Presuming that the wavefunction represents a state of definite energy E, the equation can be separated by the requirement.
Wave Mechanics is the branch of quantum mechanics with equation. Time-dependent Schrödinger equation: Separation of variables(x,t)'(x)(t)'(x)e i Et Any linear combination of stationary states (each with a different allowed energy of the system) is also a valid solution of the Schrodinger equation Stationary States In fact all possible solutions to the Schrodinger equation can be written in this way. (21) where is assumed to be a real function and represents the potential energy of the system (a complex function will act as a source or sink for probability, as shown in Merzbacher 2, problem 4.1). and given the dependence upon both position and time, we try a wavefunction of the form. The single-particle three-dimensional time-dependent Schrödinger equation is. In effect, what is says to do is 'take the second derivative of \(\psi (x)\), multiply the result by \(-(\hbar^2 /2m)\) and then add \(V(x)\psi (x)\) to the result of that. For a free particle the time-dependent Schrodinger equation takes the form.
