Schrödinger's equation is the equation of quantum mechanics. The Ψ symbol is the function that carries all the information about a particle, or particles that one can know. This equation describes how that function will change with time. The ∇2 is a differential operator when replaced by a second space derivative the resulting equation is the one dimensional version. The symbol V represents potential. This is different for different situations and as a result this equation may be very hard to solve. The equation shown is a partial differential equation, or the multi-variable version of ordinary differential equations.
Notice that this equation has an i in it. This i is a root of -1. Numbers that involve i are called complex numbers. Complex numbers were developed for mathematical reasons and it's surprising that they have a explicit role in physical equations.
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