Time-Dependent Derivation of the SE:

The time-dependent form of the Schrödinger Equation is useful for certain cases in which the state of the electron is changing as a function of time. To derivate this form, one refers back to eq.4 of the time-independent derivation:

eq.1

Each side of the equation is set equal to a constant, -k ²:

eq.2

Multiplying by c² to eliminate the constants on the right-side:

eq.3

where -b ²= -k ² * c². The solutions to z are similar to those derived previously for the function y:

eq.4

where N is a normalizing constant. To determine the values of b , one must apply the principals used to determine k in the time-independent equation by recognizing this solution describes a wave, specifically a traveling wave. The amplitude located at a point at time t, should be equal to the amplitude at the same point at time t + t , where t is the period of the wave. Representing this:

eq.5

For this equation to be true, one observes:

eq.6

Knowing t = 1/ n , where n is the frequency of the wave, one can rewrite b in terms of n (this is done for future convience when solving for energy):

eq.7

Entering this into the general form for z:

eq.8

Knowing E= h * n and h-bar=h / (2*p), one can write:

eq.9

For energy to be positive, the negative sign is chosen. Earlier, the classical function z which was a function of space and time was broken into two functions. To derive the time-dependent Schrödinger Equation, one must find the quantum version of the function z which is labeled Y and is represented as the product of the two 'sub-functions':

eq.10

To extract E from this expression and thereby arrive at the time-dependent form, the first derivative with respect to time is taken:

eq.11

Knowing -i = 1 / i:

eq.12

Rearranging:

eq.13

finally results in the time-dependent form of the Schrödinger Equation. This can also be written as:

eq.14

similar to the time-independent form.