One of the important aspects of the Schrödinger equation(s) is its linearity. For the time independent Schrödinger equation, which is usually called an eigenvalue problem, the only consequence we shall need here, is that if ϕ(x) is a eigen function (a solution for {E}_{i}) of the Schrödinger equation, so is A{ϕ}_{i}(x). This is useful in defining a probability, since we would like
|
{\mathop{\mathop{\mathop{∫
}\nolimits }}\nolimits }_{−∞}^{∞}|A{|}^{2}|{ϕ}_{
i}(x){|}^{2}dx = 1
| (3.17) |
Given {ϕ}_{i}(x) we can thus use this freedom to ”normalise” the wave function! (If the integral over |ϕ(x){|}^{2} is finite, i.e., if ϕ(x) is “normalisable”.)
Example 3.1:
As an example suppose that we have a Hamiltonian that has the function {ψ}_{i}(x) = {e}^{−{x}^{2}∕2 } as eigen function. This function is not normalised since
|
{\mathop{\mathop{\mathop{∫
}\nolimits }}\nolimits }_{−∞}^{∞}|{ϕ}_{
i}(x){|}^{2}dx = \sqrt{π}.
| (3.18) |
The normalised form of this function is
|
{1\over {
(π)}^{1∕4}}{e}^{−{x}^{2}∕2
}.
| (3.19) |
We need to know a bit more about the structure of the solution of the Schrödinger equation – boundary conditions and such. Here I shall postulate the boundary conditions, without any derivation.
|
P(x) = {|ϕ(x){|}^{2}\over
{\mathop{\mathop{\mathop{∫
}\nolimits }}\nolimits }_{−∞}^{∞}|ψ(x){|}^{2}dx}
| (3.20) |
is a probability density.