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Luke_A_Somers comments on Which Basis Is More Fundamental? - Less Wrong
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What's that Lincoln quote about ducks and calling things?
Point is, Schrodinger's Equation contains within it an implication which leads to the energy eigenstate equation. Conflating the two is bad terminology, even if it's common. I would not call the force balance equation from statics "Newton's 2nd Law" - why should I do that in quantum mechanics, calling the Energy Eigenstate Equation "Schrodinger's Equation"? My more recent textbook goes out of its way to separate the two as it was found that conflating them was impeding students' understanding of quantum mechanics (though it does so in part by eliminating the term 'Schrodinger Equation' altogether).
That's an entirely reasonable argument that it shouldn't be called that.
But it is called that, and you have to be able to communicate with those who use it thus, or have it heard it this way, even while working to change the nomenclature.