Quantum mechanics describe the evolution of the state of system given the initial conditions and the hamiltonian describing it. That's exactly what the Schrodinger equation says and why it has a partial derivate of time. Usually in introductory treatments only the "time independent Schrödinger equation" is addressed, but that's a sub product on a common technique used to solve some family of differential equations.
In classical mechanics you can interpret time as another coordinate, but with so many privileges that it's a little far reached call it in the same way as space dimensions. Without the metric introduced in special relativity there is no way to justify put them in the same footing. For example, in classical mechanics you cannot rotate in a way that makes the time coordinate fall in a space coordinate, while you can do that in relativity, and it's called a boost.
Well, the Schrödinger equation only describes the evolution of the wavefunction over time (which is indeed a similarity with classical mechanics). But how the wavefunction then further relates to the actual "state of system" is up to the interpretations of quantum mechanics. And there in lies my point: That because of the superposition the way time works in quantum mechanics (properties are simultaneously in a combination of states) is very different from classical mechanics (properties are in exactly one state at a time), even if both have a "t" in their equations.
> For example, in classical mechanics you cannot rotate in a way that makes the time coordinate fall in a space coordinate, while you can do that in relativity
That is a good point. Maybe it is fair to say that in classical mechanics we start out with an additional separate geometric dimension (which only translation / shifting is allowed in). And in special relativity it gets promoted to a full spacial dimension, allowing all sorts of transformations.
Quantum mechanics describe the evolution of the state of system given the initial conditions and the hamiltonian describing it. That's exactly what the Schrodinger equation says and why it has a partial derivate of time. Usually in introductory treatments only the "time independent Schrödinger equation" is addressed, but that's a sub product on a common technique used to solve some family of differential equations.
In classical mechanics you can interpret time as another coordinate, but with so many privileges that it's a little far reached call it in the same way as space dimensions. Without the metric introduced in special relativity there is no way to justify put them in the same footing. For example, in classical mechanics you cannot rotate in a way that makes the time coordinate fall in a space coordinate, while you can do that in relativity, and it's called a boost.