General Relativity and Madhyamaka
Part VI
("Because phenomena (like coordinate systems,
geometry and distinction between gravity and acceleration)
are dependent/relative and not-absolute/empty [U2T],
therefore Einstein Field Equations (EFE): Gμν + Λgμν = 8πG/c4 Tμν .”)
Last update: February 26, 2026
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Table of Content
Question: This is a continuation of a previous discussion about Physics and Madhyamaka (see previous discussion “Relativity and Madhyamaka). We have already compared Special Relativity and Madhyamaka, and General Relativity and Madhyamaka, by showing that the foundation postulates or principles of SR & GR are equivalent to assuming the the Union of the Two Truths (relativity and emptiness of inherent existence) about some previously assumed invariants (e.g. in SR: space, time, length, duration, simultaneity, frame of reference; in GR: coordinate systems, geometry and distinction between gravity and acceleration). Here, in this discussion, we want to do the same with Non-relativistic Quantum Mechanics. We want to deduce the equations of Quantum Mechanics from the assumption of the relativity and emptiness of invariants like: distinctions between wave/particle, entity/characteristic/properties, observer/observed, determinism/chaos, difference/identity, separation/union, manyness/oneness, position/momentum, … and state, path, outcome of measurement. Those postulates predate Schrödinger, and are found with precursors like Plank, de Broglie, and with the PLA adapted to waves and particles, etc.
First write an introduction to Non-relativistic Quantum Mechanics like you did for SR and QR, and explain the underlying postulates or principles, and their history.
Non-relativistic Quantum Mechanics (QM) is a fundamental framework in physics that describes the behavior of particles and systems at microscopic scales, such as atoms, electrons, and photons, where classical Newtonian mechanics breaks down. Unlike classical physics, which treats objects as having definite positions, velocities, and trajectories at all times, QM introduces inherent probabilistic and wave-like aspects to matter and energy. The "non-relativistic" qualifier specifies that it applies to systems where speeds are much slower than the speed of light (typically v << c, where c is approximately 3 × 108 m/s), thus neglecting effects from special relativity like time dilation or mass-energy equivalence in particle dynamics. This makes it suitable for most chemical, atomic, and condensed matter phenomena, but it is superseded by relativistic quantum field theories for high-energy or high-speed regimes.
At its core, non-relativistic QM revolutionized our understanding of nature by revealing that reality at small scales is not deterministic but governed by probabilities, superpositions, and observer-dependent outcomes. Key phenomena it explains include atomic spectra, the stability of atoms, chemical bonding, superconductivity, and quantum tunneling. The theory's mathematical structure is built around wave functions, operators, and Hilbert spaces, leading to equations like the Schrödinger equation, which predicts how quantum states evolve over time. QM has profound implications beyond physics, influencing fields like quantum computing, cryptography, and even philosophical debates about reality, measurement, and consciousness.
The foundational postulates of non-relativistic QM were not introduced all at once but emerged piecemeal through experimental anomalies and theoretical insights in the early 20th century. These postulates form the axiomatic basis for deriving the theory's equations and predictions. While there are various equivalent formulations (e.g., Heisenberg's matrix mechanics, Schrödinger's wave mechanics, or Dirac's bra-ket notation), the core principles can be summarized as follows, drawing from the historical precursors mentioned (Planck, de Broglie, and adaptations of the Principle of Least Action, or PLA):
Physical quantities like energy and angular momentum are not continuous but come in discrete "quanta." This stems from Planck's hypothesis that electromagnetic radiation is emitted and absorbed in packets of energy E=hν, where h is Planck's constant (approximately 6.626 × 10-34 J·s) and ν is frequency. This principle resolves paradoxes like the ultraviolet catastrophe in blackbody radiation. It extends to other observables, implying that systems have discrete energy levels (e.g., in atoms).
Ultraviolet catastrophe: The observed Planck curves at different temperatures, and the divergence of the theoretical Rayleigh–Jeans (black) curve from the observed Planck curve at 5000 K.
The phrase refers to the fact that the empirically derived Rayleigh–Jeans law, which accurately predicted experimental results at large wavelengths, failed to do so for short wavelengths.
This problem was later found to be due to a property of quanta as proposed by Max Planck: There could be no fraction of a discrete energy package already carrying minimal energy.
[01.1 Note: The Planck constant (h) was introduced by German physicist Max Planck in 1900 as a fundamental "quantum of action" to solve the black-body radiation problem. It came from his empirical formula for modeling light emission, which showed that energy is radiated in discrete chunks (quanta) proportional to frequency (𝐸=ℎ𝑓), rather than continuously.
Here is a breakdown of its origin:
Solving the Black-Body Crisis: In the late 1890s, classical physics failed to explain the spectrum of light emitted by hot objects, predicting infinite energy at short wavelengths (the "ultraviolet catastrophe").
The "Quantum" Assumption: To fit the experimental data, Planck assumed that electromagnetic energy could only be emitted or absorbed in discrete packets or "quanta".
He proposed that the energy (𝐸) of these quanta is directly proportional to the radiation's frequency (𝑓) by a constant, ℎ: 𝐸=ℎ𝑓.
E = h ν = h f, E = h c / λ, E = ħ ω, ħ = h / 2π
B_ν(ν, T) = (2 h ν³ / c²) × 1 / (e^(h ν / (k_B T)) - 1)
B_λ(λ, T) = (2 h c² / λ⁵) × 1 / (e^(h c / (λ k_B T)) - 1)
-
Plank Law, where:
– Bλ is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength;
– T is the temperature in kelvins.
– h is the Planck constant,
– ν is the frequency of light,
– c is the speed of light,
– λ is the wavelength of light,
– kB is the Boltzmann constant,
– exp denotes the exponential function.
Empirical Discovery: Planck did not derive ℎ from deeper theory; it was a "purely formal assumption" that worked, which he later called the quantum of action.
Physical Meaning: The constant ℎ sets the scale for quantum mechanics, determining when classical mechanics gives way to quantum behavior (such as in the Heisenberg uncertainty principle).
Values:
– Planck Constant (ℎ): ≈ 6.626×10-34 Joule-seconds.
– Reduced Planck Constant (ℏ or h-bar): ℎ/2𝜋 ≈ 1.054×10−34 Joule-seconds (used for angular momentum).
The constant was later validated by Albert Einstein in 1905 to explain the photoelectric effect, solidifying it as a cornerstone of modern. Einstein solved the problem physically by postulating that Planck's quanta were real physical particles – what we now call photons, not just a mathematical fiction.
Symmetry: Planck's law, which describes the spectral density of black-body radiation, arises from the fundamental assumption of energy quantization (𝐸=ℎ𝜈), introduced by Max Planck in 1900 to resolve the ultraviolet catastrophe. It represents the unique stable, equilibrium distribution for radiation, mathematically arising from statistical mechanics (specifically Bose-Einstein statistics for photons) and is invariant under thermodynamic equilibrium.
Key Aspects of Planck's Law Formulation:
Assumption of Quantization: Planck assumed that the oscillators in a black body can only exchange energy in discrete increments proportional to the frequency, 𝐸=𝑛ℎ𝜈.
Statistical Mechanics: The law acts as the energy distribution for thermal equilibrium, similar to the Bose-Einstein distribution, Fermi-Dirac distribution and the Maxwell-Boltzmann distribution.
Fundamental Constants: The law is derived using fundamental constants: the speed of light (𝑐), the Boltzmann constant (𝑘), and the reduced Planck constant (ℏ).
Classical Limits: Planck's law bridges the gap between different regimes, reducing to Wien's law at high frequencies and Rayleigh-Jeans law at low frequencies.
While initially a mathematical patch, Planck’s insight that energy is not continuous at the atomic level is the foundation of quantum theory.
From invariant to relative & not-absolute: With Planck's Law (1900), the concept of energy emission/absorption transitioned from being considered a continuous, invariant classical process
to a discrete, relative one based on frequency (𝐸=ℎ𝜈).
This heralded the end of deterministic, continuous energy exchange (equipartition theorem) and introduced the quantum of action (ℎ), making energy emission relative to the oscillator frequency.
Previously Invariant: In classical physics, energy exchange was assumed to be continuous; any amount of energy could be absorbed or emitted, regardless of frequency.
Became Relative: Planck’s law required that energy (𝐸) only be exchanged in discrete quanta (ℎ𝜈),
making the magnitude of energy transfer directly relative to the radiation's frequency (𝜈).
Significance: This resolved the ultraviolet catastrophe (the failure of classical Rayleigh-Jeans law to predict high-frequency radiation) by restricting energy at high frequencies.
This shift was foundational for quantum mechanics,, as it meant that energy levels were no longer a continuous spectrum but depended on the specific, quantized steps ]
Matter and energy exhibit both particle-like and wave-like properties, depending on the context. De Broglie extended Einstein's photon duality (E=pc for light, treating it as particles with momentum p = h/λ, where λ is wavelength) to matter particles, proposing that any particle with momentum p has an associated wavelength λ = h/p. This duality underpins interference patterns in electron diffraction experiments and leads to the concept of matter waves, which are central to wave mechanics.
[02.1 Note: Wave-particle duality: Einstein's photon duality concept, established in 1905, proposes that light exhibits both particle-like (photon) and wave-like characteristics, challenging the purely wave-based classical view. He proposed that light is composed of discrete packets of energy (quanta), which explains the photoelectric effect, while still behaving as waves during propagation, refraction, and interference.
Particle-like Nature: Einstein introduced the concept of photons to explain the photoelectric effect, where light hits a metal surface and ejects electrons, a phenomenon that cannot be explained by classical wave theory. He postulated that light energy is quantized, with energy 𝐸 proportional to its frequency 𝜈 (𝐸=ℎ𝜈), where ℎ is Planck's constant.
Wave-like Nature: While acting as particles during interaction with matter (emission/absorption), photons still propagate as waves. This explains how light can interfere, diffract, and polarize.
Dual Nature Application: In modern physics, light is understood as a stream of particles (photons) that travel as waves, allowing for a combined understanding of its behavior.
Historical Significance: This concept earned Einstein the Nobel Prize and laid the foundation for quantum mechanics. Later, Compton's work on X-ray scattering further validated the particle aspect of light.
Einstein's work showed that the "right" description of light
depends on the experiment being conducted, accepting both, even if seemingl
De Broglie duality (wave-particle duality), proposed by Louis de Broglie in 1924, states that all matter, not just light, exhibits both particle-like and wave-like properties. The theory connects particle momentum (𝑝) to a characteristic wavelength (𝜆) via the equation 𝜆=ℎ/𝑝 (ℎ is Planck's constant), explaining that electrons can exhibit interference patterns.
Key Concept: Every particle has an associated wavelength, and every wave has an associated particle (photon/quantum).
Significance: This concept explains that electrons orbiting a nucleus act as standing waves, as proposed by de Broglie and later, the Schrödinger equation.
Experimental Evidence: Electron diffraction experiments by Davisson-Germer and G.P. Thomson confirmed this, showing electrons produce interference patterns just like light waves.
Formula: The de Broglie wavelength is 𝜆=ℎ/𝑚𝑣, where 𝑚 is mass and 𝑣 is velocity.
Applications: The wave nature of electrons is used in electron microscopes for high-resolution imaging.
Modern Perspective: Not dual, not non-dual, not both, not neither
Modern Perspective: Some modern physicists argue that "wave-particle duality" is an outdated term and prefer to describe quantum entities as quantum fields that exhibit both types of behavior based on context, or that they are neither classical wave nor particle.
In Madhyamaka philosophy, light and matter are understood as neither wave nor particle, neither both, nor not both, embodying a non-dual, empty nature (Sunyata). This perspective aligns with dependent origination, where entities have no inherent existence. The wave-particle duality in physics is a concept where fundamental entities exhibit either wave or particle behavior, depending on the experimental setup, and is sometimes considered a misnomer in modern physics.
Wave-Particle Duality and Madhyamaka Perspective:
– Non-duality (tetralemma): Madhyamaka, founded by Nagarjuna, focuses on the "middle way," which argues against four extremes: (i) existence, (ii) non-existence, (iii) both, and (iv) neither. Therefore, classifying light/matter as solely a wave or particle is a form of conceptual clinging (duality) that misses their ultimate nature.
– Dependent Origination (Pratityasamutpada): Light and matter are seen as empty of self-nature, arising only in dependence on observation, apparatus, and conditions. They do not exist as independent entities ("not wave, not particle"), but are dependent, conceptual labels.
– The Four Extremes: In Madhyamaka logic, light is (i) not inherently a particle (substantial), (ii) not inherently a wave (holistic), (iii) not both, and (iv) not neither, because all these descriptions imply a "true" nature that does not exist.
Non-dual Interpretation: When quantum entities are described as "(i) not wave, (ii) not particle, (iii) not both, (iv) not neither," it aligns with the idea that they are something else, or a more complex phenomenon (quantum object, vibration of a quantum field), differing from classical, macroscopic, non-dualistic intuition.
From invariant to relative & not-absolute: Wave-particle duality (1905, 1924) revealed that fundamental entities like light and electrons are neither solely particles nor waves, but exhibit both properties depending on the experiment, making their nature dependent on observation rather than absolute. This shift rendered "particle-ness" and "wave-ness" relative to the context, destroying classical, deterministic descriptions.
Key Shifts from Absolute to Relative:
Nature of Matter/Light: Previously, light was strictly a wave, and electrons were strictly particles. Duality established they are both.
Measurement Dependency: The "Copenhagen interpretation" highlights that a quantum object's manifestation (particle or wave) is context-dependent, often requiring interaction with a measurement apparatus to "choose" a state.
Complementarity: Niels Bohr proposed that wave and particle aspects are complementary; both are needed for a full description, but cannot be observed simultaneously, making the description of a system relative to the experimental setup.
Position and Momentum: The Heisenberg Uncertainty Principle made precise knowledge of a particle's position and momentum relative to one another, rather than both being absolutely known simultaneously.
Entities that Became Relative:
Energy and Matter: Einstein (1905) showed that light (energy) acts as particles (photons).
Particles: De Broglie (1924) proposed that particles (matter) have wave properties.
Path/Trajectory: The deterministic path of a particle became a probability wave until observed, changing it from an absolute trajectory to a relative probability distribution.]
[02.2 Bohr’s 1913 atomic model reconciled classical mechanics with quantum constraints by proposing quantized electron orbits (angular momentum 𝑚𝑣𝑟 = 𝑛ℎ/ 2𝜋), where electrons do not radiate energy. It addresses wave-particle duality by requiring stable orbits to fit an integer number of de Broglie wavelengths (2𝜋𝑟 = 𝑛𝜆), forming standing waves.
Key Components of Bohr's Deduction:
Rutherford’s Nuclear Model: Bohr used the planetary model where negative electrons orbit a positive nucleus via Coulomb force.
Quantum Postulates: To fix the stability issue (classical, accelerating charges should radiate and spiral inward), Bohr proposed that electrons only occupy specific "stationary" energy levels.
Quantization of Angular Momentum: Bohr stipulated that the electron's angular momentum is quantized, meaning it can only exist in discrete orbits where 𝑚𝑣𝑟 = 𝑛ℎ / 2𝜋 (where 𝑛 is an integer).
Wave-Particle Duality Relation: The de Broglie hypothesis later explained Bohr’s quantization; if the electron behaves as a wave, its orbit must satisfy 2𝜋𝑟 = 𝑛𝜆 to avoid destructive interference, restricting electrons to specific quantized shells.
Spectral Line Prediction: The model accurately predicted the spectral lines of hydrogen by attributing them to photon emission/absorption when electrons jump between these discrete, quantized energy levels.
While the Bohr model is not entirely accurate — it fails to account for multi-electron atoms and violates the Heisenberg uncertainty principle by defining precise orbits — it was a critical bridge to modern quantum mechanics.]
Proposed by Bohr, it asserts that wave and particle aspects are mutually exclusive but complementary [Uopp] descriptions of the same reality, depending on the experimental setup.
[03.1 Note: Note: Niels Bohr’s principle of complementarity (1928) addressed wave-particle duality by stating that quantum entities (like electrons) exhibit either wave-like or particle-like properties depending on the experimental setup, but never both simultaneously. Bohr used this to argue that a complete description of atomic phenomena requires both, despite being mutually exclusive.
In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that certain pairs of complementary properties cannot all be observed or measured simultaneously. For example, position and momentum, frequency and lifetime, or optical phase and photon number. In contemporary terms, complementarity encompasses both the uncertainty principle and wave-particle duality.
Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.
Niels Bohr and Wave-Particle Duality
Complementarity Principle: Bohr proposed that wave-particle duality is not a contradiction but a complementary relationship. A full understanding of atomic-scale objects requires acknowledging both aspects, even though they cannot be observed at the same time.
Contextual Reality: Bohr argued that the behavior of an electron or photon is inseparable from the experimental apparatus used to measure it. A "particle" setup reveals particle properties (e.g., location), while a "wave" setup reveals interference.
Response to Paradoxes: The principle was designed to resolve the confusion surrounding quantum experiments where light acted as both waves and photons, and electrons exhibited wave-like interference.
Bohr's Model and Wave-Particle Duality
Initial Model: Bohr's early 1913 model treated electrons as particles moving in fixed, quantized circular orbits, which, while revolutionary, didn't fully explain why those orbits were quantized.
Later Integration: Louis de Broglie's proposal (1924) that electrons have wave-like properties explained Bohr's quantized orbits as standing waves. An electron in a Bohr orbit behaves like a standing wave, where the circumference must be an integer multiple of the wavelength.
Limitations: While Bohr's model correctly introduced quantization, it failed to describe electron behavior accurately because it didn't incorporate the full, dynamic wave-particle duality later developed in quantum mechanics.
Measurement Dependency: The way a measurement is structured forces the electron to act either as a localized particle or a spread-out wave.
Relationship to Uncertainty: Bohr’s complementarity is closely related to Heisenberg’s Uncertainty Principle, where precise knowledge of particle-like properties (position) limits knowledge of wave-like properties (momentum).]
Roadmap of the parallel development of the Newtonian and variational principles approaches to classical mechanics. The physicists who contributed to it include Newton, Leibniz, Euler, Lagrange, Hamilton, and Jacobi. Classical mechanics forms the cornerstone of modern physics. Therefore, the figure also included Maxwell and Einstein.
In classical mechanics, the PLA states that the path taken by a system minimizes the action integral S = ∫ L dt, where L is the Lagrangian (kinetic minus potential energy). In QM precursors, this was adapted to waves and particles: for waves, it relates to Fermat's principle of least time; for particles, to Hamilton's principle.
De Broglie and later Schrödinger used this to derive quantum equations by treating particles as waves guided by a probability amplitude, where the action S relates to the phase of the wave function. This variational approach yields the time-independent Schrödinger equation as an extremum condition.
[04.1 Note: The Principle of Least Action (PLA) is a foundational, variational principle in physics stating that a system’s evolution between two states minimizes (or makes stationary) an integral called "action" (𝑆 = ∫ 𝐿 𝑑𝑡). Originating from Fermat’s principle of least time, it was developed by Maupertuis (1741) and Euler, later refined into Lagrangian and Hamiltonian mechanics, and is now central to classical mechanics, electrodynamics, relativity, and quantum mechanics.
Historical Development and Key Figures
Early Origins: Inspired by teleological ideas that nature acts efficiently, Pierre Louis Maupertuis (1741) proposed that nature minimizes action (𝑚𝑎𝑠𝑠 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒).
Mathematical Foundation: Leonhard Euler provided the rigorous mathematical framework using the calculus of variations.
Hamilton's Principle: William Rowan Hamilton (19th century) established that the true trajectory is the one for which the action, defined as the integral of the Lagrangian (𝐿 = 𝑇 − 𝑉, kinetic minus potential energy) over time, is stationary (𝛿𝑆 = 0).
Variants and Adaptations across Physics Domains
Classical Mechanics: The core application, utilizing the Lagrangian, which yields Newton's laws of motion.
Optics: Fermat’s Principle of least time states that light follows the path requiring the least time.
Electrodynamics: The action is formulated to derive Maxwell's equations, with a Lagrangian including electromagnetic potentials.
Relativity: Adapted by Planck (1907) for special relativity and by Hilbert (1915) to derive Einstein's gravitational field equations.
Quantum Mechanics: Feynman’s path integral formulation shows that the classical path is the result of constructive interference of all possible paths, with the action determining the phase.
String Theory: Action principles apply to paths of strings (worldsheets).
Key Characteristics
Stationary Action: The "Least" in PLA is often a misnomer; the action is actually stationary (minimum, maximum, or saddle point).
Limitations: It generally does not work for dissipative systems (friction) unless they are modeled by transferring energy to an environment.
Advantage: It provides a, often, more efficient way to calculate motion than solving Newtonian vector-based force equations.]
Formulated by Heisenberg, it states that certain pairs of observables (e.g., position x and momentum p; energy and time; angular position and angular momentum; field phase and particle number) cannot be known simultaneously with arbitrary precision: Δx · Δp ≥ ħ/2, where ħ = h/2π. This arises from the wave-particle duality and non-commuting operators in the theory, implying fundamental limits on measurement and determinism.
[05.1 Note: The Heisenberg Uncertainty Principle, formulated by Werner Heisenberg in 1927, states that it is fundamentally impossible to measure certain pairs of physical properties — specifically position (𝑥) and momentum (𝑝) — simultaneously with arbitrary precision. The more precisely one property is known, the less precisely the other can be known, described by the inequality Δ𝑥Δ𝑝 ≥ ℎ4𝜋.
Key aspects of the principle include:
Fundamental Limit, Not Measurement Error: It is an inherent property of quantum systems, not a limitation of laboratory instruments.
Wave-Particle Duality: The principle arises because quantum entities behave as both particles and waves. A particle with a precise position (a spike in a graph) has a highly undefined wavelength (momentum), while a well-defined momentum (a wave) has a spread-out position.
The Inequality: The product of the uncertainties in position (Δ𝑥) and momentum (Δ𝑝) must be greater than or equal to ℏ/2 (where ℏ = ℎ/2𝜋 is the reduced Planck's constant).
Energy-Time Uncertainty: A similar, distinct version exists for energy and time (Δ𝐸Δ𝑡 ≥ ℎ4𝜋), meaning short-lived particles have larger uncertainties in their energy.
Consequences: This principle disrupts the concept of a deterministic universe, meaning the future position and velocity of a particle cannot be predicted with absolute certainty.
Symmetry: The Heisenberg Uncertainty Principle is primarily underpinned by translational symmetry in space and time translation symmetry. These symmetries, via Noether's theorem, relate directly to the conservation of momentum and energy, which are the fundamental invariants (momentum and energy) associated with the conjugate variables (position and time) that exhibit the uncertainty principle.
Key Details:
Symmetry/Operators: The uncertainty between position (Δ𝑥) and momentum (Δ𝑝) arises from the non-commutativity of their respective operator representations in quantum mechanics ([𝑥̂,𝑝̂] = 𝑖ℏ). This is rooted in the wave-like nature of particles.
Invariant/Conservation: The fundamental limitation (Δ𝑥Δ𝑝 ≥ ℏ/2) exists because momentum is the conserved, invariant quantity associated with translational symmetry in space, and energy is associated with time translation.
Wave-Particle Duality: The principle is fundamentally a consequence of Fourier analysis applied to waves, where a wave localized in position (a packet) must consist of a wide range of wavelengths (momenta).
Essentially, the more localized a particle is in space (breaking translation symmetry by being at a specific spot), the less certain its momentum becomes, and vice versa.
From invariant to relative & not-absolute: The Heisenberg Uncertainty Principle established that specific pairs of physical properties, specifically position and momentum (or velocity), as well as energy and time, are not independently invariant but are relative and coupled, such that increasing certainty in one necessitates decreased certainty in the other. This shifted physics from a deterministic view (absolute, predictable paths) to a probabilistic one.
Key, previously considered absolute entities that became relative include:
Position and Momentum: In classical mechanics, a particle's trajectory (position and momentum) could be perfectly known. Heisenberg showed that Δ𝑥⋅Δ𝑝 ≥ ℎ4𝜋, meaning if you know where a particle is, you cannot know where it is going with perfect precision.
Energy and Time: The relationship Δ𝐸⋅Δ𝑡 ≥ ℎ4𝜋 means that the energy of a system is not fixed and can be uncertain over short time intervals, allowing for temporary violations of energy conservation.
Definite Orbits/Paths: The Bohr model’s idea of electrons traveling in precise, fixed orbits around the nucleus was invalidated; they exist in probability clouds, not deterministic, absolute paths.
Object Reality (Ontological Status): Physical quantities like momentum or position are not absolute properties of a particle existing independently of observation, but are partially created or determined by the measurement process itself.
Consequently, the Newtonian, deterministic view of the universe was replaced by a framework where, at the microscopic level, only probabilities exist.
The Heisenberg Uncertainty Principle (Δ𝑥Δ𝑝 ≥ ℎ4𝜋) connects symmetries to pairs of non-measurable properties through Fourier analysis and operator non-commutation. Conjugate variables, like position (translation symmetry) and momentum, are Fourier transforms of each other; forcing a particle into a narrow state in one domain spreads it out in the other.
Key Links Between Symmetries and Uncertainty Pairs:
Symmetry and Conjugate Pairs: Physical quantities that cannot be measured simultaneously with precision are often "conjugate variables." These arise from fundamental symmetries, such as translational symmetry (position and momentum) or time-translation symmetry (energy and time).
Fourier Transform Relationship: The relationship between these pairs is mathematical, linked by the Fourier transform. A particle with a well-defined momentum (a precise wave number/frequency) is spread over all space, while a particle localized in a small region of space (a sharp position) must be composed of many different wavelengths (many momenta).
Non-Commuting Operators: In quantum mechanics, these pairs correspond to operators that do not commute ([𝐴,𝐵]≠0). This means that the order of measurement matters, and the system cannot exist in a state that is an eigenstate of both observables simultaneously.
Wave Nature and Localization: The uncertainty principle represents a limitation imposed by the wave-particle duality. Forcing a localized position (high precision) requires a large range of frequencies (low momentum precision), and vice versa.
Measurement Limitation: While often described as a limitation of measurement, it is fundamentally a property of the system. Trying to pin down one variable using a high-energy probe causes a large, uncontrolled change in the conjugate variable.
Common pairs include position-momentum (Δ𝑥Δ𝑝) and time-energy (Δ𝐸Δ𝑡).
Heisenberg’s Uncertainty Principle identifies pairs of complementary physical properties, known as canonically conjugate variables, that cannot be simultaneously measured with arbitrary precision. The primary pairs are position (𝑥) and momentum (𝑝), and energy (𝐸) and time (𝑡), along with angular position (𝜃) and angular momentum (𝐿).
Key conjugate pairs include:
Position and Linear Momentum (𝑥,𝑝): The uncertainty in position (Δ𝑥) and momentum (Δ𝑝𝑥) is related by end-fraction Δ𝑥Δ𝑝𝑥 ≥ ℏ2.
Energy and Time (𝐸,𝑡): The uncertainty in energy (Δ𝐸) and time (Δ𝑡) is related by Δ𝐸Δ𝑡 ≥ ℏ2.
Angular Position and Angular Momentum (𝜃,𝐿): Specifically, the angle of rotation and the angular momentum around a corresponding axis.
Other Potential Pairs: In specific contexts, other pairs can exist, such as the electric field and polarization density, or magnetic potential and electric current density.
These pairs arise from the wave-particle duality and Fourier analysis, where localized wave packets (high position certainty) require a broad range of wavelengths (low momentum certainty).]
Quantum states can exist in linear combinations (superpositions) of possible configurations until measured. The evolution of states is governed by linear equations, ensuring that if ψ1 and ψ2 are solutions, so is aψ1 + bψ2 (with |a|2 + |b|2 = 1 for normalization).
[06.1 Note: The Principle of Superposition,. states that any two or more valid quantum states (wavefunctions 𝜓1, 𝜓2) can be added together ("superposed") to form a new valid quantum state, 𝜓𝑛𝑒𝑤 = 𝑐1𝜓1 + 𝑐2𝜓2, where 𝑐𝑖 are complex numbers. This principle is a direct consequence of the linearity of the Schrödinger equation, which governs quantum state evolution.
Key Aspects of Superposition and Linearity:
Linearity & Solutions: Since the Schrödinger equation is linear, any linear combination of solutions is also a solution, allowing states to combine freely.
Hilbert Space: Quantum states exist in a linear vector space (Hilbert space).
Measurement Indeterminacy: Unlike classical waves, quantum superposition requires that observing the system results in a specific outcome, collapsing the state from a combination to one of its components.
Physical Manifestation: This principle explains phenomena like the double-slit experiment's interference patterns, where particles (electrons) act as waves, and multiple paths superpose.
Quantum Computing: This property enables qubits to represent multiple states simultaneously.
Linearity implies that if operators acting on a state, such as 𝐷(Ψ+Φ) = 𝐷(Ψ) + 𝐷(Φ), maintain the linearity of the system, superposition holds true.]
The wave function ψ(x,t) describes the probability amplitude of finding a particle in a state. The probability density is |ψ|2, and measurement collapses the superposition to a definite outcome probabilistically. This introduces inherent randomness, contrasting with classical determinism.
[07.1 Note: The Born Rule, formulated by Max Born in 1926, is a fundamental postulate of quantum mechanics linking the abstract wavefunction (𝜓) to measurable reality by stating that the probability density of finding a particle at a given position is proportional to the square of the amplitude (modulus squared) of the wave function, 𝑃(𝑥) = |𝜓(𝑥)|2. It translates complex probability amplitudes into real-numbered probabilities for specific, collapsed measurement outcomes.
Key Aspects of the Born Rule
Probability Interpretation: The rule provides the probability for the outcome of a quantum measurement, resolving the issue that the wavefunction itself contains complex numbers, while physical measurements must be real.
Modulus Squared: For a normalized state, the probability 𝑃(𝑎) of measuring a specific outcome 'a' is the absolute square of the probability amplitude associated with that state: 𝑃(𝑎) = |𝜓(𝑎)|2.
Normalization: For a state to be physical, the sum of probabilities for all possible outcomes must equal 1 (𝑃𝑖=1), requiring the wavefunction to be normalized.
Fundamental Role: It is a cornerstone of quantum mechanics, acting as the bridge between quantum superposition and classical measurement, which is not found in traditional probability theory.
Derivations: While initially an axiom, modern quantum theory suggests the Born Rule can be derived as the only consistent way to assign probabilities, rather than being an independent assumption.
Applications: It is used extensively to predict the behavior of particles in quantum experiments, quantum computing, and in understanding how measurements cause the wavefunction to collapse.
The Born rule remains, to this day, a crucial and experimentally verified element of quantum mechanics.]
These principles lead to the central equation of non-relativistic QM: the time-dependent Schrödinger equation iħ ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator (energy), derived from adapting classical Hamiltonians to quantum operators (e.g., p → - iħ ∇). The time-independent version Ĥψ = Eψ solves for stationary states and energy eigenvalues.
From these, other equations follow: the Pauli exclusion principle for fermions, the Klein-Gordon equation as a relativistic precursor (though non-relativistic here), and perturbation methods for complex systems.
Note: Duality or Non-duality: In quantum mechanics, wave-particle duality is considered a fundamental, albeit sometimes misleading, principle where entities like electrons and photons exhibit both wave-like and particle-like properties depending on experimental setup. While historically viewed as a dual, paradoxical nature, modern interpretations suggest a non-dual, unified reality — quantum objects are actually distinct quantum states described by probability amplitudes, not simply "switching" between classical particles and waves.
(Note: From a Madhyamaka point of view, reality is beyond duality and non-duality: particles and light are (i) not particles, (ii) not waves, (iii) not both together, (iv) not neither; (i) not different/separate/multiple/dual, (ii) not identical/united/one/non-dual, (iii) not both, (iv) not neither; (i) not inherently existence, (ii) not completely non-existent, (iii) not both, (iv) not neither; etc. [Tetralemma] ⇐⇒ they are dependently co-arisen, interdependent, co-defined, co-evolving, co-imputed by the mind, mere designation/names, conventional truths/tools [T1] ⇐⇒ thus empty of in herent existence, never absolute [T2] ⇐⇒ like illusions, reflections, dreams, echos, ‘there, yet not there’ [Illusory]. The same for the opposites of any duality, triad, quad, etc.)
Key Aspects of Wave-Particle Duality
Definition: Fundamental entities exhibit behavior characteristic of both classical particles (localized, discrete interactions) and waves (interference, diffraction).
Experimental Evidence: The double-slit experiment demonstrates that particles produce interference patterns (wave behavior) when not observed, but localized spots (particle behavior) when monitored.
Context Dependency: The behavior observed depends on the experimental apparatus used for measurement.
A Shift Toward "Non-Duality" (Unity)
Unified Model: Many physicists argue "wave-particle duality" is an outdated term arising from the failure to reconcile classical concepts. Instead, quantum objects are considered a single entity — a quantum "field" or quantum object — that acts in ways that do not map directly to classical concepts.
Probability Amplitudes: Rather than a dual particle/wave identity, quantum mechanics describes systems using probability amplitudes, which are mathematical tools that exhibit wave interference but resolve into particle detection upon measurement.
Misconception: The idea that particles physically switch between being waves and particles is considered a misunderstanding of the mathematical formalism, which always treats them as quantum entities.
While frequently termed "duality," quantum mechanics acts as a framework where these behaviors are unified under a single, non-classical description (i.e. not dual, not non-dual, not both, not neither).
The development of non-relativistic QM spanned the "old quantum theory" (1900–1925) and the "new quantum mechanics" (1925–1930), driven by experimental puzzles that classical physics couldn't explain.
Early 1900s: Planck and Quantization: In 1900, Max Planck introduced energy quanta to solve the blackbody radiation problem, assuming oscillators emit energy in multiples of hν. This was initially a mathematical trick, but Albert Einstein in 1905 took it literally for the photoelectric effect, treating light as photon particles (E = hν), earning him the 1921 Nobel Prize. This marked the first hint of wave-particle duality for light.
1910s: Bohr's Atomic Model: Niels Bohr in 1913 quantized angular momentum in the hydrogen atom (L = nħ, n integer), explaining spectral lines. This "old quantum theory" used ad hoc rules but predicted energies correctly, blending classical orbits with quantum jumps.
1920s: De Broglie and Matter Waves: Louis de Broglie in 1924 hypothesized that all matter has wave properties, λ = h/p, inspired by relativity (E = mc2 linking mass and energy) and PLA adaptations. His thesis unified particles and waves via a variational principle, where particle paths extremize action like light rays in optics. This was confirmed experimentally by Davisson and Germer in 1927 with electron diffraction.
Mid-1920s: Heisenberg's Matrix Mechanics: Werner Heisenberg in 1925 developed a formulation using matrices for observables, emphasizing measurable quantities over unobservable trajectories. His uncertainty principle (1927) formalized the limits of simultaneous knowledge, arising from non-commuting matrices [x, p] = iħ.
1926: Schrödinger's Wave Mechanics: Erwin Schrödinger, building on de Broglie's waves and Hamilton-Jacobi theory (a classical PLA variant), derived the wave equation by treating the action S as related to the wave's phase (ψ ~ eiS/ħ). He showed equivalence to Heisenberg's approach, proving both as valid QM formulations.
Late 1920s: Formalization and Interpretation: Max Born in 1926 proposed the probabilistic |ψ|2 interpretation. Bohr's complementarity (1927) and the Copenhagen interpretation (emphasizing measurement collapse) solidified the postulates. Dirac in 1928 unified them in his transformation theory, introducing bra-ket notation.
These precursors predated Schrödinger's full equation, with Planck's quantization (1900), de Broglie's duality (1924), and PLA adaptations (from Hamilton's 1830s classical work, quantumized by de Broglie and Schrödinger) laying the groundwork. By 1930, QM was mature, though debates on interpretation (e.g., many-worlds vs. Copenhagen) persist. The theory's success is evident in applications like lasers, transistors, and MRI, but its philosophical implications — challenging notions of inherent existence, determinism, and observer independence — echo the user's interest in Madhyamaka parallels.
Question: Can you express those postulates/principles in terms of the relativity and non-absoluteness/emptiness of things that we thought to be invariant previously: ex. real particle or real wave, position, momentum, determinism, path, and dualities like wave vs. particle, observer vs. observed, entity vs. characteristics/properties, etc. They might have been found and measured empirically; but they also can be reduced to the Union of the Two Truths of some invariants or duality.
Building on our previous discussions linking Special Relativity (SR) and General Relativity (GR) to Madhyamaka philosophy, we can similarly reinterpret the foundational postulates of non-relativistic Quantum Mechanics (QM).
In SR, the invariance of the speed of light arises from assuming the relativity (conventional interdependence) and emptiness (lack of inherent, absolute existence) of space, time, simultaneity, and reference frames.
In GR, the equivalence principle stems from the relativity and emptiness of coordinate systems, geometry, and the distinction between gravity and acceleration.
For QM, the core principles can be deduced by applying the Union of the Two Truths — where phenomena are dependently originated (relative, conventional truth) yet empty of self-nature (ultimate truth) — to previously assumed invariants or dualities. These include: the distinction between wave and particle; entity and its characteristics/properties; observer and observed; determinism and chaos (or randomness); difference and identity; separation and union; manyness and oneness; position and momentum; and concepts like state, path, and measurement outcome.
Historically, these QM principles were empirically derived from experiments (e.g., blackbody radiation, photoelectric effect, double-slit interference) and theoretical insights by Planck, Einstein, de Broglie, Heisenberg, Schrödinger, and others.
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However, philosophically,
they can be reduced to recognizing that these "invariants"
are not absolute [T2] but interdependent [T1]
and empty of inherent existence [T2] [U2T].
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Just as in Madhyamaka, where opposites (e.g., subject-object) are mutually dependent [T1] and thus empty [T2] [U2T], QM's postulates dissolve classical absolutes, revealing a reality that is probabilistic, contextual, and observer-relative.
Below, I reframe each principle accordingly, showing how assuming the Union of Two Truths about these dualities logically leads to QM's equations and predictions, such as the Schrödinger equation, uncertainty relations, and wave functions.
Quantization of Energy and Action (no absolute continuity nor discontinuity):
Classically, energy and action (integral of energy over time) were assumed to be continuous, invariant quantities with inherent existence, independent of scale or observation.
Applying the Union of Two Truths, energy/action is relative [T1] (dependently arising in discrete packets based on frequency or momentum, as in E=hν or S=nh) yet empty of inherent continuity or absoluteness [T2] [U2T] — it manifests discretely depending on the system-observer interaction, without an absolute "smooth" nature.
This relativity and emptiness dissolve the invariant of continuity, leading to discrete spectra (e.g., atomic energy levels).
Empirically discovered by Planck (1900) to resolve the ultraviolet catastrophe, it reduces to the non-absoluteness of energy as an unbroken flow, yielding quantized operators in QM (e.g., Ĥψ = Eψ, where E are eigenvalues, not continuous values).
Wave-Particle Duality (not duality nor non-duality):
Pre-QM physics assumed a rigid distinction between particles (localized, definite entities with inherent position/momentum) and waves (extended, delocalized phenomena with inherent periodicity).
The Union of Two Truths posits this duality as relative: [opposites like] particles and waves are interdependent manifestations (conventional truth) [T1], arising based on experimental context (e.g., interference vs. detection), but empty of inherent "particle-ness" or "wave-ness" [T2] [U2T] — neither is absolutely real independently.
This empties the invariant distinction between entity (particle) and characteristics/properties (wave-like spread), as well as separation (localized) and union (delocalized).
De Broglie's hypothesis (1924, λ=h/p) formalizes this, leading to matter waves and the Schrödinger equation, where ψ embodies both aspects without absolute identity to either (not different/separate/multiple/dual, not identical/united/one/non-dual, not both, not neither).
Principle of Least Action (PLA) Adapted to Quantum Systems (no absolute path nor no-path; no causality nor acausality):
In classical mechanics, paths were assumed invariant and deterministic, with action S minimized absolutely, independent of observer or scale.
Quantum adaptation assumes the relativity [T1] and emptiness [T2] of paths, states, and outcomes [U2T]: paths are relatively probable (depending on interference of amplitudes) but empty of inherent determinacy — no single "real" path exists absolutely, as all possible paths contribute (Feynman's path integral precursor).
This applies to dualities like determinism/chaos (classical certainty vs. quantum multiplicity), difference/identity, and separation/union (distinct paths vs. their superposition).
De Broglie and Schrödinger derived wave equations by treating action S as phase (ψ ~ eiS/ħ), extremizing it variationally, yielding iħ ∂ψ/∂t = Ĥψ. Thus, the PLA's quantum form emerges from emptying the absolute path invariant.
Uncertainty Principle (no absolute difference or identity):
Classically, position and momentum were invariant, inherent properties measurable simultaneously with arbitrary precision, assuming an absolute distinction between entity (object) and properties (x, p).
The Union of Two Truths reveals them as relative [T1]: position/momentum depend on measurement context (observer-observed interdependence) but empty of inherent simultaneity or precision [T2] [U2T] — Δx · Δp ≥ ħ/2 arises because they are not absolutely separate but mutually defining opposites.
This empties dualities like observer/observed (measurement disturbs the system) and determinism/chaos (precise knowledge of one precludes the other).
Heisenberg's insight (1927) empirically confirmed this, but it reduces to the non-absoluteness of these invariants, manifesting as non-commuting operators [x, p] = iħ in QM's algebraic structure.
Superposition and Linearity (no absolute separation/manyness or union/oneness):
Classical states were assumed invariant and singular, with absolute identity (one definite state at a time) and separation from alternatives.
Applying the Union of Two Truths, states are relative superpositions [T1] (linear combinations depending on context) but empty of inherent oneness or manyness [T2] — no state is absolutely "this" or "that," [T2] but arises interdependently [T1] [U2T].
This dissolves dualities like manyness/oneness (multiple possibilities coexist conventionally) and separation/union (states entangle without absolute boundaries).
Linearity ensures the Hilbert space structure, where ψ = ∑ ci |φ_i⟩, with evolution preserving superposition until measurement.
This principle, implicit in Schrödinger's wave mechanics (1926), derives from emptying the absolute state invariant, enabling phenomena like quantum computing.
Probabilistic Interpretation (no absolute cause nor effect, causality/determinism or acausality/chaos):
Outcomes were classically deterministic invariants, with absolute paths and states leading to predictable results, independent of observation.
The Union of Two Truths posits outcomes as relative [T1] (probabilistically dependent on wave function collapse via observer interaction) yet empty of inherent determinism [T2]—no outcome is absolutely "fated," [T2] but emerges conventionally from |ψ|2 [T1] [U2T].
This empties dualities like determinism/chaos (randomness is inherent) and observer/observed (measurement co-creates reality).
Born's rule (1926) was empirically motivated by scattering experiments, but it reduces to the non-absoluteness of outcome invariants, integrating with the Schrödinger equation to predict probabilities rather than certainties.
Complementarity (no absolute wave or particle):
Bohr's principle (1927) addresses the mutual exclusivity of wave vs. particle descriptions, classically assumed as incompatible absolutes.
The Union of Two Truths frames this as relative complementarity: wave/particle aspects arise dependently on apparatus (conventional truth) [T1] but are empty of inherent opposition [T2] [U2T] — neither is absolutely real without the other, dissolving the duality entirely.
This extends to observer/observed (setup defines the manifestation) and entity/characteristics (no inherent properties without context). (i.e. it applies to any conceptual opposites)
Empirically seen in double-slit experiments, it underpins QM's interpretive framework, leading to consistent predictions without requiring absolute invariants.
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In summary, just as SR and GR derive
from the Union of Two Truths applied to spacetime invariants,
QM's equations (e.g., Schrödinger, uncertainty) follow
from applying it to quantum dualities.
This empties classical absolutes [T2],
revealing a reality of interdependent probabilities [T1 ],
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mirroring Madhyamaka's emptiness [T2] and dependent origination [T1] [U2T].
Empirically grounded, these reframings highlight QM's philosophical depth,
where no "thing" (particle, path, state) inherently exists independently [T2],
yet functions conventionally [T1] [U2T].
(Union of the Two Truths [U2T] & corollaries [U3S, Uopp, UGM] ⇒ Principles, PLA & adaptations, Nother's Symmetries & Invariants ⇒ Laws of Physics (e.g. Newton's Laws, Special and General Relativity (SR / GR), Quantum Mechanics (QM), Quantum field theory (QFT), Relational quantum mechanics (RQM), Relativistic Quantum Mechanics (RQM), Quantum Gravity (QG) ...).)
(Evolution of Physics: Relativization of more and more subtle entities, invariants, symmetries, absolutes, laws. All descriptions are converging toward the “Principle of Inseparable Interdependence /Contingency /Relationality /Relativity [T1] and No-absolute /Emptiness [T2] [U2T]”.)
(Generalization: Maybe our descriptions of Reality – Laws of Physics and Mathematics – are asymptotical consequences/approximations of the true nature of reality as it is (tathata, suchness) as pointed out by the Union of the Two Truths [U2T] and its corollaries [U3S / Uopp / UGM / U2T-2T].)
Question: To use a formulation we have developed with SR & GR, "because phenomena like particles, waves, properties/characteristics, energy/action, determinism, path, state, outcomes, and the dualities particle/ware, observer/observes, difference/identity, separation/union, etc. are relative [T1] and empty [T2] [U2T / Uopp], therefore Schrödinger equation: 𝑖ℏ 𝜕/𝜕𝑡 |𝜓(𝑡)⟩ = 𝐻̂ |𝜓(𝑡)⟩ and other QM equations”
In our ongoing exploration of parallels between modern physics and Madhyamaka philosophy, we have established a consistent framework:
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foundational physical principles and equations
can be deduced by applying the Union of the Two Truths [U2T]
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— phenomena are conventionally relative
(dependently originated, interdependent) [T1]
yet ultimately empty of inherent, absolute existence [T2] —
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to concepts previously assumed as invariants or absolutes.
This dissolves rigid dualities
and reveals emergent laws that govern appearances.
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For Special Relativity (SR), because space, time, length, duration, simultaneity, and frames of reference are relative (interdependent on observer motion) [T1] and empty (lacking inherent separateness or absoluteness) [T2] [U2T], therefore the invariance of the speed of light (c) and the Lorentz transformations Δt′ = γ(Δt−vΔx/c2), Δx′ = γ(Δx−vΔt), etc., which unify space and time into spacetime [Uopp / U2T-opp].
For General Relativity (GR), because coordinate systems, geometry, and the distinction between gravity and acceleration are relative (depending on local frames and curvature) [T1] and empty (no inherent flatness or privileged coordinates) [T2] [U2T], therefore the equivalence principle (gravity ≡ acceleration) and Einstein's field equations Gμν + Λgμν = 8πG/c4 Tμν, which describe gravity as spacetime curvature [Uopp] (the interdependence [T1-opp] and emptiness [T2-opp] of gravity/acceleration and space-time [U2T-opp]).
Now, extending this to non-relativistic Quantum Mechanics (QM), we apply the same logic to quantum invariants and dualities. Classically, phenomena like particles (as definite entities), waves (as extended fields), properties/characteristics (inherent attributes), energy/action (continuous quantities), determinism (fixed outcomes), paths (singular trajectories), states (definite configurations), and outcomes (predictable results) were assumed to have inherent, absolute existence. Dualities such as particle/wave, observer/observed, difference/identity, separation/union, manyness/oneness, position/momentum, and determinism/chaos were treated as absolute oppositions.
However, because these phenomena and dualities are relative (conventionally interdependent, arising based on context, measurement, or scale) [T1] and empty (ultimately lacking self-nature, independence, or absoluteness) [T2] [U2T], therefore the Schrödinger equation iℏ ∂/∂t ∣ψ(t)⟩ = Ĥ | ψ(t)⟩
and other QM equations (e.g., uncertainty relations, commutation rules, and the Born rule).
These equations emerge as the mathematical description of a reality where no "thing" is inherently fixed [T2], but probabilities and superpositions govern interdependent manifestations [T1] [U2T].
Below, we deduce this step by step,
showing how the Union of Two Truths logically necessitates QM's formalism.
Classically, a particle is an absolute entity with inherent localization (definite position/momentum as properties), distinct from a wave's delocalized, periodic characteristics.
But [conceptual opposites like] particles and waves [or characterized and characteristic/property; or observer and observed, or cause and effect …] are relative [T1-opp]: a system's manifestation depends on the observational setup (e.g., double-slit interference shows wave-like behavior, while detection shows particle-like).
They are empty of inherent particle-ness or wave-ness [T2-opp] — neither exists absolutely without the other, as de Broglie's relation λ = h/p unifies them.
Because this duality is relative [T1-opp] and empty [T2-opp] [U2T-opp], no absolute distinction between entity (localized "thing") and properties (extended attributes) holds.
Therefore, states must be described by a wave function ψ(x,t), which encodes both aspects: amplitude for probability (particle-like localization upon measurement) and phase for interference (wave-like delocalization).
This leads to the need for a dynamical equation governing ψ's evolution, as absolute entities would require deterministic Newtonian laws F = ma,
but emptiness demands a probabilistic (no absolute), linear framework.
Position (x) and momentum (p) were classically invariant properties, allowing deterministic paths and states (e.g., exact trajectories via x(t) = x0 + vt + 1/2 at2).
But [conceptual opposites like] position and momentum are relative [T1-opp]: precise knowledge of one depends on (and precludes) the other, as measurement contexts interlink them.
They are empty of inherent simultaneity [T2-opp] [U2T-opp] — no absolute "exact position" or "exact momentum" exists independently, dissolving determinism into chaos (inherent randomness).
Because [conceptual opposites like] paths and states are relative (multiple possibilities interfere dependently) [T1-opp] and empty (no single inherent trajectory or configuration) [T2-opp] [U2T-opp], therefore the uncertainty principle Δx ⋅ Δp ≥ ħ/2, derived from non-commuting operators [x^, p^] = iℏ.
Paths become superpositions of all possibilities (precursor to path integrals), and states |ψ⟩ in Hilbert space allow linear combinations, emptying oneness (singular state) into manyness (superposition).
Determinism's emptiness implies probabilistic outcomes,
necessitating an equation that evolves states unitarily without collapse until observation.
The observer (subject) and observed (object) were classically separate absolutes, with measurements revealing inherent identities without disturbance.
But they are relative [T1-3S]: observation co-creates the outcome (e.g., wave function collapse), depending on the apparatus.
They are empty of inherent separation [T2-3S] [U2T-opp] — no absolute "independent object" exists apart from the observer, unifying them in entanglement (union over separation).
Because difference (distinct entities) and identity (self-same properties) are relative and empty [U2T-opp], systems lack inherent boundaries, leading to complementarity: wave/particle aspects depend on the observer-observed interplay. Therefore, the Born rule P = | ⟨ϕ∣ψ⟩ |2, where measurement projects onto eigenstates, emptying absolute outcomes into relative probabilities. This interdependence requires operators O^ for observables, with eigenvalues as possible manifestations, not inherent truths.
Energy (E) and action S = ∫ Edt were continuous invariants, yielding definite outcomes.
But they are relative [T1]: quantized discretely (E = nhν) depending on system interactions.
Empty of inherent continuity [T2], they manifest as eigenvalues, dissolving manyness (infinite possibilities) into oneness (discrete levels) conventionally, yet empty ultimately.
Because outcomes are relative (dependent on state preparation) [T1] and empty (no predetermined result) [T2] [U2T], therefore quantization via via H^∣ψn⟩ = En∣ψn⟩, the time-independent Schrödinger equation. Action's emptiness adapts the classical Principle of Least Action (PLA): instead of one absolute path minimizing (S), quantum action phases eiS/ℏ sum over paths, but in wave mechanics, this variational extremum yields the wave equation.
Synthesizing these: because all listed phenomena and dualities are relative [T1] and empty, [T2] [U2T / U2T-opp] no absolute classical description suffices. Instead, a complex-valued field ψ (or ket |ψ⟩) must represent interdependent possibilities, evolving linearly to preserve emptiness (no preferred state inherently).
From Emptiness of Paths and Determinism: Classical Hamilton-Jacobi equation ∂S/∂t + H = 0 (where H = p2/2m + V) assumes absolute action (S). Emptying this, (S) becomes a phase, and ψ = A eiS/ℏ (de Broglie). Substituting into a variational principle (adapted PLA) for waves/particles, and requiring consistency with relativity of position/momentum, yields iℏ ∂ψ/∂t = - ℏ2/2m ∇2ψ + Vψ, the position-space Schrödinger equation.
Abstract Form: In Hilbert space, emptiness of states demands Ĥ as the generator of time evolution, giving iℏ ∂/∂t |ψ(t)⟩ = Ĥ |ψ(t)⟩, with Ĥ = T^ + V^ (kinetic and potential operators).
Other Equations:
Uncertainty Δx Δp ≥ ℏ/2 from emptiness of simultaneous invariants.
Commutation [x^,p^] = iℏ from relativity of observer/observed.
Time-energy uncertainty ΔE Δt ≥ ℏ/2 from emptiness of energy/action.
Born rule from emptiness of outcomes, ensuring probabilities sum to 1.
This derivation mirrors Madhyamaka: emptiness doesn't negate conventional function; it enables QM's predictive power, where interdependent appearances (relative truth) arise without inherent essence (ultimate truth). Just as in SR/GR, QM's "therefore" equations formalize this insight, empirically validated yet philosophically profound.
Continuing our exploration of foundational physics through the lens of Madhyamaka philosophy, we now turn to Noether's Theorem within the framework of non-relativistic Quantum Mechanics (QM). Noether's Theorem is a profound principle that links continuous symmetries of a physical system to conserved quantities, providing a deep connection between invariance and dynamics. In classical mechanics, it explains why symmetries like time translation, spatial translation, and rotation lead to conservation of energy, momentum, and angular momentum, respectively. In non-relativistic QM, the theorem adapts to the quantum realm, where symmetries manifest through unitary operators on Hilbert space, and conservation laws are expressed via commuting observables with the Hamiltonian.
Noether's Theorem underscores the interdependence of symmetry (as a form of relativity or invariance under transformation) and conservation (as emptiness of change in certain quantities). This aligns seamlessly with our prior discussions: just as we deduced the equations of SR, GR, and QM from the Union of the Two Truths applied to assumed invariants, Noether's Theorem in QM can be viewed as emerging from the relativity [T1-opp] and emptiness [T2-opp] of symmetries, transformations, and their associated quantities [Uopp / U2T-opp]. Classically, symmetries were seen as absolute invariants of the system, independent of context. In QM, they are relative (depending on the state's preparation and measurement) [T1] and empty (lacking inherent permanence [T2] [U2T], as quantum fluctuations and uncertainties dissolve absolutes).
Noether's Theorem, formulated by Emmy Noether in 1918, states that for every continuous symmetry of the action (or Lagrangian) of a system, there is a corresponding conserved current or quantity. In non-relativistic QM, this translates to the quantum domain via the Schrödinger picture or path-integral formulations. The theorem is crucial for understanding why quantities like energy, momentum, and charge are conserved in quantum systems, and it underpins the structure of quantum field theories (though we focus here on non-relativistic particle QM).
Key applications in QM include:
Time-translation symmetry: If the Hamiltonian Ĥ is time-independent, energy is conserved, meaning ⟨Ĥ⟩ is constant for evolving states.
Spatial-translation symmetry: Leads to conservation of total momentum P^ = - iħ ∑ ∇i.
Rotational symmetry: Conserves angular momentum L^ = r × p^.
Gauge symmetries: In quantum electrodynamics (a relativistic extension), local phase invariance leads to charge conservation, but in non-relativistic QM, global phase symmetry of the wave function ψ → eiα ψ implies probability conservation (unitarity).
Historically, Noether's work predated QM but was quickly integrated: Heisenberg and Schrödinger's formulations (1925–1926) implicitly used symmetries, and Dirac's relativistic QM (1928) formalized quantum Noether currents. In modern QM, the theorem is derived using commutators: a symmetry generator Q^ (e.g., momentum operator for translations) commutes with Ĥ if conserved, [Ĥ, Q^] = 0, ensuring d/dt ⟨Q^⟩ = 0.
Phenomena explained include atomic stability (angular momentum conservation), selection rules in transitions, and the absence of perpetual motion in isolated systems. Philosophically, it reveals that "invariants" like conserved quantities are not absolute [T2] but arise from deeper symmetries [T1], echoing Madhyamaka's Union of emptiness [T2] and dependent origination [T1] [U2T].
Noether's Theorem in QM rests on:
Symmetry of the Action or Hamiltonian: The system's description (Lagrangian (L) or Ĥ) remains invariant under continuous transformations (e.g., t → t + δt).
Variational Principle: Derived from the Principle of Least Action (PLA), where paths extremize S = ∫ L dt, adapted quantumly as in our prior discussion.
Unitarity and Linearity: Transformations are unitary operators U^ = e-iG^ϵ/ħ, where G^ is the infinitesimal generator.
Commutativity for Conservation: Conservation follows if the symmetry commutes with dynamics.
These were empirically motivated by conservation laws observed in experiments (e.g., momentum in collisions) but theoretically unified by Noether.
Applying our established formulation: phenomena in QM, such as symmetries (invariance under transformations), conserved quantities (energy, momentum, etc.), time, space, action, paths, states, and dualities like change/invariance, transformation/identity, symmetry/asymmetry, generator/observable, and conservation/variation, were classically assumed as absolute invariants.
However, because these phenomena and dualities are relative (conventionally interdependent, arising based on system context, observer, or scale — e.g., symmetry depends on the Hamiltonian's form, conservation on isolation) [T1] and empty (ultimately lacking inherent, absolute existence [T2] [U2T] — no symmetry is intrinsically unchanging [T2] without relational context [T1], no quantity inherently conserved without emptiness of isolated "self-nature"), therefore Noether's Theorem in QM: for a continuous symmetry generated by Q^, if [Ĥ, Q^] = iħ Q^/∂t (generally 0 for time-independent Q^), then d/dt ⟨Q^⟩ = 0, and associated quantum equations like the continuity equation ∂ρ/∂t + ∇ ∙ j = 0 for probability current.
This deduction follows step by step:
Relativity and Emptiness of Symmetry/Asymmetry and Change/Invariance:
Symmetries are relative (e.g., time-translation symmetry holds if Ĥ lacks explicit time dependence) but empty of inherent invariance—quantum fluctuations (e.g., vacuum energy) dissolve absolute unchanging nature. Because symmetry is not absolute, it must couple to dynamics via generators Q^, leading to unitary transformations δ |ψ⟩ = - iϵQ^ |ψ⟩/ħ. This empties the duality of change (evolution) and invariance (conservation), necessitating that invariance implies no change in certain expectations.
Relativity and Emptiness of Time/Space, Transformation/Identity:
Time and space are relative (as in SR/GR extensions, but non-relativistically approximate) and empty (no inherent flow or extension). Transformations (e.g., boosts x → x + δx) depend on context but lack absolute identity — systems aren't inherently "the same" post-transformation without relational symmetry. Therefore, for spatial translation, Q^ = p^, and invariance requires [Ĥ, p^] = 0, conserving momentum. Similarly for time: emptiness of absolute time yields energy conservation if ∂Ĥ/∂t = 0.
Relativity and Emptiness of Action/Paths, Generator/Observable:
From our QM discussion, action (S) and paths are relative superpositions but empty of determinacy. Noether extends the PLA: variations δS=0 under symmetry imply conserved currents. In QM, this quantumizes to operator relations. Because generators Q^ (observables) are relative (measurement-dependent) and empty (no inherent value), their commutation with Ĥ (from Schrödinger equation) ensures conservation: dQ^/dt = iħ[Ĥ, Q^] + ∂Q^/∂t = 0.
Relativity and Emptiness of Conservation/Variation, States/Outcomes:
Conserved quantities are relative (e.g., energy conserved in closed systems) but empty of absolute permanence—quantum uncertainty allows temporary violations (e.g., ΔE Δt ≥ ħ/2). States evolve unitarily, but outcomes vary probabilistically. Therefore, Ehrenfest's theorem d/dt⟨Q^⟩ = i/ħ ⟨[Ĥ, Q^]⟩ + ⟨∂Q^/∂t⟩, which reduces to Noether's conservation for symmetries.
Tie to Schrödinger Equation:
Since the Schrödinger equation iħ ∂/∂t |ψ⟩ = Ĥ |ψ⟩ already emerges from emptiness of paths/states (as deduced previously), Noether's application within it follows naturally: symmetries of Ĥ preserve the equation's form, yielding conserved operators. For example, in the position basis, probability conservation (global phase symmetry) gives the continuity equation from ∂ |ψ|2}/∂t = -∇ ࢈ (ħ/2mi (ψ* ∇ψ - ψ∇ψ*)).
In summary, Noether's Theorem in non-relativistic QM is deduced from the Union of Two Truths applied to symmetry-related invariants, mirroring our SR/GR/QM derivations. It reveals conservation as an emergent conventional truth from the interdependence and emptiness of transformations, without inherent essence — much like Madhyamaka's insight that phenomena arise dependently [T1] yet are ultimately empty [T2] [U2T]. This framework not only unifies QM's structure but highlights its philosophical resonance with relativity of appearances.
Non-relativistic Quantum Mechanics (QM) represents a paradigm shift in understanding the microscopic world, where classical notions of definite particles following predictable paths give way to probabilistic waves, superpositions, and observer-dependent realities. Developed in the early 20th century through the works of pioneers like Planck, Einstein, de Broglie, Heisenberg, and Schrödinger, QM addresses phenomena inexplicable by Newtonian mechanics, such as atomic spectra, electron diffraction, and quantum tunneling. At its heart lies the wave function, a mathematical entity that encapsulates all possible states of a system, evolving according to the Schrödinger equation, and collapsing upon measurement to yield probabilistic outcomes. This framework not only powers technologies like semiconductors and lasers but also challenges our intuitive grasp of reality, suggesting that the universe at small scales is inherently indeterminate and interconnected.
In paralleling QM with Madhyamaka philosophy, we extend the approach used for Special Relativity (SR) and General Relativity (GR), where foundational principles were deduced from the Union of the Two Truths [U2T] — conventional relativity (dependent origination) [T1] and ultimate emptiness (lack of inherent existence) [T2] — applied to assumed invariants and dualities.
For QM, we apply this to quantum dualities and invariants like wave/particle, position/momentum, determinism/chaos, observer/observed, and path/state/outcome. Classically, these were treated as absolute, inherent truths, but QM reveals them as context-dependent [T1] and devoid of self-nature [T2] Û2T], mirroring Madhyamaka's insight that phenomena arise interdependently [T1] yet are empty of essence [T2] [U2T]. This reframing allows us to derive QM's equations not merely as empirical fits but as logical consequences of dissolving these absolutes, much like how SR's Lorentz transformations emerge from emptying space-time invariants.
The depth of QM's connection to Madhyamaka lies in its philosophical implications: the theory dissolves the subject-object divide [U3S / U2T-3S] through measurement, entangles entities beyond separation, and replaces determinism with potentiality, echoing the Madhyamaka view that all dharmas (phenomena) lack svabhava (inherent existence) [T2] while functioning conventionally [T1] [U2T]. Precursors like de Broglie's matter waves and the adapted Principle of Least Action prefigure this by blending classical invariants with quantum relativity, setting the stage for Schrödinger's equation as an expression of interdependent evolution. By exploring this, we uncover how QM's "weirdness" — superposition, entanglement, uncertainty — arises from the same emptiness that Madhyamaka posits as the ground of all reality, offering a unified lens for physics and philosophy.
In reframing non-relativistic QM via Madhyamaka, we began by identifying classical invariants — such as definite particles, deterministic paths, inherent properties, and rigid dualities like wave/particle or observer/observed—as conventionally relative [T1] yet ultimately empty [T2] [U2T]. This Union of the Two Truths dissolves their absoluteness: particles and waves interdepend on context (relative truth) but lack independent essence (empty truth), leading to de Broglie's duality and the wave function ψ as a probabilistic amplitude. Similarly, position and momentum's interdependence yields Heisenberg's uncertainty, emptying simultaneous precision, while paths' emptiness implies superposition over singular trajectories, adapting the Principle of Least Action to quantum variability.
From this foundation, QM's core equations naturally emerge: because states, outcomes, and determinism are relative [T1] and empty [T2] [U2T], the Schrödinger equation iℏ ∂/∂t |ψ(t)⟩ = Ĥ |ψ(t)⟩ governs unitary evolution, preserving interdependence [T1] without inherent fixity [T2] [U2T].
Complementary principles like the Born rule and complementarity further arise — probabilities from emptiness of outcomes, mutual exclusivity of aspects from relativity of observer-observed.
This deduction mirrors our SR and GR analyses, where emptying invariants (e.g., space-time in SR, geometry in GR) yields their equations; in QM, it unifies empirical postulates into a philosophical derivation, explaining quantization, linearity, and non-commutativity as consequences of dependent origination and voidness.
The extension to Noether's Theorem in QM reinforces this: symmetries and conservations (energy, momentum) are relative (system-dependent) [T1] and empty (no absolute invariance amid fluctuations) [T2] [U2T], thus commuting operators [Ĥ, Q̂] = 0 ensure conservation, tying back to the Schrödinger dynamics.
Overall, this approach highlights QM's depth beyond mechanics — its resonance with Madhyamaka reveals quantum "paradoxes" as illusions of inherent existence, where emptiness [T2] enables the conventional play of probabilities, entanglements, and measurements [T1] [U2T], bridging Eastern wisdom with Western science.
This exploration of non-relativistic QM through Madhyamaka concludes that the theory's revolutionary features — indeterminacy, duality, and interdependence — are not mere anomalies but inevitable outcomes of applying the Union of the Two Truths to classical invariants.
By recognizing the relativity and emptiness of particles/waves, paths/states, and observer/observed, we deduce the Schrödinger equation and ancillary relations as formal expressions of a reality free from inherent essences, yet functionally coherent.
This parallels our SR and GR derivations, affirming a consistent pattern: physics' deepest laws arise from dissolving assumed absolutes, aligning with Madhyamaka's negation of svabhava to reveal pratītyasamutpāda (dependent arising).
The implications extend philosophically:
– QM's measurement problem, often debated in interpretations like Copenhagen or Many-Worlds, finds resolution in emptiness—collapse isn't a "real" event but a conventional manifestation of interdependent observation, empty of ultimate mechanism.
– Entanglement, defying separation, embodies non-dual union, while uncertainty empties determinism into open potentiality, echoing Madhyamaka's freedom from extremes of existence/non-existence.
Thus, QM not only predicts empirical phenomena but philosophically liberates from reified views, much as Madhyamaka does through śūnyatā (emptiness).
Ultimately, this synthesis invites broader inquiry: if QM, like SR/GR, stems from the Two Truths, perhaps all physics does, suggesting a unified "Madhyamaka physics" where emptiness is the ground state, and relative truths manifest as laws. For practitioners, this bridge fosters contemplative science—meditating on quantum emptiness could deepen insight, while physics inspires Madhyamaka analysis. In closing, non-relativistic QM, reframed thus, transcends its equations to reveal the interdependent, empty nature of reality, harmonizing ancient wisdom with modern discovery.
