QM, the convergence of the harmonic oscillator function.

In summary, Griffiths finds that for coefficients a of the function ##h(z)##, very far up (k large), a_{k+2} \approx \frac{2}{k} a_{k}##. The next equation suddenly states the following: ##a_{k} \approx \frac{C}{(k/2)!}##.
  • #1
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1. After finding out that the wave function ##\Psi(z) \sim Ae^{\frac{-z^{2}}{2}}## in the limit of plus or minus infinity Griffiths separates the function into two parts ##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}##

My question will be about a certain aspect of the function ##h(z)##

After solving the ODE series method, one finds a certain recursion relationship for the coefficients of h(z) with an even and odd part.

As Griffiths notes for coefficients ##a## of this function, very far up (k large), ##a_{k+2} \approx \frac{2}{k} a_{k}##

The next equation suddenly states the following: ##a_{k} \approx \frac{C}{(k/2)!}##

I don't understand this step. Let me show you why my reasoning leads me to the wrong conclusion:

Let's start again from what we know: ##a_{k+2} \approx \frac{2}{k} a_{k}##. Applying the same thing for ##a_{k}## and plugging in will give: ##a_{k+2} \approx \frac{1}{k/2} \frac{1}{k/2 - 1} a_{k-2}##

Keep doing this and I find that ##a_{k+2}=\frac{B}{(k/2)!}## In the book this where I have k+2 stands k and I can't figure out why it's more correct than what I have here.
 
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  • #2
Hello again,

I'm pretty convinced you won't be pleased with this, but hear (read) me out... :)

What matters here isn't the exact value of this ak (or ak+2, for that matter), but the ratio of these coefficients for really big k.

Anything to do with the precise value can be swept in what Griffiths calls C ("for some constant C" -- anything you don't want to be bothered with: throw it in there. As long as it doesn't keep growing with k you're fine).

Mind you, you want to be way above j = K/2 before you come even close to aj+2 / aj ##\ \approx\ ## 2/j !

(I find I'm unconsciously switching to Griffiths' indices j now, sorry..)And even after all these years, I find it almost miraculous that from this kind of reasoning one is simply forced to conclude that K can NOT assume arbitrary values, but MUST be exactly equal to 2j + 1 for some j, however big. A hair difference and the power series runs away like ##e^{x^2}##. Awesome !
 
  • #3
You were right, I was hoping for having made a stupid reasoning mistake or having overlooked something. The real reason seems to be less elegant than I would have liked but what can you do. Nothing to do with your answer, it was great, thanks.
 

1. What is the quantum mechanical (QM) description of the harmonic oscillator function?

The quantum mechanical description of the harmonic oscillator function is a mathematical model that describes the behavior of a particle in a potential well that is symmetric and parabolic. It is commonly used in quantum mechanics to study systems such as atoms, molecules, and simple solids.

2. How does the harmonic oscillator function converge?

The harmonic oscillator function converges through the use of a numerical method called the shooting method. This method involves solving a second-order differential equation by taking a series of initial guesses for the solution and adjusting them until the solution converges to the desired accuracy.

3. What is the significance of the convergence of the harmonic oscillator function in quantum mechanics?

The convergence of the harmonic oscillator function is significant in quantum mechanics because it allows us to accurately model and predict the behavior of particles in a potential well. This is important for understanding the behavior of atoms, molecules, and other quantum systems.

4. What factors can affect the convergence of the harmonic oscillator function?

There are several factors that can affect the convergence of the harmonic oscillator function, including the complexity of the potential well, the accuracy of the initial guesses, and the chosen numerical method. In some cases, the function may not converge at all, indicating a problem with the model or the chosen method.

5. How does the quantum mechanical description of the harmonic oscillator function differ from the classical description?

The main difference between the quantum mechanical and classical descriptions of the harmonic oscillator function is that the quantum mechanical version takes into account the wave-like nature of particles and the concept of quantization. In the classical description, the particle is treated as a point mass and its motion is described using classical mechanics equations.

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