Proof of the preservation of normalizability of the wavefunction

[Ahmad Kishki]

p. 12 Introduction to Quantum Mechanics by Griffiths
Equation 1.25: the differential operatot was factored. This to me seems like a mathematical trick or due to amazing foresight, but is there any underlying/guiding theory for this factorisation?
Equation 1.27: the wavefunction was assumed to be zero at infinity which to me seems a bit weird, since by this assumption we enforced normalizability and got the answer we fabricated, so is my criticism valid? Is there a more rigorous proof?

The image of the book is enclosed below.

Thanks :)

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[stevendaryl]

I'm not exactly sure what your question is. We assume that initially,

$\int |\psi|^2 dx = 1$

So we assume that it starts off normalized, and that $\psi \rightarrow 0$ as $x \rightarrow \pm \infty$

Then we prove, based on the assumption that it is true at time $t=0$, that it will be true for all time.

 

[vanhees71]

The question is, how to prove that the wave function is normalized to 1 for all times, if it was normalized to 1 at the initial time $t=0$.

The answer is: That follows from the Schrödinger equation and the self-adjointness of the Hamiltonian. Working in position representation ("wave mechanics") it reads
$$\mathrm{i} \partial_t \psi(t,\vec{x})=\hat{H} \psi(t,\vec{x}).$$
Here and in the following I use "natural units", where $\hbar=1$.

The self-adjointness of the Hamiltonian means that for all square-integrable wave functions
$$\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi_1(\vec{x})^* \hat{H} \psi_2(\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} [\hat{H} \psi_1(\vec{x})]^* \psi_2(\vec{x}).$$
Now you can prove that the norm of the wave function is conserved under the time evolution following from the Schrödinger equation:
$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi^*(t,\vec{x}) \psi(t,\vec{x}) = \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \left [\partial_t \psi^*(t,\vec{x}) \psi(t,\vec{x}) + \psi^*(t,\vec{x}) \partial_t \psi(t,\vec{x}) \right ].$$
Now from the Schrödinger equation and its complex conjugate this gives
$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi^*(t,\vec{x}) \psi(t,\vec{x}) = \mathrm{i} \int_{\mathbb{R}^3} \left [(\hat{H} \psi)^* \psi-\psi^* \hat{H} \psi \right ]=0.$$
Thus the norm is preserved under time evolution and thus, if the wave function is normalized to 1 at any time it stays so. This must be so from the physical point of view, because the probality to find the particle somewhere in the whole universe is always 1 (because in non-relativistic physics particles usually are not destroyed in interactions in contrast to relativistic physics, where the creation and annihilation of particle-antiparticle pairs and similar processes are the usual business :-)).