From: 3blue1brown
The Heisenberg uncertainty principle in quantum mechanics states that the more precisely a particle’s position is known, the less certain one can be of its momentum, and vice versa [00:00:03]. This principle is a specific instance of a much broader trade-off observed in everyday, non-quantum wave phenomena [00:00:21]. Understanding this general uncertainty principle involves exploring the relationship between frequency and duration [00:00:50].
Core Idea: Frequency and Duration
An intuitive understanding of this trade-off can be seen in common scenarios:
- Turn Signals: Observing a car’s turn signals flashing in sync for a few seconds gives low confidence in their exact frequency, as they could fall out of sync later [00:01:02]. However, observing them in sync for a full minute provides much greater certainty about their frequencies [00:01:24]. This demonstrates that certainty about frequency requires an observation spread out over time [00:01:33].
- Musical Notes: The shorter a musical note lasts, the less certain one can be about its exact frequency [00:01:51]. For example, a clap or shockwave has no discernible pitch [00:01:57]. Conversely, a more definite frequency necessitates a longer duration signal [00:02:04]. A short signal correlates with a wider range of frequencies, while a signal correlating with only a narrow range of frequencies must last for a longer time [00:02:13].
The Fourier Transform and Signal Analysis
The Fourier transform is a mathematical tool for analyzing the frequencies present within any signal [00:03:32]. It allows viewing a signal not by its intensity over time, but by the strength of its various frequencies [00:02:55].
The process involves “winding” the signal around a circle at different frequencies [00:03:05]. When the winding frequency matches the signal’s dominant frequency, peaks and valleys align, shifting the “center of mass” of the wound-up graph [00:03:42]. The position of this center of mass encodes the strength and phase of that frequency in the original signal [00:04:02].
The uncertainty principle is revealed in how the signal’s duration affects its Fourier transform:
- Long-duration signal: If a signal persists over a long period, even slight differences in winding frequency cause the signal to balance out when wrapped around the circle [00:05:58]. This results in a very sharp drop-off in the Fourier transform’s magnitude as the frequency shifts away from the dominant one, indicating a concentrated frequency representation [00:06:09].
- Short-duration signal: If a signal is localized to a short period, it doesn’t have enough time to balance out when the winding frequency is adjusted [00:06:22]. This leads to a much broader peak in the Fourier transform, meaning the signal correlates with a wider range of frequencies [00:06:42].
In essence, a signal concentrated in time must have a spread-out Fourier transform (correlating with a wide range of frequencies), and a signal with a concentrated Fourier transform must be spread out in time [00:06:51].
Doppler Radar Analogy
The general uncertainty principle is tangibly present in Doppler radar [00:07:04]. Radar involves sending out radio wave pulses and analyzing their echoes to determine object distance and velocity [00:07:08].
- Position and Time: The time it takes for an echo to return indicates an object’s distance [00:07:11].
- Velocity and Frequency: The Doppler effect causes the frequency of the echo to shift based on the object’s velocity (e.g., higher frequency for objects moving towards the radar) [00:07:23].
A radar operator faces a dilemma due to this trade-off [00:08:44]:
- Precise Position: To achieve precise understanding of object distances, a very quick, short pulse is needed [00:09:24]. However, a short pulse’s Fourier transform is necessarily spread out in frequency [00:09:34]. This means Doppler-shifted echoes from multiple objects are more likely to overlap in frequency space, making it ambiguous to determine individual velocities [00:09:45].
- Precise Velocity: To obtain a clean, sharp view of velocities, an echo occupying a small amount of frequency space is required [00:10:02]. But for a signal to be concentrated in frequency space, it must be spread out in time [00:10:11]. Such a long echo would overlap with echoes from other objects, making their exact locations ambiguous [00:09:04].
This is a clear example of the Fourier trade-off: crisp delineation for both position and velocity is impossible [00:10:16].
The Quantum Case: Particles as Waves
The Heisenberg uncertainty principle extends to the quantum realm through the concept of wave-particle duality. Louis de Broglie, in his 1924 PhD thesis, proposed that all matter exhibits wave-like properties [00:10:40]. Crucially, he concluded that the momentum of any moving particle is proportional to the spatial frequency of its associated wave (how many times the wave cycles per unit distance) [00:10:52].
De Broglie’s reasoning involved an analogy to the Doppler effect, but applied to spatial frequency and in the context of special relativity [00:11:55]. He considered the energy of a particle as an oscillating phenomenon, similar to photons [00:12:48]. Due to relativistic effects, observing this oscillating energy while moving relative to it causes the oscillations to appear out of phase over space [00:13:03]. This “relativistic Doppler effect” implies that changes in a mass’s movement correspond to changes in its spatial frequency, thereby linking momentum to spatial frequency [00:14:15].
Applying the general Fourier trade-off to particles:
- If a particle is described as a wave packet concentrated in space (meaning its position is well-defined) [00:15:03], its Fourier transform (which represents its momentum) must be spread out [00:15:06].
- Conversely, if a particle’s momentum is well-defined (its momentum wave is concentrated) [00:15:11], then its position wave must be spread out [00:15:11].
Unlike the Doppler radar case, where the ambiguity arises from measurement, in quantum mechanics, the particle is the wave [00:15:30]. Therefore, the spread in position and momentum is fundamental to the particle itself, not an artifact of imperfect measurement techniques [00:15:34].
”Unsharpness Relation” and Probability
The Heisenberg uncertainty principle is often misinterpreted as solely a statement about unknowability or randomness [00:15:50]. However, it is fundamentally a trade-off between how concentrated a wave and its frequency representation can be, applied to the premise that matter is a wave [00:16:03].
The German term for the principle, which translates more directly to “unsharpness relation,” better captures the Fourier trade-off without implying limitations on what can be known [00:17:03].
In quantum mechanics, when a particle’s wave is concentrated in a region of space, it means there is a higher probability of finding the particle in that region, making its location more certain [00:16:36]. But because this concentration implies a spread-out Fourier transform, the momentum wave will also be spread out, meaning there is no narrow range of momenta the particle has a high probability of occupying [00:16:48]. The core fascination of the Heisenberg uncertainty principle lies in the fact that position and momentum have the same underlying mathematical relationship as sound and frequency [00:17:33].