Myself and a user on another board have come to the following hurdle we can't overcome:(adsbygoogle = window.adsbygoogle || []).push({});

Mathematically we represent an arbiarty matter wave as a superposition of plane waves; using the theory of Fourier analysis. We can write an arbitary wavepacked as a Fourier integral of the form:

[tex]$\psi (x,t) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {A(p_x )e^{\frac{i}{\hbar }(p_x x - Et)} dP_x } $

[/tex]

What this means is we can mathematically constuct a localised complace wavefunction by adding up (integrating) a large number of plane waves each with different momentum. The relaive amplitde of the component waves is determined by the function A(p[x]) which is given by:

[tex]$A(p_x ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {\psi (x,t)e^{\frac{i}{\hbar }(p_x x - Et)} dx} $

[/tex]

The functions psi(x,t) and A(p[x])) are called Fourier transofrms of one another.

How could we use this practically? Isn't there a circular argument there? To get teh amplitudes, we need the wavefunction, but to get teh wavefunction we need the amplitudes. The only answer I could give was that the amplitudes may fall naturally out of the system being studied. However having not 'offically' studied Fourier series before, is there a simpiler explanation?

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# Fourier Analysis in QM

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