The one-particle, * time-dependent Schrodinger equation* is a partial differential equation whose solutions are the one-particle, time-dependent wave functions of quantum-mechanical systems.

Even though the equation is widely regarded as a postulate, we can derive it using a general travelling wave equation . Since cosine is an even function, , which in the complex square-integrable form is: . Since , we have . Substituting Planck’s relation and de Broglie’s hypothesis in the wave equation, which is a mathematical description of the properties of a quantum-mechanical particle, we have , where .

The total energy of the particle is: , and so

To develop an expression for , we find the partial derivative of with respect to :

As for , we find the the 2^{nd}-order partial derivative of with respect to :

Substituting eq55 and eq56 in eq54, we have

Eq57 is the one-particle, one-dimensional, time-dependent Schrodinger equation, which has the general solution .

###### Question

Show that is a solution to eq57.

###### Answer

For LHS of eq57

So,