Time-Independent Derivation of the SE:

The Schrödinger Equation is based upon two formulas: the classical wave function as derived from Netwon's Second Law, and the DeBroglie wave expression. To begin deriving the time-independent form of the Schrödinger Equation, the formula for a classical wave is used first:

eq.1

The letter z stands for the amplitude of the wave which is a function of x and t: z(x,t), c is the speed of the wave through the medium. The function z is broken down into two functions each being only dependent on one independent variable, either x or t.

eq.2

The first function y represents the spatial part of the original function and is dependent only on x, the second function function z represents the temporal part of the original function and is dependent only on t. Using this, the following is obtained:

eq.3

Notice only one function is differentiated in each term. Rearranging in order to separate y and z, produces:

eq.4

In order for this to be true, both sides of the equation must equal each other for all values of x and t. Therefore, each side can be set equal to an arbitrary constant, allowing each function to be analyzed separately. In this case, the arbitrary constant utilized will be -k ², which allows for the greatest possible ease when manipulating the second-order differential equations present. The left-hand side is the time-independent part and produces the time-independent form of the Schrödinger Equation. Dealing with this side first makes it easier later on to deal with the full, time-dependent form of the Schrödinger Equation. Concentrating now on the left side:

eq.5

Rearranging:

eq.6

The general form of y in this equation is:

eq.7

where N is a normailzing constant (important in dealing with actual numerical problems). Now it is important to determine the expression to which kappa is equal so that a definite set of wave functions can be ascertained. Realizing that this function describes the amplitude of a wave, one can assume that the amplitude at one point should equal that at a different point one wavelength removed. Representing this:

eq.8

where l is the wavelength of the wave. In order for this statement to be true, one observes:

eq.9

Now the DeBroglie wave expression mentioned earlier is utilized, which is:

eq.10

Substituting this into eq. 9, produces our representation of the values for k :

eq.11

Then, using this, one can return to eq. 6 to continue working towards the time-independent Schrödinger Equation which relates wavefunction with energy. Doing so:

eq.12

A convention is made by use of a symbol called h-bar and is represented as:

eq.13

This symbol appears so often in quanutm chemistry, that this convention is an absolute necessity. Rewritting:

eq.14

Isolating y on the right-hand side, produces:

eq.15

Using the expression for kinetic energy,:

eq.16

eq.15 can be rewritten, so that the relationship between a wavefunction and its energy is explicitly shown

eq.17

thereby resulting in the time-independent form of the Schrödinger Equation. This form can be modified to show realistic behavior by adding a potential energy term on the left side as well as replacing d²/dx² with the Laplacian operator (del²) which represents taking the full second derivative of y :

eq.18

Using operator notation, the Schrödinger Equation can be represented as:

eq.19

H is called a hamiltonian operator or simply the hamiltonian. A mathematical operator is a set of instructions that acts on a function, an eigenfunction, to produce a value that is characteristic of that function, an eigenvalue. In this case, the hamiltonian operator acts on y , the wave function, and produces a characteristic value, energy. The explicit form of H can vary between situations. For more complex manipulations, it condenses the work expended to simply utilize H to represent several operations. Operators are demonstrated in greater detail later on.