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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.
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.