Waves

At the core of this discussion lies a simple concept: all types of signals or functions, no matter how intricate, can be expressed as superpositions of sine waves—pure waves characterized by a constant wavelength and amplitude.

A superposition signifies that all waves interact, and their collective sum (termed interference) results in a more complex signal. Essentially, we can decompose a function into its simpler, pure wave components, akin to calculating Fourier coefficients for Fourier series, though this process applies beyond just periodic functions.

This phenomenon is well illustrated in music; for instance, when a guitar produces sound, it generates overtones that overlap with the primary frequency of the plucked string. Consequently, the sound emitted by a guitar (or any musical instrument, including the human voice) comprises pure sine waves of varying frequencies and amplitudes.

When characterizing such a complex signal, we can opt for two equivalent representations—essentially, two different units of expression. We might describe how all the waves that create our interference pattern interact over time, or alternatively, we could detail the frequencies of the pure waves that compose it.

The ability to express this in two equivalent manners is known as a dual relationship.

It would be advantageous to have a mathematical instrument to articulate this dual relationship between time-based signals and frequency-based signals. As you might have guessed, we possess such a tool.

The Fourier Transform

The mathematical tool that facilitates this dual relationship is known as the Fourier transform. It is undeniably one of the most influential and widely utilized instruments in mathematics.

Before delving into its properties, we should grasp a general understanding of this transform. The Fourier transform is an integral transform (an operator) that takes a function and yields another function.

As an operator acting on a function space, it can be regarded as a pure mathematical construct, yet it often carries a meaningful physical interpretation alongside it, applicable in both contexts.

Today, we will primarily focus on its physical implications, although it remains rooted in pure mathematics.

Throughout our discussion, we will presume that the integrals converge.

Let f be an integrable function. The Fourier transform of f is defined by the following integral:

If f represents a sound wave as a function of time, then the Fourier transform reveals the frequencies that constitute this sound wave, making f a function of frequency.

Below is an animated illustration showcasing how a sound wave (a unit pulse in this instance) comprises numerous pure sine waves, resulting in the sinc function (i.e., sin(?s)/?s).

It is crucial to recognize that every signal possesses these two equivalent forms of representation. They are equivalent because knowing one uniquely determines the other, and we possess formulas to compute it. The choice of representation is a matter of preference.

The unique inverse Fourier transform is given by:

Properties of the Fourier Transform

Exploring this subject is a multi-course endeavor, and we can only scratch the surface in this article. Nevertheless, we cannot discuss the Fourier transform without mentioning and proving some of its remarkable properties.

We begin with translation or shifting. Suppose h(t) = f(t + a). By employing a change of variable, we find:

Thus, a shift in time (a delay in the signal) corresponds to a phase shift in the frequency. Now, what about scaling?

Assume that h(t) = f(at). We can differentiate between the cases of a < 0 and a > 0.

Notice that the variable change employed was u = at. Let’s analyze what occurs when a < 0.

It is noteworthy that the numeric value is taken solely from a outside the function. This can be succinctly expressed in one formula:

What is the physical interpretation of this?

> The scaling property of the Fourier transform indicates that compressing a signal in time corresponds to expanding the signal in frequency (horizontally) and vice versa.

This property is incredibly significant, as we will soon observe.

One intuitive way to comprehend this is through dimensional analysis. Time is measured in seconds (unit s), while frequency is measured in reciprocal seconds (unit 1/s). It seems reasonable that if we extend time, we compress frequency, and vice versa.

If you are curious about the origin of the output unit of frequency, that is entirely understandable. The s in the Fourier transform ultimately dictates the periods of the pure waves that comprise the signal. You can grasp this by either utilizing Euler’s formula to expand the complex exponential into sines and cosines or by viewing the Fourier transform as a continuous set of Fourier coefficients.

The Fourier transform possesses numerous intriguing properties, but since this article is not solely focused on the transform, we will move on, allowing interested readers to explore this further on their own.

One practical property that readers may find useful or fascinating is that the Fourier transform converts derivatives into products with constants. This implies that differential equations in one space can correspond to algebraic equations in another space.

Consequently, some differential equations may be solvable by transforming them, solving them algebraically, and then applying the inverse Fourier transform to obtain the solution.

The Wave Function and Heisenberg’s Uncertainty Principle

Quantum physicists describe a quantum system (for instance, a particle) through potential quantum states. The family of functions modeling this phenomenon is termed wave functions.

The square of the magnitude of a wave function corresponding to position yields a probability distribution associated with the particle. Consequently, the wave function can be interpreted as generating a probability wave that indicates the likelihood of a particle being located in a specific spatial region. A wave function representing a particle’s position can, therefore, be perceived as a spatial wave rather than a temporal one.

> When we apply a Fourier transform to this position wave (the wave function of position), we derive a space-frequency wave that turns out to be the wave function for the particle’s momentum.

This conclusion is not surprising, especially considering that if we regard light as a wave packet or matter wave, then the momentum is dictated by the frequency of the light.

Specifically, we have ? = h/p and f = E/h, illustrating this relationship. Here, ? is the wavelength, h is Planck's constant, p is momentum, f is frequency, and E is energy.

The more confident we are that a particle is confined within a narrow interval, the more localized (horizontally compressed) the position wave function will become. Since the momentum wave function is the Fourier transform of the position wave function, it will be horizontally stretched, resulting in greater uncertainty regarding the momentum. This phenomenon is a consequence of the aforementioned scaling property of the Fourier transform.

This encapsulates Heisenberg’s uncertainty principle! It is essentially the Fourier transform in action.

It asserts that:

?x?p ? h/4?

where h represents Planck's constant, and ?x and ?p are the uncertainties (standard deviations) in position and momentum, respectively.

General Uncertainty

When a function g is the Fourier transform of the function f, we refer to f and g as conjugate variables or a conjugate pair. For every pair of conjugate functions, an uncertainty principle exists.

Heisenberg’s uncertainty principle is merely a specific instance of this broader and more profound phenomenon involving conjugate variables.

Why should the uncertainty principle for conjugate variables hold, especially from a mathematical perspective? The rationale is as follows: Short signals, such as bursts of sound, necessitate numerous waves to maintain amplitudes at zero outside a given interval. In contrast, the more sine-like a signal is, with a pure wave extending across space, the fewer frequencies are required to describe it.

When you hear a brief burst of sound, distinguishing the involved frequencies becomes challenging. However, if you hear a pure signal resonating for an extended duration, you can discern the various frequencies with ease. This illustrates the uncertainty principle.

Similarly, the better we understand the distance to a radar target, the less we can ascertain about its exact velocity, and vice versa. This exemplifies the uncertainty in Doppler measurements and range.

Another pair of conjugate variables includes energy and time. Consequently, there is another formulation of Heisenberg’s uncertainty principle for simultaneous measurements of energy and time. The inequality representing this relationship resembles that of the classical uncertainty principle:

?E?t ? h/4?.

Numerous other conjugate variables exist, leading to additional uncertainty principles, but they all share a commonality: the fundamental laws that govern them are not inherently physical but rather mathematical! The mathematics of waves inherently limits the amount of information we can extract from any quantum system.

The Effects of Heisenberg’s Uncertainty Principle Are Real

When a laser is directed toward a narrow slit, causing some light to be obstructed while allowing some to pass through, a fascinating phenomenon occurs.

The light appears to disperse on the wall behind the slit, and as the slit narrows, the dispersion becomes more pronounced. This seems counterintuitive—restricting it results in broader spreading.

This phenomenon arises from Heisenberg’s uncertainty principle. As we reduce the slit size, we compel the position wave (the wave function) to become increasingly localized (narrow), and according to the uncertainty principle, the momentum wave function broadens, making an increasing number of directions probable.

Since momentum is a vector with a direction, this means that the angle at which the photon can travel after passing through the slit becomes larger, creating a beautiful wide wave pattern on the wall.

Uncertainty can also elucidate why the Sun shines and even why the space-time phenomenon of Hawking radiation is causing black holes to shrink.

I hope my point is clear by now:

Uncertainty is fundamentally a mathematical phenomenon, but because quantum systems manifest aspects of this mathematical theory, uncertainty is also a physical principle.