From: 3blue1brown
Quantum mechanics emerged from observations that revealed certain restrictions on the energy levels of waves, how they behave when measured, and quantum entanglement [01:27:00]. A fundamental aspect of this field is the concept of discrete energy levels and the existence of photons.
Foundations in Classical Wave Mechanics
To understand quantum mechanics deeply, it’s essential to grasp the mathematical description of waves [00:51:00]. Many ideas central to quantum mechanics, such as describing states as superpositions with various amplitudes and phases, are first encountered in the context of classical waves [01:14:00].
Light as an Electromagnetic Wave
By the late 1800s, light was understood as waves in the electromagnetic field [01:38:00]. This understanding is largely based on Maxwell’s equations, which describe how electric and magnetic fields interact and cause changes in each other [02:37:00]. A natural consequence of this interplay is the propagation of electromagnetic waves, where electric and magnetic fields oscillate perpendicularly to each other and to the direction of propagation [03:18:00]. Light, radio waves, and other forms of electromagnetic radiation are examples of these propagating waves [03:35:00]. The energy of a classical wave is proportional to the square of its amplitude [10:51:00].
Classical Superposition and Polarization
Classical waves can be described using concepts like amplitude and phase shift [05:35:00]. If two distinct waves satisfy Maxwell’s equations (in a vacuum), their sum also forms a valid wave, a principle known as superposition [06:23:00]. For example, a horizontally polarized light wave (electric field oscillating horizontally) and a vertically polarized light wave (electric field oscillating vertically) can superpose [06:06:00], creating a new wave that oscillates in a diagonal direction [07:34:00]. If these components are out of phase, their sum might trace out an ellipse, leading to circularly polarized light [07:43:00].
The choice of directions to describe a wave (e.g., horizontal and vertical vs. diagonal and anti-diagonal) is called a “basis” [08:49:00]. This choice depends on the specific analysis, such as when light passes through a polarizing filter [08:55:00].
The Quantum Leap: Discrete Energy
Physicists in the late 19th and early 20th centuries observed that the energy of electromagnetic waves does not change continuously but rather comes in discrete amounts [11:38:00]. This energy is always an integer multiple of Planck’s constant (h) multiplied by the wave’s frequency (f) [11:46:00]. This means there is a minimal non-zero energy level for waves of a given frequency, hf [12:09:00]. An electromagnetic wave with this minimal possible energy is called a photon [12:51:00].
While this concept of discrete energy is unusual for waves in free space, it is common in constrained classical waves, like harmonics in pipes or instrument strings [12:39:00]. The “weirdness” in quantum mechanics is that electromagnetic waves exhibit this behavior even when unconstrained [12:45:00].
Photons and Probabilistic Interpretation
The mathematical framework used to describe classical electromagnetic waves carries over to describing a photon [12:56:00]. A photon, for instance, with a 45-degree diagonal polarization, can be described as a superposition of horizontal and vertical states, each with an amplitude and phase [13:03:00]. These components are often encoded using a single complex number, which naturally represents both amplitude and phase [10:12:00].
However, the interpretation of this superposition differs fundamentally from the classical case [13:26:00].
Probability vs. Energy Contribution
Classically, if a diagonal wave has an amplitude of one, its horizontal and vertical components would each have an amplitude of the square root of one-half [13:35:00]. The energy, being proportional to the square of the amplitude, would be split, with half the energy in the horizontal component and half in the vertical [13:52:00].
For a photon, this classical interpretation is incorrect because photons have discrete, indivisible energy chunks [14:05:00]. Instead, the squares of the amplitudes of the components in a photon’s superposition indicate the probabilities of finding the photon in a particular state upon measurement [18:46:00].
For example, if a diagonally polarized photon (amplitude 1) were to pass through a vertically oriented polarizing filter:
- Classically, 50% of its energy (from the horizontal component) would be absorbed [14:37:00].
- Quantically, since energy comes in indivisible photon packets, the photon either passes through with all its energy or is entirely absorbed [14:45:00].
- Experimentally, about half the time the photon passes through entirely, and about half the time it is absorbed entirely, seemingly at random [14:52:00]. If it passes through, its polarization changes to align with the filter’s direction [15:05:00].
Measurement and State Collapse
This phenomenon is analogous to the concept of Schrödinger’s cat [15:14:00]. A photon exists in a superposition of states (e.g., diagonal polarization being a superposition of horizontal and vertical). When a measurement is made, forcing it to interact with an observer in a way where each state would behave differently, the superposition “collapses” to be entirely in one state or another (e.g., horizontal or vertical) [15:17:00].
A common experiment demonstrates this:
- Two polarizing filters rotated 90 degrees relative to each other block out all light [15:46:00], as vertically polarized photons passing the first filter have a 0% chance of passing a horizontally oriented filter [16:00:00].
- However, inserting a third filter at a 45-degree angle between them allows more light through [16:10:00]. This happens because 50% of the vertically polarized photons pass the diagonal filter, changing their state to purely diagonal polarization. Once in this diagonal state, they have a 50/50 chance of passing the final 90-degree filter [16:18:00]. This results in 25% of the original photons passing through all three filters [16:44:00], an outcome unexplainable if the middle filter didn’t force the photons to change their states [16:49:00]. This experiment and the probabilities involved are further explored in a video about Bell’s inequalities [16:55:00].
Broader Implications
The principle that the square of the amplitude of a component gives the probability of a measurement outcome extends beyond just photon polarization. It applies to quantum mechanical states generally, where states are described as superpositions of multiple base states, with each component having an amplitude and phase (often as a complex number) [19:52:00].
Photons represent a case where something initially understood as a wave (due to Maxwell’s equations) was later found to behave as individual particles or “quanta” [19:32:00]. Conversely, particles like electrons, initially thought of as discrete packets, were discovered to be governed by similar wavy quantum mechanics [19:42:00]. If the wave nature of particles had been discovered first, quantum mechanics might have been named “harmonic mechanics,” emphasizing the wave equations that govern particles [20:25:00].