Quantum Mechanics Works

Abstract: I will attempt to explain how position and momentum form a Fourier transform pair using elementary quantum state equations. Showing this is a Fourier pair entails the uncertainty principle. The implicit math involved is first order ordinary differential equations with a sprinkling of linear algebra.

Introduction: Position and momentum form a Fourier transform pair. Professor Johnson's link on Fourier transforms explained how Fourier mappings move a function from the time domain to its pair in the frequency domain and back. 

Similarly, position and momentum are inverse Fourier transforms mapping from momentum space to position space:

Calculations:

Let  indicate position and indicate momentum  

Momentum State (in momentum space):

 

(I won't show how to explicitly calculate this, but it essentially involves using Schrodinger's equation for a defined wave function, eigenfunction  , and momentum operator. We can convert between momentum frequency because quantum mechanical particles have frequencies proportional to their momenta)

Position State (in position space):

Note the negative exponents which indicate that they are a Fourier pair.

We can Fourier transform between these pairs as follows: 

Adding up all possible momentum states for a particle, we have the wave function in position space:

and adding up all possible position states for a particle we have the wave function in momentum space:

 

Conclusion:

 

The interdependence between the integrals suggests the Heisenberg uncertainty principle. Recall that intervals of length I between successive peaks in a function correspond to intervals or length 1/I in that function's Fourier inverse. This means that, the higher accuracy we know either position or momentum (the smaller the I) then the less accurately we know its Fourier inverse (larger 1/I). That is identically the uncertainty principle.

Acknowledgments:

I would like to thank Casey Handmer (2010 Ph 12a TA) for first suggesting this during recitation, http://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle and  http://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html as well. I'm also indebted to my gradually receding ability to concentrate, and to viewers like you. But not necessarily in that particular order.

For a more interesting explanation of this post: