Quantum mechanics and wave function

From: veritasium

Introduction to Quantum States

In classical mechanics, knowing a system’s state—such as a particle’s position and velocity—allows for the calculation of its future actions using equations like Newton’s second law [00:00:05]. Quantum mechanics operates differently: if the quantum state of a particle, known as its wave function, is known, the Schrödinger equation can be used to predict its future behavior [00:00:16]. Typically, a particle’s wave function spreads out over time [00:00:26].

The Measurement Problem

A significant challenge in quantum mechanics is reconciling the smoothly evolving, spread-out wave function with the observation that when measured, a particle is found at a single point in space [00:00:46]. Early quantum theory founders prioritized the measurement as more real, as it matched observable experience [00:01:07].

Interpretation of the Wave Function: The Born Rule

Erwin Schrödinger developed his wave equation because scientists like Louis de Broglie suspected that matter exhibited wave-like properties [00:01:26]. However, it was Max Born who proposed how to interpret the wave function [00:01:35]. He suggested that squaring the complex amplitude of the wave function at any point in space yields the probability of finding the particle there [00:01:48]. This “Born rule,” though initially a footnote in Born’s paper, introduced probability into the core understanding of reality, making the universe no longer strictly deterministic [00:02:01]. This probabilistic nature made many scientists, including Einstein, uneasy, but the Born rule remains central to quantum mechanics due to its success in predicting experimental outcomes [00:02:12].

Wave Function Collapse

The traditional understanding of quantum mechanics involves two sets of rules [00:02:26]:

1. Evolution: When not observed, the wave function evolves smoothly according to the Schrödinger equation [00:02:32].
2. Collapse: Upon measurement, the wave function suddenly and irreversibly collapses, with the probability of a particular outcome given by the squared amplitude of the wave function associated with that outcome [00:02:35].

Schrödinger’s Cat Thought Experiment

Schrödinger himself disliked this two-rule formulation, leading him to devise the famous Schrödinger’s Cat thought experiment [00:02:51].

• Setup: A cat is placed in a box with a radioactive atom, a radiation detector, and a vial of poisonous cyanide gas [00:02:57]. If the atom decays, the detector triggers the release of the poison, killing the cat [00:03:28]. If the atom does not decay, the cat remains alive [00:03:35].
• Entanglement: The state of the cat and detector are directly tied to the state of the atom, meaning they are entangled [00:03:42].
• Superposition: According to quantum mechanics, the atom can exist in a superposition of both decayed and not decayed states simultaneously (assuming no measurements) [00:03:51]. This superposition then extends to the detector and the cat [00:04:04].
• The Paradox: The wave function of everything inside the box becomes a superposition of “atom not decayed, poison not released, cat alive” and “atom decayed, poison released, cat dead” [00:04:13]. Thus, the cat is both alive and dead at the same time [00:04:26]. Only when the box is opened and a measurement is made does the wave function collapse, and the cat becomes definitively dead or alive [00:04:31].
• Schrödinger’s Intent: While often used to illustrate the strangeness of quantum mechanics, Schrödinger’s actual goal was to show that the standard formulation was flawed [00:04:40].

Re-examining Key Concepts

To find a “better way” to understand Schrödinger’s cat and quantum mechanics, it’s helpful to examine superposition, entanglement, and measurement.

Superposition

Superposition is the idea that quantum objects can exist in multiple states simultaneously [00:05:17].

• Evidence: The double-slit experiment demonstrates superposition: individual electrons fired through two slits produce an interference pattern, implying a single electron passes through both slits simultaneously [00:05:26].
• Analogy to Waves: This is easily understood with waves, which are spread out in space and can interfere [00:05:51]. Since electrons are represented by a wave function when not observed, the double-slit experiment confirms the reality of this wave, allowing electrons to pass through both slits at once [00:06:04]. Thus, superposition is considered a solid concept [00:06:18].

Entanglement

Entanglement occurs when two or more particles become linked such that the quantum state of each particle cannot be described independently of the others [00:07:09].

• Example: When two electrons scatter off each other, their trajectories are described by spread-out wave functions [00:06:22]. If the momentum of one electron is measured, the momentum of the other is instantaneously known (due to conservation of momentum), even if they are light-years apart [00:06:37]. Before measurement, each electron’s momentum was in a superposition of states; measuring one immediately collapses the wave function of the other [00:06:51].
• Single Wave Function: After interaction, entangled electrons are described by a single, shared wave function [00:07:09]. More rigorously, there is ultimately only one wave function: that of the entire universe [00:07:28]. Entanglement is the result when particles interact with something else [00:07:44].

Measurement as Entanglement

The measurement postulate was added to connect the mathematics of quantum mechanics to observable reality [00:08:11]. However, it seems odd to have different rules for system evolution based on whether or not we are observing [00:08:21]. Measurement can be understood as the interaction of one quantum system (e.g., electrons, photons) with another [00:08:30]. If the separate measurement rule is discarded, then in the Schrödinger’s Cat experiment, when an observer opens the box, there is no wave function collapse [00:08:44]. Instead, the observer (who is also quantum mechanical) simply becomes entangled with the state of everything inside the box [00:08:57].

The Many-Worlds Interpretation

This interpretation, formulated by Hugh Everett, resolves the measurement problem by removing the concept of wave function collapse [00:11:12].

• Branching: When a quantum object in a superposition becomes entangled with its environment, it undergoes environmental decoherence [00:09:51]. This causes the wave function of the universe to “branch,” essentially splitting into slightly different copies [00:09:56].
• Schrödinger’s Cat in Many-Worlds:
  1. The radioactive atom is in a superposition of decayed and not decayed [00:10:09].
  2. The detector becomes entangled with this superposition [00:10:17].
  3. The detector interacts with environmental particles (air molecules, photons), causing it to decohere and branch the wave function [00:10:23].
  4. At this moment, the observer is split into two identical copies, each entangled with one outcome of the experiment (e.g., one seeing a live cat, one seeing a dead cat) [00:10:43].
  5. Both outcomes “happen,” but to separate, non-interacting versions of the observer [00:10:56].
• Implications:
  • The branching of the wave function is happening constantly, possibly infinitely often [00:11:18].
  • The Many-Worlds interpretation takes the mathematical implications of quantum mechanics seriously [00:11:30].
  • It offers a cleaner and more elegant formalism, where only wave functions evolving under the Schrödinger equation are needed [00:11:50].
  • It suggests that the wave function is the complete picture of reality, and our measurements are just tiny fractions of it [00:12:00].
  • The universe becomes deterministic again, as every outcome happens 100% of the time, just in different branches [00:12:15].

Common Objections and Clarifications (with Sean Carroll)

Sean Carroll, a physicist and author on Many-Worlds, addresses common questions [00:12:39]:

• Energy Conservation: Energy is 100% conserved for the total wave function of the universe [00:13:04]. The universe is not duplicated but “subdivided” or “sliced” into pieces, with each branch perceiving a certain amount of energy [00:13:22].
• Frequency of Branching: Branching occurs frequently whenever a quantum system in superposition becomes entangled with its environment [00:13:50]. For example, radioactive decays in one’s body (around 5,000 times per second) cause branching [00:14:03]. It’s unknown if branching happens infinitely often, as this depends on unexplored aspects of quantum gravity and cosmology, but the number of branches is “humongous” [00:14:26].
• “Everything That Could Possibly Happen, Happens”: This is a misconception [00:14:52]. Many-Worlds means the wave function obeys the Schrödinger equation, which predicts many potential outcomes but not everything [00:14:59]. For example, an electron will not convert into a proton because the Schrödinger equation assigns zero probability to such an event [00:15:09].
• Instantaneous Branching: Whether branching happens instantly across all space or spreads at the speed of light is a matter of description [00:17:25]. Both descriptions yield the exact same predictions for what is observed [00:17:42]. This is because “branches” are a convenient human way to understand the universe’s wave function, not an inherent part of reality itself [00:17:58], similar to how temperature describes air instead of listing every molecule’s position and velocity [00:18:21].