# Fourier transforms and the fundamental source of uncertainties in Quantum Mechanics

## Motivation:

The textbook answer to the question of why Quantum Mechanics has a probabilistic description goes along the lines of the Heisenberg Uncertainty Principle. This is true even in Landau’s famous account of Quantum Mechanics [1]. In essence, we can’t simultaneously know both the position and momentum of quanta.

However, it may be demonstrated that the HUP is nothing more than a mathematical consequence of the wave description which is in itself not more than a useful approximation for engineering applications. This suggests that we may need to analyse actual quantum mechanical phenomena in order to understand the origins of these uncertainties.

## Defining uncertainties within the context of wave equations:

Given the De Broglie hypothesis and the Everettian interpretation of quantum mechanics, the physical behaviour of any object is described by a wave function and therefore the universe itself may be identified with a wave function \(\Psi (x,t)\).

If \(x\) denotes the position of an object and \(p\) its momentum, its associated wave function may generally be expressed as the sum of many orthogonal waves:

\begin{equation} \Psi(x) \propto \sum_n a_n \cdot \Psi_n = \sum_n a_n \cdot e^{i p_n x/\hbar} \end{equation}

where \(\lvert \Psi(x) \rvert^2 = \sum_n \lvert a_n \rvert^2\) may be interpreted as the probability that an object is localised at \(x\).

Now, if we bring (1) to the continuum limit:

\begin{equation} \Psi(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} \phi(p) \cdot e^{i p x/\hbar} dp \end{equation}

so \(\phi(p)\) is the Fourier transform of \(\Psi(x)\).

Since \(\lvert \Psi(x) \rvert^2\) is the probability density for position, we may calculate its standard deviation and use this as a representation of our statistical uncertainty. In fact, we shall demonstrate that the uncertainty principle is a direct consequence of basic facts in Fourier analysis.

## Deriving the uncertainty principle via Fourier analysis:

If we subtract the mean from each variable, we may define the variances for position and momentum as follows:

\begin{equation} \sigma_x^2 = \int_{-\infty}^{\infty} x^2 \cdot \lvert \Psi(x) \rvert^2 dx \end{equation}

\begin{equation} \sigma_p^2 = \int_{-\infty}^{\infty} p^2 \cdot \lvert \phi(p) \rvert^2 dp \end{equation}

Now, if we define the function:

\begin{equation} f(x) = x \cdot \Psi(x) \end{equation}

which may be interpreted as a vector in a certain function space, we may define the variances as inner-products:

\begin{equation} \sigma_x^2 = \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx = \langle f,f \rangle \end{equation}

Given that \(\Psi(x)\) and \(\Phi(p)\) are Fourier transforms of each other, it may be shown that:

\begin{equation} g(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} p \cdot \phi(p) e^{-ipx/\hbar} = -i \hbar \frac{d}{dx} \cdot \Psi(x) \end{equation}

By applying Parseval’s theorem, we have:

\begin{equation} \sigma_p^2 = \int_{-\infty}^{\infty} \lvert p \cdot \phi(p) \rvert^2 dp = \int_{-\infty}^{\infty} \lvert g(x) \rvert^2 dx = \langle g,g \rangle \end{equation}

Now, given that for any \(z \in \mathbb{C}\):

\begin{equation} \lvert z \rvert^2 = \text{Re}(z)^2 + \text{Im}(z)^2 \geq \text{Im}(z)^2 = \Big(\frac{z -\bar{z}}{2i}\Big)^2 \end{equation}

if we let \(z = \langle f,g \rangle\) and \(\bar{z} = \langle g,f \rangle\) we have:

\begin{equation} \lvert \langle f,g \rangle \rvert^2 \geq \Big(\frac{\langle f,g \rangle -\langle g,f \rangle}{2i}\Big)^2 \end{equation}

By evaluating the inner products, we find:

\begin{equation} \langle f,g \rangle -\langle g,f \rangle = i \hbar \end{equation}

Finally, from (9) and (10) it follows that:

\begin{equation} \sigma_x^2 \cdot \sigma_p^2 \geq \lvert \langle f,g \rangle -\langle g,f \rangle \rvert^2 \geq \Big(\frac{\langle f,g \rangle -\langle g,f \rangle}{2i}\Big)^2 = \frac{\hbar^2}{4} \end{equation}

and so we may deduce that the product of standard deviations yields:

\begin{equation} \sigma_x \cdot \sigma_p \geq \frac{\hbar}{2} = \frac{h}{4 \pi} \end{equation}

## Discussion:

As demonstrated here, the Heisenberg Uncertainty Principle is a useful mathematical observation but it does not provide any profound epistemological insights into the fundamental source of uncertainty in quantum mechanics. That said, I believe it is possible to make important progress in this direction by carefully analysing what Feynman considered the essential mystery in quantum mechanics. The double-slit experiment.

I plan to proceed with such an analysis using the tools of information theory as any serious account of uncertainty is of an information-theoretic nature. Moreover, such an information-theoretic investigation into the foundations of quantum mechanics was originally proposed by John Wheeler.

## References:

- L. D. Landau & L. M. Lifshitz. Quantum Mechanics-Nonrelativistic Theory. Butterworth-Heinemann. 1981.
- Wheeler. INFORMATION, PHYSICS, QUANTUM: THE SEARCH FOR LINKS. 1989.
- Feynman. The Feynman Lectures in Physics. 1963.
- Wikipedia. The Uncertainty Principle. 2021.