David Hilbert's Prediction of Schrödinger Equation

As @Emilio points out, your premise is aggressively wrong. The two pictures are equivalent, and one does not dominate the other, any more than the left-hand side of an equation dominates the right-hand one. The matrix formulation was first, chronologically, cf J Bernstein's 2005 article, and algebraists prize it on account of the intuitive overview it provides: Recall how Pauli derived the quantum spectrum of the Hydrogen atom before the advent of the Schroedinger equation, and the nightmare of special functions involved.

Born, its inventor, had already used matrix methods 4 years before 1925, and it was only a year after that (1926) that von Neumann, Hilbert's assistant, coined the term "Hilbert space" for the infinite-dimensional eigenvectors space of differential and integral operators. For history, you might try HSM.SE. Hilbert merely reminded them that the integrodifferential operators he was studying had discrete spectra (like drumheads) and could be handled elegantly by linear algebraic methods, matrix techniques. He suggested they go backwards, to possibly get something more out of that. He then recognized this general setup he was emphasizing then in the Schroedinger pedagogical revolution, when that came along, that's all.

Physicists were more comfortable with differential equations than with linear algebra, then; chemists, in fact, have stuck to differential operators and never looked back. But he did not "predict" the Schroedinger equation--not even anticipate it. At best he suggested looking for one.

I'll focus on your physics question.

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Once you have solved a problem in the matrix formulation, there is little incentive to get the Schroedinger picture transliteration of it, as you probably learned in your introductory QM, where the Dirac ladder operator solution of the quantum harmonic oscillator is explained after the orgy of Hermite functions' in Schroedinger's picture convinces you that there must be a simpler way... Supersymetric quantum mechanics purveyors of isospectral potentials go that route on occasion.

Here, I'll just remind you by Dirac's paradigmatic oscillator example of what you already know. I non-dimensionalize (absorb) all pointless constants, $hbar, m,omega$, setting them, reversibly, to 1, and ignore ancillary normalization issues. I'm only reminding you of the trail map.

So the conceit is you have used $[a,a^dagger]=1$ to find the spectrum of $H=a^dagger a +1/2$, $$ a^dagger|nrangle = sqrt{n + 1} | n + 1rangle a|nrangle = sqrt{n} | n - 1rangle, qquad a|0rangle=0, ~~leadsto H|nrangle = (n+1/2)|nrangle. $$ Implicitly, this amounts to specification of any and all matrix elements in the energy eigenstate basis. Explore alternate bases, by suitable rotations.

Now, can you rotate your algebra by a right angle in phase space to perhaps get to an equivalent but more evocative basis? Indeed, defining $$ x={1over sqrt{2}} (a+a^dagger) ,~~~ -ipartial_x= {1over sqrt{2}} ( a^dagger-a ),~~~leadsto [x,-ipartial_x]=i, $$ elementary calculus.

So you just need to find a function $psi_0(x)$ annihilated by $x+partial_x$ ! It is obviously $$ (x+partial_x) e^{-x^2/2} =0 ~~~~~leadsto langle x |0ranglepropto psi_0(x), ~~~0=(x+partial_x)langle x |0rangle =langle x|a|0rangle. $$ You may then move on to $$ (x-partial_x)langle x |0rangle =langle x|a^dagger|0rangle= langle x|1rangle propto psi_1(x), $$ and so on, given the wonderful recursive property of the Hermite functions, $$ psi_{n+1}(x)propto (x-partial_x)psi_n(x) . $$ I'll let you take care of the normalizations.

You realize then that you have constructed the eigenfunctions of the differential operator $$ H= 1/2 + (x-partial_x)(x+partial_x)/2= (-partial_x^2 + x^2)/2, $$ the quantum oscillator Hamiltonian, without going through the pain of solving differential equations (the heart of the algebraic methods of Hilbert and Courant).

You (well, following the great Dirac) have just changed basis. Hilbert had the framework right: integrodifferential operators, matrices, what's the difference?