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I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of the SE's normalization with the evolution of time. I was wondering if anyone can help me understand the physical and mathematical side towards answering the question of how SEs even preserves the normalization in the first place.

I hope someone out there can help me out in me research... Because my teacher isn't giving me anything useful.

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  • $\begingroup$ Physically, it's because the probability of a particular state existing 'as is' remains constant, because of unitary time evolution. See my answer here, for example: physics.stackexchange.com/a/434912/133418 $\endgroup$ – Avantgarde May 16 at 12:17
  • $\begingroup$ Since this "wave function" is interpreted as a probability amplitude and probability must equal 1 when summed over all possible states this must be imposed as a constraint on the wave functions. The time evolution preserves this, as pointed out in the answer. $\endgroup$ – ggcg May 16 at 13:48
  • $\begingroup$ The more mature approach to quantum mechanics starts by postulating that time evolution is unitary. We stipulate unitarity because we'd like Born's rule (norm squared of the wavefunction is a probability distribution) to hold true; then from this one can prove that Schrodinger equation is the unique and general relation satisfying this condition of unitarity. Check out my blog for a further explanation. $\endgroup$ – Omar Nagib May 16 at 17:58
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To be honest, it is designed to do that for a very profound property that we expect the Nature to follow, time-translation symmetry. Given this symmetry, Wigner's theorem tells us that the time evolution of a state should be described by unitary linear operators and the Schrodinger equation is just that statement. There may exist multiple different such qualified unitary operators and they describe different allowed systems in Nature such as a simple harmonic oscillator or a free-particle, etc. (or, you can say they describe different possible laws for the Nature as a whole if you consider the time evolution operator of the universe itself).

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For a time independent Hamiltonian $H$, The wavefunction evolves like $$|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle$$ Using this we see that

$$\langle\psi(t)|\psi(t)\rangle=e^{iHt/\hbar}e^{-iHt/\hbar}\langle\psi(0)|\psi(0)\rangle=\langle\psi(0)|\psi(0)\rangle$$

Therefore if $\langle\psi(0)|\psi(0)\rangle=1$ then for all time $\langle\psi(t)|\psi(t)\rangle=1$

This is true for unitary transformations.

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  • $\begingroup$ This proof is only usable for time-independent Hamiltonian. $\endgroup$ – Ruslan May 16 at 15:36
  • $\begingroup$ A high-schooler will understand none of this. You have to aim for a level without operators and Hamiltonians. $\endgroup$ – AtmosphericPrisonEscape May 16 at 19:03
  • $\begingroup$ @AtmosphericPrisonEscape If the OP wants to study QM then I am going to answer with what is usually used in QM. If you think this is too complicated and not useful you are more than welcome to downvote or post your own answer using the approach you want. $\endgroup$ – Aaron Stevens May 17 at 2:01
  • $\begingroup$ @DvijMankad Yes, I see this explanation in your answer. Thanks for the additional information. $\endgroup$ – Aaron Stevens May 17 at 2:04
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    $\begingroup$ @DvijMankad It probably just gets converted to the model organism of your choice. So you would probably get Schrodinger's bacteria, yeast, fruit fly, zebra fish, mouse, etc. Basically Schrodinger's zoo. $\endgroup$ – Aaron Stevens May 17 at 3:04

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