Quantum Mechanical Position, Momentum and Fourier Transform

For people familiar with quantum physics, it’s a known fact that a ‘waveform’ \left|\Psi\right\rangle can be essentially ‘measured’ in either ‘position’ x or ‘momentum’ p basis. We define the inner products using the following notations:

\left\langle x | \Psi \right \rangle = \Psi_p(x)

and

\left\langle p | \Psi \right \rangle = \Psi_x(p)

The point to be noted is that momentum is the time derivative of position. In such a case,

\Psi_p(x) = \mathfrak{F}\{\Psi_x(p)\} \\    \Psi_x(p) = \mathfrak{F}\{\Psi_p(x)\}

This can be proved as follows. A transformation from the x basis to the p basis can be made as follows.

The \hat{p} operator is

\hat{p} = -i\hbar\frac{\partial}{\partial \hat{x}}.

To find eigenstates of \hat{p}, we can call \langle x|p\rangle = f_p(x)

-i\hbar\frac{\partial}{\partial x}f_p(x)=pf_p(x)

which yields to f_p(x) = e^{ipx/\hbar}.

Now, to pass from a basis to the other we can write

\langle p|\psi\rangle= \int \langle p|x\rangle\langle x|\psi\rangle dx

or

\Psi_x(p) = \int e^{-ipx/\hbar}\Psi_p(x)dx

which is a Fourier transform.

Thus \Psi_p(x) and \Psi_x(p) form a Fourier pair. Simply speaking, a waveform that is measured in the time derivative of a certain basis will be the Fourier transform of the waveform being measured in that basis.

Now, coming to the signal processing scenario in which the representation of a ‘signal’ f is effectively a ‘measurement’ in either ‘time’ t or ‘frequency’ \omega basis.

The correspondence between these two models:

x \mapsto t

\frac{p}{\hbar} \mapsto \omega

Classically, time and frequency are inverses of each other.

\omega = \frac{2\pi}{T}
where T is the time period (measured in time domain).

Similarly, going by de Broglie wave equation,

\frac{p}{\hbar} = \frac{2\pi}{\lambda}
where \lambda is the wavelength (measured in position domain).

In the next post, I will elaborate on the uncertainty relation in both signal processing and quantum physics. The whole idea behind creating an analogy between the two is to see if certain tools that have already been developed in quantum physics, can be adapted to enhance signal processing techniques.

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