Zettili Oscillator Wave Function and Hermite Polynomials

Question

I was studying the algebric method of solving the wavefunction for a harmonic oscillator from Quantum Mechanics by Zettili and the part where he finally brings the wavefunction derived from analytic method equal to one he got from ladder operator method (Page 246) , he writes

At this level, we can show that the wave function (4.165) derived from the algebraic method is similar to the one obtained from the
ﬁrst method (4.118). To see this, we simply need to use the
following operator identity:
$$displaystyle e^{frac{-x^2}{2}}(x-frac{d}{dx})e^{frac{x^2}{2}}= -frac{d}{dx}$$
$$e^{frac{-x^2}{2 x_{0}^{2}}}(x- x_0^2 frac{d}{dx})e^{frac{x^2}{2 x_{0}^{2}}}=-x^2_0frac{d}{dx}$$

I want to know what identity is this cause if you solve this (LHS) you get zero as an answer.
Here is the image link

Philip · Accepted Answer

Here's a golden rule whenever you're dealing with differential operators: always use a test function $f(x)$.
If you call the operator on the left hand side $hat{O}$, then what you want to show is that for any generic function $f(x)$,
$$hat{O} f(x) = -frac{text{d}f}{text{d}x}.$$
In other words, you need to show that
$$e^{-x^2/2} left(x - frac{text{d}}{text{d}x} right)left( e^{x^2/2} fright) = -frac{text{d}f}{text{d}x}.$$
I leave it to you to complete the proof.

Zettili Oscillator Wave Function and Hermite Polynomials

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