.com/input/?i=-h^2/(2*m)*y''+++x^2*y+=+k*y

Note: k == Energy, I just used k because it changed E -> e.

If you scroll down and see the solution, it is in terms of the parabolic cylinder function D which is the solution to weber equations.
https://en.wikipedia.org/wiki/Parabolic_cylinder_function
I am not really getting what to do with this.
I'm not sure on the notation used, and the link is no help.
https://en.wikipedia.org/wiki/Abramowitz_and_Stegun

Can someone give me advice on how to numerically approximate this function?

 

[DrDu]

First, I don't see that you are calculating a quadratic barrier but rather a simple harmonic oscillator. For a barrier, the x^2 term has to have negative sign.
Further note that you didn't specify that your solution shall be normalizable. Hence you get two solutions for every value of k. Only for special values of k, at least one solution is normalizable. In the case of the parabolic cylinder function, this happens when the confluent hypergeometric functions in its definition, which are basically defined via their Fourier series, is a polynomial of finite order. These polynomials are the Hermite polynomials and you recover the ordinary solutions of the harmonic oscillator equation. What you can do if you want to study tunnelling, is to shift the HO to the left and right and use the left shifted version for x<0 and the right shifted version for x>0. To obtain a solution for the full range, you have to glue together the correspondingly shifted Weber functions at x=0 so that the overall solution is normalizable.

 

[BiGyElLoWhAt]

Why would the quadratic potential barrier be negative? P^2/2m + V = E. The V is positive, so if my potential is x^2, shouldn't that he positive in my equation?

 

[DrDu]

Because it has a minimum and not a maximum at x=0.

 

[BiGyElLoWhAt]

You're losing me.

x^2 has a minimum at x = 0
-(x^2) has a maximum at x=0

So I want a minimum at x = 0 for the PE, which means I want V(x) = x^2. If this is wrong correct me, but otherwise I have no idea what you're saying.

 

[DrDu]

You said you are interested in barriers. For me a barrier is a maximum in potential energy, not minimum.

 

[BiGyElLoWhAt]

I'm thinking V=0 at x=0 V=1 at x=1,-1, etx, confining the particle to near x=0. "Confining".

Sort of like a potential well, but quadratic in nature.

 

[DrDu]

Ok, so this sounds like an ordinary harmonic oscillator problem. I still don't quite get why you are talking of a quadratic "barrier".

 

[BiGyElLoWhAt]

V (x) = x^2
I plugged in the diff eq to wolfram and got an answer in terms of the parabolic cylinder function.

 

[DrDu]

Yes, parabolic cylinder functions are the general solutions of the differential equation. But only for special values of k, these functions are normalizable, i.e. are functions belonging to the Hilbert space of normalizable functions. In this case, the cylinder functions can be expressed in terms of Hermite polynomials. I tried to explain this already in #2.
I still don't understand why you are talking of a barrier.

 

[BiGyElLoWhAt]

I didn't plan for this to be the haronic oscillator. I simply picked a value for Vthat was reasonably different from what we've done in class. I suppose this relates to the oscillator when k=2.

Is a quadratic well more suiting?

Anyways, I read about them on wofram, and didn't see much other than how to type it into mathematica. I read the wikipedia page and didn't understand the notation that was being used. I more or less want to know how to calculate these functions so I can make a simulation for this.

 

[DrDu]

You have to understand that it is difficult to answer your questions when we do not even know what kind of classes you are taking, nor what your background in quantum mechanics is. Do you use some specific textbook on QM? Every textbook contains a section on how to solve the harmonic oscillator and some books even on parabolic cylinder functions (Landau & Lifshitz, quite certainly).

 

[BiGyElLoWhAt]

We have a book that we don't really use much. An intro to qm by griffiths. It's basically been a reference, and I can count the number of times I've actually used it on one hand.