For the different possible states there is a wave function Ψ associate to each one. This wave function is governed by the Schrodinger Wave Equation. For a particle moving in 3 – D the equation is

(1)Where m is the mass of the particle and V(r,t) is the potential of the system (is the external force acting on the particle). Restricting the equation for a particle moving in 1 – D

(2)To simplify, let assume V(x) be a static potential (time independent). Now with this assumption is possible to find a solution to the Schrodinger Wave equation using separation of variable. We are looking for a solution like

(3)X(x) is function of the position and T(t) of the time. Replacing on the SWE, we got

(4)Dividing by X / T

(5)Now the left term side is a function of time and the right term side is a function of x. These mean that these two terms has to be equal to a constant E.

(6)Where E is a constant

The solution to this equation is

Setting the potential V(x) to zero, meaning that we have a free particle, the right side become

(8)for more detail about potential well refer to [ http://electrons.wikidot.com/particle-in-a-box]

Introducing a new constant $p^2=2mE$ the equation become

(9)The solution is

(10)Combining both solution,

(11)This is the solution to the Schrodinger Equation for a free particle, where $E = p^{2}/2m$

The superposition of two solutions is also a solution of the SWE

(12)Generalizing

(13)Different set of Amplitude will give us different set of Ψ.

Corresponding to every physical observable quantity like position, momentum and energy, there is a Hermitian Operator associate to it.

An operator â act in a function to give another function, an example

(14)If an operator â acts in a function Ψ, and this return the old function multiply by a number λ, this number is an Eigen value of the function and is an Ψ Eigenfunction.

(16)For example if $\frac{d}{dx}$ is an operator and $e^{-ikx}$ is a function,

(17)Where ik is an Eigen value of the Eigen function $e^{-ikx}$.

Going back to Hermitian Operator

Hermitian Operator is an operator where the Eigens values are real.