Yes, it is entirely possible and reveals the beauty and precision of quantum mechanics when statistically understood.
We recall how the time-dependent heat equation is solved in its most general case without needing the heat PDE itself.
The question arises what are the details of similarity between the transition matrix B used in classical physics and the transition matrix Q used in quantum physics?
This question is not just about classical physics, quantum physics, or mathematics, but a rigorous and comprehensive combination or incorporation of all three.
I'm sure many researchgate contributors can answer this question as well as I, if not better.
In fact, it is possible to solve the Schrödinger equation via a statistical transition matrix without needing the Schrödinger equation itself in the same way as solving the heat diffusion PDE, in his case the most general, without needing the heat equation itself.
The only requirement is the ability to imagine nature in 3D Cartesian geometry (solid geometry) and then, in a later stage, in 4D x-t unit block space.
The energy field in 4D x-t unit space has been solved in B-matrix chains and can also be solved through any other suitable statistical transition matrix.
My recommendation here is not to listen to those "HDR scientists" who dismiss the whole matter as insignificant due to their own lack of imagination and confused understanding.
While waiting for your answers or suggestions, I am preparing my own answer based on a modified B-Transition matrix (we call it Q-matrix).
Therefore, in a following answer, we provide our own numerical answers (where I assume you can personally provide answers that are adequate or better than mine) calculations of the outputs of the Schrödinger equation in an infinite potential well via the technique of the Q-matrix for the quantum transition.
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