Fold Function and Quantum Mechanics


Introduce a fold function and investigate it’s properties. For a given finite barrier or well starting at x_1 and ending at x_2, that it, tending to zero.


Define a fold function \(f_*(x)\) as a transform from a defined function \(f(x)\) as \[f_*(x)=\frac{ \int_{x}^{x_2} f(s) \;ds}{ \int_{x_1}^{x} f(s) \;ds}\]

Which should result in a function normalised to the interval \([-1,1)\) ??? what needs doing to make this. This is something like a simple auto-correlation function although only in concept, it’s properties will be quite different. The function should higlight the ’area centre’ of some function, such that the area on the left side is that on the right side. For a given function with an ’area centre’ of 0, the function may mor may not be symetric about the y axis. For simple functions the derivative of the fold function equal to zero will yeild a location for this cetre point.

Porbably more realistic to scale down the \(\infty\) to some large value for terminating functions etc. numeric analysis.

Moved on to study the transform \[F_0(x)=2f_{max}\bigg[ \frac{ \int_{x}^{x_2} f(s) \;ds}{ \int_{x_1}^{x} f(s) \;ds} + \frac{ \int_{x_1}^{x} f(s) \;ds}{ \int_{x}^{x_2} f(s) \;ds}\bigg]^{-1} \\ F_0(x)=\frac{2f_{max}}{f_*(x)+f_*^{-1}(x)}\]

There then appears to be a higher order of interest \[F_1(x)=\bigg[ \frac{ \int_{x+w}^{x_2} f(s) \;ds}{ \int_{x_1}^{x+w} f(s) \;ds} + \frac{ \int_{x_1}^{x+w} f(s) \;ds}{ \int_{x+w}^{x_2} f(s) \;ds}\bigg]^{-1} - \bigg[ \frac{ \int_{x-w}^{x_2} f(s) \;ds}{ \int_{x_1}^{x-w} f(s) \;ds} + \frac{ \int_{x_1}^{x-w} f(s) \;ds}{ \int_{x-w}^{x_2} f(s) \;ds}\bigg]^{-1}\]

where w is a quarter width of the well.

Let us turn our attention to the harmonic oscillator potential function given by [Rae] as \[f(x)=\frac{1}{2}m\omega_c^2x^2\]

The constants will drop out when taking the fold of this potential \[f_*(x)=\frac{A}{ \frac{x_2^3-x^3}{x^3-x_1^3}+ \frac{x^3-x_1^3}{x_2^3-x^3} }\]

which is \[f_*(x)=\frac{A(x^3-x_1^3)(x_2^3-x^3)}{ (x_2^3-x^3)^2 + (x^3-x_1^3)^2 }\]

becoming reminiscant of the form \[\frac{k_1k_2}{k_1^2 +k_2^2}\]

Start Here:

We are seeking a functional transformation, a functional on \(V(x)\) such that for any given \(V(x)\) we can predict the wavefunction of a system. That is, the direct relationship \[\psi(x)=F(V(x))\]

We use the folding transforms as candidate functions for \(F()\).

Fold Function (maroon) fitting to the wavefuntion (red) of the first excited state for the harmonic oscillator potential.

Fold function for a finite potential well matched to the wavefunction (red) as an approximation for an infinite potential well. The change of function is expected as there is a non zero probability of being found outside the box which appears to decay in a fashoin like exp() function.