Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

Question

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, we have: $$a_{j+2}\approx\frac{2}{j}\,a_j.$$ Up to this point I totally agree (one just takes the limit).

However, the subsequent derivation of solution $a_{j}$ and $h(\xi)$ I attached from the Griffiths' textbook are very confusing.

• How did it go from $a_{j+2}\approx\frac{2}{j}a_j$ to the solution of $a_{j}$?
• Also, how do the second and third approximations work in $h(\xi)$?

My questions are mainly mathematical. I very much hope someone can provide a derivation or refer a link where these questions may already be answered.

Answer 1

Looking at even indices, from $a_{2j_0+2} \approx \frac{1}{j} a_{2j_0}$, we get : $$a_{2j_0+2k} \approx \frac{(j_0-1)!}{(j_0+k-1)!}a_{2j_0} = \frac{1}{(j_0+k)!} C \tag{1}$$, with $C=(j_0+k)\,(j_0-1)! \,\,a_{2j_0}$

So, taking $n= 2(j_0+k)$, we have finally $a_{n} \approx \frac{1}{(n/2!)} \, C$

Finally, $h_{even}(\xi) \approx \sum\limits_{n= j_0 + 1}^{+\infty}a_{2n} \xi^{2n}= C\sum\limits_{n= j_0 + 1}^{+\infty}\frac{1}{n!} (\xi^2)^n $

The behaviour of $h_{even}(\xi)$ for high $\xi$, is then the same as the behaviour of $C\sum\limits_{n= 0}^{+\infty}\frac{1}{n!} (\xi^2)^n = C e^{\xi^2}$

You have the same demonstration for $h_{odd}(\xi)$, so finally, the behaviour of $h(\xi)$ for high $\xi$ is $C e^{\xi^2}$, too.

UPDATE (to answer to the OP's comments)

a) You have : $a_{2j_0+2} \approx \frac{1}{j} a_{2j_0}$, then you apply the formula to $(j_0+k)$ : $a_{2j_0+2k} \approx \frac{1}{j_0+k-1} a_{2j_0+2k-2}$, so finally, you have : $a_{2j_0+2k} = \frac{1}{(j_0+k-1)(j_0+k-2)...(j_0)}a_{2j_0}$. You multiply numerator and denominator by $(j_0+k)(j_0-1)!$, and you get the formula $(1)$

b) The identity is $e^y = \sum\limits_{i=0}^{+\infty}\dfrac{y^n}{n!}$

c) You have to choose some particular $j_0$ as a departure point for the recurrence relation described in a). I used this notation to show that $j_0$ is not a variable, it is fixed.

d) The identity described in b) is true for all $y$, because the serie is convergent for all $y$