Quantum Harmonic Oscillator- Solving the Differential Equation at a Limit

The eigenvalue equation for the quantum harmonic oscillator is $$\langle y | E\rangle '' +(2\epsilon-y^2)\langle y| E \rangle=0$$ where $$\epsilon = \frac{E}{\hbar\omega}$$ and $$y=\sqrt{\frac{\hbar}{m\omega}}x$$. In Shankar's book, he starts to solve this by taking the limit at infinity, making the equation $$\langle y | E\rangle '' -y^2\langle y| E \rangle=0$$ Apparently, the solution to this equation in the same limit is $$\langle y | E\rangle = Ay^me^{\pm\frac{1}{2}y^2}$$. I know how to work with kind of equation normally, but how do I solve this equation in the limit $$y \to \infty$$?

• – Cosmas Zachos Sep 11 '19 at 3:03

Define $$\langle y | E\rangle = \psi_(y)$$ and study the asymptotics of your ODE, $$(\partial_y^2-y^2)~\psi (y)=0.$$ That means the leading behavior of ψ for large y; so , for example, if you had a polynomial in y, you just keep the highest order thereof: If you have an order-m polynomial, you'd just keep the $$y^m$$ monomial, since it dominates all lower powers in y at large y.
So, does $$Ay^me^{\pm\frac{1}{2}y^2}$$ satisfy your differential equation to leading order in y? Does $$\partial_y^2 (y^me^{\pm\frac{1}{2}y^2}) = (y^{m+2}+ O(y^{m+1} )~ ) e^{\pm\frac{1}{2}y^2} ~~?$$ Let's see: $$\partial_y^2 (y^me^{\pm\frac{1}{2}y^2}) = \partial_y \Bigl ((m/y\pm y) y^me^{\pm\frac{1}{2}y^2}\Bigr )\\ = ((m/y\pm y)^2-m/y^2\pm 1)~y^m e^{\pm\frac{1}{2}y^2}\\ = (y^2 \pm 2m + m^2/y^2 -m/y^2 \pm 1 )~~y^m e^{\pm\frac{1}{2}y^2},$$ So, indeed, the leading behavior of the parenthesis on the r.h.s. is $$y^2$$, your desideratum.
When you solve the full equation at all y, beyond asymptotics, $$(\partial_y^2- y^2 + 2m +1)\psi_m(y)=0,$$ and you dismiss the solutions blowing up for large y, you find a celebrated solution, the Hermite functions, $$\psi_m(y)\propto (-)^m e^{y^2/2} ~\partial_y^m e^{-y^2} = e^{-y^2/2} \Bigl (2y -\partial_y \Bigr)^m \cdot 1 ,$$ which, presumably, your text is building up motivation for. They are the eigenfunctions of the Fourier transform.