Schrödinger's Time-Dependent Wave Equation

The question is, how do we proceed to construct a dynamical equation for quantum objects which are represented by wave functions ? Clearly, we must choose a basic physical law and try to express it in terms of wave functions and operators, rather than classical concepts. Accordingly, we choose the Law of Conservation of Energy :

$$\mbox{Total Energy} \qquad = \qquad \mbox{Kinetic Energy} + \mbox{Potential Energy}$$ (30)

Expressed classically

$$E = \frac{p^2}{2m} + U(x,t)$$ (31)

and expressed quantum mechanically, using wave functions and operators

$$\mathbf{\mathsf{E}}\Psi = \frac{\mathbf{\mathsf{p}}^2}{2m}\Psi + U(x,t)\Psi$$ (32)

$$\mbox{or}$$

$$i \hbar \frac{\partial }{\partial t}\Psi = - \frac{\hbar^2}{2m} \frac{\partial^2 }{\partial x^2} \Psi + U(x,t)\Psi$$ (33)

Generalising to three dimensions, we achieve finally -
Schrödinger's Time Dependent Wave Equation :

$$i \hbar \frac{\partial }{\partial t}\Psi = - \frac{\hbar^2}{... ...ac{\partial^2 }{\partial z^2} \right) \Psi + U(x,y,z,t)\Psi.$$ (34)

In general, the procedure for studying physical systems with Schrödinger's time dependent wave equation is as follows:

Determine the Potential Energy function $U(x,y,z,t)$.
Insert it into Schrödinger's time dependent wave equation.
Solve the resulting equation to obtain the wave function for the particle in the specified potential.
Calculate observables using the wave function.

Because Schrödinger's wave equation determines the mechanical behaviour of quantum particles, the new physics that results is known as Quantum Mechanics.

You may feel that we have simply assumed a wave function for a free particle ($U(x,y,z,t)=$ const) and then used this to write down a quantum mechanical form of the law of conservation of energy using operators. This is correct, we can only assume. Schrödinger's wave equation cannot be derived. It is postulated (with some intuition !). It is nearly a century of success in describing known physics and vastly improved insight into hitherto undreamt of phenomena that has given a good measure of confidence in Quantum Mechanics.