From: veritasium
Historically, physicists believed that objects followed a single, well-defined path through space [00:06:00]. However, this understanding was challenged with the advent of quantum mechanics, which revealed that everything, from light to electrons, explores all possible paths simultaneously [00:17:00], [02:16:00]. This concept is central to wave-particle duality and was significantly advanced by Louis de Broglie’s hypothesis.
Early Problems Leading to Quantum Mechanics
Blackbody Radiation and Planck’s Quantum
At the turn of the 20th century, a major challenge in physics involved understanding blackbody radiation, specifically how much visible light a hot filament would emit [03:44:00]. Scientists at Germany’s PTR institute observed that above 500°C, all materials glowed similarly, with the peak emission shifting to shorter wavelengths as temperature increased [04:08:00], [04:18:00].
The theoretical model at the time, the Rayleigh-Jeans law, predicted an “ultraviolet catastrophe,” suggesting an infinite amount of energy would be emitted at the shortest wavelengths, which contradicted experimental data [06:35:00], [06:42:00].
Max Planck, despite being advised that physics was a “complete science” [07:07:00], spent three years attempting to find a theoretical explanation [07:21:00]. In an “act of desperation” [07:44:00], he proposed that the energy of an electromagnetic wave was not continuous but could only come in discrete multiples of a “quantum” [08:07:00]. He formulated , where is Planck’s constant [08:21:00]. This revolutionary idea correctly matched experimental data, solving the ultraviolet catastrophe by showing that at higher frequencies (shorter wavelengths), fewer atoms had enough energy to emit a quantum of radiation [09:01:00], [09:19:00]. Planck initially viewed this as a mathematical trick, troubled by the introduction of a new physical constant without clear reason [09:47:00].
Einstein and Photons
In 1905, Albert Einstein provided a physical interpretation of Planck’s theory, claiming that light itself consists of discrete packets, or photons, each with energy [11:11:00]. This explained the photoelectric effect, where light can eject electrons from metal only if its frequency is high enough, regardless of intensity [11:24:00]. This reinforced the concept of light exhibiting both wave-like and particle-like properties.
Bohr’s Quantized Orbits
Eight years later, Niels Bohr applied the idea of quantization to the structure of atoms [11:44:00]. He sought to explain why electrons orbiting a nucleus do not spiral inwards, radiating energy. Bohr proposed that an electron’s angular momentum is quantized, meaning it can only exist in discrete integer multiples of Planck’s constant divided by 2π (nh-bar) [12:10:00], [12:38:00]. While this “ad hoc” [13:11:00] assumption successfully predicted the energy levels and spectrum of the hydrogen atom [12:50:00], [13:03:00], it lacked a deeper physical explanation [12:44:00].
De Broglie’s Hypothesis
In 1924, for his PhD, Louis de Broglie contemplated these discoveries [13:29:00]. His key insight was that if light, traditionally considered a wave, could behave like a particle, then perhaps matter particles could also behave like waves [13:35:00], [13:39:00].
De Broglie proposed that everything – electrons, basketballs, people – possesses a wavelength ([13:44:00], [13:46:00], [13:52:00]). He defined this wavelength as Planck’s constant () divided by the particle’s momentum ():
This hypothesis provided a fundamental physical reason for Bohr’s quantization condition [14:51:00]. If an electron in an atom acts as a wave, it must exist as a standing wave to remain bound to the nucleus [14:03:00]. This requires that a whole number of wavelengths fit around the circumference of the electron’s orbit [14:12:00].
Mathematically, if the circumference () must equal an integer () times the wavelength (), then:
Substituting de Broglie’s wavelength ():
Rearranging this equation yields:
This expression () is precisely the angular momentum, and the equation is Bohr’s quantized angular momentum condition [14:46:00]. Thus, de Broglie’s hypothesis elegantly explained why electron orbits are quantized: they are stable because they correspond to standing waves that experience constructive interference [14:55:00], [15:00:00].
Implications for Particle Paths
The wave nature of quantum objects means they do not follow a single trajectory through space [15:10:00]. Instead, they “explore all possible paths” [15:13:00].
Feynman’s Path Integral Formulation
Richard Feynman famously conceptualized this by extending the double-slit experiment [16:54:00]. If a particle can pass through two slits simultaneously, then logically, it must be able to pass through an infinite number of slits, or indeed, any path in empty space [17:14:00], [17:18:00]. This includes paths that are seemingly impossible, like those that travel faster than light, go back in time, or extend to the moon and back [18:00:00].
Feynman’s approach to quantum mechanics suggests that anything moving from one point to another is connected in every possible way [18:36:00], with each path contributing an amplitude (represented as a spinning vector or “stopwatch” that measures phase) [20:17:00], [20:33:00].
The Role of Action
The rate at which this “stopwatch” turns, or the phase changes for each path, is determined by a quantity called the action [23:31:00], [23:36:00]. This classical concept, initially proposed by Maupertuis and formalized by Hamilton, is equivalent to the integral over time of kinetic energy minus potential energy [02:59:00], [03:03:00], [23:21:00].
The phase change for a path is given by the action for that path divided by Planck’s constant (specifically, h-bar) [22:50:00]. Since Planck’s constant (h-bar) is extremely tiny (approximately Joule-seconds) [23:43:00], the action of everyday objects on ordinary paths is vastly larger [23:50:00]. This causes the phase vector to spin around zillions of times, pointing in a random direction for most “crazy” paths [23:58:00], [24:20:00].
Emergence of Classical Paths
When the amplitudes of all possible paths are added up, most of them cancel out due to destructive interference [24:29:00], [24:32:00]. The only paths that contribute significantly are those extremely close to the path of least action [24:35:00]. For these paths, tiny changes in the trajectory result in negligible changes in action, causing their phase vectors to point in approximately the same direction and constructively interfere [24:46:00], [24:57:00].
This explains why we perceive objects as following single, well-defined trajectories: these are the paths of least action where wave contributions add up [25:03:00], [25:11:00]. This also illustrates how classical mechanics emerges from quantum mechanics [25:24:00]. For massive particles, the action is so large relative to h-bar that only an extremely narrow range of paths around the least action path survive, making them appear distinctly particle-like [25:40:00]. For smaller particles like electrons or photons, the action is smaller, leading to a greater spread in possible trajectories and more pronounced wave-like behavior [25:51:00].
Experiments using diffraction gratings demonstrate this principle, showing that by selectively interfering with certain paths, light can be made to reflect in “crazy” or unexpected ways, validating the idea that it explores all paths [26:15:00], [31:00:00].
Significance of Action
The principle of least action and the concept of action are fundamental to modern physics [31:33:00]. Theoretical physicists often use action and the related concept of the Lagrangian (a function from which action is derived) as the basis for describing physical laws, rather than focusing solely on energy or forces [31:42:00], [32:09:00]. The hunt for a unified “theory of everything” in physics is essentially a search for the correct Lagrangian that can generate all known laws of the universe [32:34:00], [32:41:00].