Understanding Heisenberg's Uncertainty Principle: A Mathematical Perspective
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Chapter 1: Introduction to Heisenberg's Uncertainty Principle
You may want to take a seat for this discussion.
It's common to hear the assertion that Heisenberg's uncertainty principle involves observers interacting with electrons via photons, affecting the momentum of the photons in the process. While it's true that an observer may influence an electron's quantum state to observe it, this is not the core reason behind the uncertainty principle!
To clarify our understanding, let's define Heisenberg's uncertainty principle. In the realm of quantum mechanics, this principle (often referred to as Heisenberg's uncertainty principle) encompasses various mathematical inequalities that establish a fundamental limit on how precisely we can determine pairs of physical quantities of a particle, such as its position and momentum, from initial conditions.
— Wikipedia
A widely accepted interpretation suggests that at any specific moment, one cannot precisely measure both a particle's momentum and its position simultaneously. This limitation is not due to inadequate instruments or measurement challenges; rather, no matter how advanced our tools become, we will always face this uncertainty.
First, it's important to recognize that multiple uncertainty principles exist, many of which manifest in our macroscopic surroundings. In fact, you may encounter these principles frequently without realizing it.
Moreover, the fundamental cause of Heisenberg's uncertainty principle is not rooted in quantum physics but in mathematics. The essence of these equivalent principles stems from a mathematical truth that all waves (conjugate variables) must adhere to, including matter. We will delve deeper into this shortly.
Music, radar technology, energy, and light also obey their own "uncertainty principles," and as we will soon discover, the underlying laws that govern these phenomena are fundamentally mathematical.
The Nature of Waves
Ultimately, everything boils down to a straightforward concept: all types of signals or functions, no matter how intricate, are essentially superpositions of sine waves—pure waves with well-defined wavelengths and amplitudes.
Superposition refers to the interaction of all waves, resulting in an interference pattern that constitutes a more complex signal. In other words, we can decompose a function into its simpler constituent components (pure waves). This process is similar to calculating Fourier coefficients for Fourier series, although it applies to non-periodic functions as well.
This concept is well-illustrated in music. For instance, when a guitar is played, it produces overtones that interfere with the primary sound wave (the frequency of the plucked string). Thus, the sound generated by a guitar (and any other musical instrument, including your voice) comprises pure sine waves with varying frequencies and amplitudes.
When describing such a complex signal, we can choose between two equivalent representations. We can depict how all the waves contributing to our interference pattern are interacting simultaneously using a time axis, or we can represent it through the frequencies of the pure waves that comprise it. This capability to express it in two ways is known as a dual relationship.
It would indeed be advantageous to have a mathematical framework to describe this dual relationship between the signal in time and its frequency representation. As you might have surmised, such a tool exists: the Fourier Transform.
Chapter 2: The Fourier Transform
The Fourier transform serves as a powerful mathematical tool that describes the dual relationship mentioned earlier. It is undoubtedly one of the most significant and widely used tools in mathematics.
Before discussing its properties, we should grasp the fundamental concept of this transform. The Fourier transform is an integral transform (acting as an operator) that takes a function and produces another function.
While we can view it as a piece of pure mathematics, it also has a meaningful physical interpretation that often accompanies it. Today, we will primarily consider its applications in physics, though it remains fundamentally a mathematical concept.
To illustrate, let ( f: mathbb{R} to mathbb{C} ) be an integrable function. The Fourier transform of ( f ) can be expressed through the following integral:
If ( f ) represents, for example, a sound wave as a function of time, the Fourier transform will represent the frequencies that constitute this sound wave, thus making ( f ) a function of frequency.
The first video titled "Your Daily Equation #18: Heisenberg's Uncertainty Principle: Math not Meth" provides insights into the mathematical aspects of Heisenberg's principle.
Moving on, below is a GIF illustrating how a sound wave (depicted as a unit pulse) is composed of numerous pure sine waves, resulting in the sinc function (i.e., ( frac{sin(pi s)}{pi s} )).
It is crucial to comprehend that a signal has these two equivalent modes of expression. They are deemed equivalent because knowledge of one uniquely determines the other, and there exists a formula for calculating it. Ultimately, it comes down to how we wish to describe or represent that signal.
The unique inverse Fourier transform can be expressed as:
Properties of the Fourier Transform
Studying the Fourier transform is a complex endeavor, and in this article, we will only scratch the surface of its properties. Nevertheless, we cannot discuss the Fourier transform without acknowledging some of its remarkable features.
One of the initial concepts is translation or shifting. Suppose ( h(t) = f(t + a) ). By applying a change of variable, we find:
Consequently, a time shift (a delay in the signal) corresponds to a phase shift in the frequency.
What about scaling? Assume ( h(t) = f(at) ). We'll examine the cases where ( a < 0 ) and ( a > 0 ). Notice that the change of variable used was ( u = at ).
The physical interpretation of this phenomenon is significant. The scaling property of the Fourier transform indicates that compressing a signal in time corresponds to expanding it in frequency, and vice versa.
This property holds immense importance, as we shall soon see. A high-level and intuitive perspective on this can be gleaned from dimensional analysis. Time is measured in seconds (unit: s), while frequency is quantified in reciprocal seconds (unit: 1/s). It stands to reason that if time is extended, frequency must be contracted, and vice versa.
For clarity on where the output unit of frequency originates, it is understandable. The ( s ) in the Fourier transform ultimately determines the periods of the pure waves forming the signal. You can grasp this concept by employing Euler's formula to expand the complex exponential into sines and cosines or viewing the Fourier transform as a continuous set of Fourier coefficients.
Numerous fascinating properties exist within the Fourier transform. However, since this is not solely an exploration of the transform itself, we will shift our focus and allow interested readers to delve deeper independently.
One particularly convenient property that readers may find practical is the ability of the Fourier transform to convert derivatives into products by constants. This means that differential equations in one space can correspond to algebraic equations in the alternate space. As such, some differential equations can be solved by transforming them, addressing them algebraically, and then using the inverse Fourier transform to return the solution.
Chapter 3: The Wave Function and Uncertainty
In quantum physics, the description of a quantum system (like a particle) is accomplished using potential quantum states. The collection of functions modeling this is known as wave functions. The square of the magnitude of a wave function related to position provides a probability distribution concerning the particle.
Thus, we can view the wave function as generating a probability wave that indicates the likelihood of a particle being present in a specific spatial region. A wave function that describes a particle's position can hence be interpreted as a wave in space rather than time.
When we apply a Fourier transform to this position wave (the wave function of position), we obtain a space-frequency wave that turns out to be the wave function for the particle's momentum.
This relationship should not come as a surprise when considering light as a wave-packet or matter-wave, wherein momentum is represented by the light's frequency. Specifically, we have:
[
gamma = frac{h}{p} quad text{and} quad f = frac{E}{h}
]
where ( gamma ) is the wavelength, ( h ) is Planck's constant, ( p ) is momentum, ( f ) is frequency, and ( E ) is energy.
The more certain we are about a particle's location within a small interval, the more localized (horizontally compressed) the wave function of its position will be. Since the momentum wave function is the Fourier transform of the position wave function, it becomes more spread out horizontally, leading to greater uncertainty in momentum. This phenomenon can be attributed to the aforementioned scaling property of the Fourier transform.
Thus, we arrive at Heisenberg's uncertainty principle! It is simply the Fourier transform at work, articulated mathematically as:
[
Delta x Delta p geq frac{h}{4pi}
]
where ( h ) is Planck's constant, and ( Delta x ) and ( Delta p ) denote the uncertainties (standard deviations) in position and momentum, respectively.
General Uncertainty
When a function ( g ) is the Fourier transform of another function ( f ), we refer to ( f ) and ( g ) as conjugate variables or a conjugate pair. For any conjugate pair of functions, an uncertainty principle exists.
Heisenberg's uncertainty principle is merely a specific instance of this broader, deeper phenomenon concerning conjugate variables.
From a mathematical standpoint, why should the uncertainty principle of conjugate variables hold? The answer lies in this: brief signals, such as bursts of sound, necessitate numerous waves to maintain zero amplitudes outside a particular interval. Conversely, the more sine-like the signal (with a pure wave permeating all space), the fewer frequencies are required to describe it.
When you perceive a short burst of sound, discerning the involved frequencies becomes challenging, but if you hear a prolonged pure signal resonating, you can identify the different frequencies. This exemplifies the uncertainty principle.
Similarly, the more accurately we ascertain a radar target's distance, the less we can determine its precise velocity of approach or retreat, and vice versa. This represents uncertainty in Doppler and range measurements.
Another pair of conjugate variables is energy and time. Thus, there exists another form of Heisenberg's uncertainty principle for simultaneous measurements of energy and time, represented by the inequality:
[
Delta E Delta t geq frac{h}{4pi}
]
Numerous other conjugate variables and, consequently, uncertainty principles exist, but they all share a commonality: the underlying laws 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 Real Implications of Heisenberg's Uncertainty Principle
When you direct a laser toward a small slit, blocking part of the light while allowing some to pass, an astonishing phenomenon occurs. The light appears to disperse on the wall behind the slit, and if you narrow the slit further, the dispersion increases. This seems counterintuitive; how can restricting the light lead to more spreading?
This behavior is a direct consequence of Heisenberg's uncertainty principle. As the slit narrows, we compel the position wave (the wave function) to become increasingly localized (narrow), and due to the uncertainty principle, the momentum wave function expands, making an increasing range of directions probable.
Since momentum is a vector with direction, this means that the angle at which the photon can travel on the other side of the slit broadens, resulting in a beautifully wide wave pattern on the wall.
Uncertainty also provides insight into why the Sun shines and explains the space-time phenomenon of Hawking radiation, which leads to the shrinking of black holes.
I hope my message is clear at this point: uncertainty is fundamentally a mathematical phenomenon. Yet, as quantum systems embody aspects of this mathematical theory, uncertainty also emerges as a physical principle.