Quantum mechanics

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms.^[2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Quantum mechanics can describe many systems that

Quantum systems have bound states that are quantized to discrete values of energy, momentum, angular momentum, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves (wave–particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).

Quantum mechanics

A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. This is known as the Born rule, named after physicist Max Born. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.^{[7]: 67–87}

One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.^{[7]: 427–435}

Another consequence of the mathematical rules of quantum mechanics is the phenomenon of

When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".^[13] Quantum entanglement enables quantum computing and is part of quantum communication protocols, such as quantum key distribution and superdense coding.^[14] Contrary to popular misconception, entanglement does not allow sending signals faster than light, as demonstrated by the no-communication theorem.^[14]

Another possibility opened by entanglement is testing for "

It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects.^[17][18] Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.

Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ${\displaystyle \psi }$ belonging to a (separable) complex Hilbert space ${\displaystyle {\mathcal {H}}}$. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ${\displaystyle \langle \psi ,\psi \rangle =1}$, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ${\displaystyle \psi }$ and ${\displaystyle e^{i\alpha }\psi }$ represent the same physical system. In other words, the possible states are points in the

${\displaystyle L^{2}(\mathbb {C} )}$

${\displaystyle \mathbb {C} ^{2}}$

Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are

: in the simplest case the eigenvalue ${\displaystyle \lambda }$ is non-degenerate and the probability is given by ${\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}$, where ${\displaystyle {\vec {\lambda }}}$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ${\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }$, where ${\displaystyle P_{\lambda }}$ is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the

probability density

.

After the measurement, if result ${\displaystyle \lambda }$ was obtained, the quantum state is postulated to

collapse

to ${\displaystyle {\vec {\lambda }}}$, in the non-degenerate case, or to ${\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}$, in the general case. The

Time evolution of a quantum state

The time evolution of a quantum state is described by the Schrödinger equation:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}$

Here ${\displaystyle H}$ denotes the

total energy

of the system, and ${\displaystyle \hbar }$ is the reduced

${\displaystyle i\hbar }$

The solution of this differential equation is given by

${\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}$

The operator ${\displaystyle U(t)=e^{-iHt/\hbar }}$ is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is deterministic in the sense that – given an initial quantum state ${\displaystyle \psi (0)}$  – it makes a definite prediction of what the quantum state ${\displaystyle \psi (t)}$ will be at any later time.^[20]

Some wave functions produce probability distributions that are independent of time, such as

Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form.^[21][22][23]

However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy.^[7]: 793 Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.^[7]: 849

Uncertainty principle

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.^[24][25] Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator ${\displaystyle {\hat {X}}}$ and momentum operator ${\displaystyle {\hat {P}}}$ do not commute, but rather satisfy the canonical commutation relation:

${\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar .}$

Given a quantum state, the Born rule lets us compute expectation values for both ${\displaystyle X}$ and ${\displaystyle P}$, and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have

${\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}$

and likewise for the momentum:

${\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}$

The uncertainty principle states that

${\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}$

Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.^[26] This inequality generalizes to arbitrary pairs of self-adjoint operators ${\displaystyle A}$ and ${\displaystyle B}$. The commutator of these two operators is

${\displaystyle [A,B]=AB-BA,}$

and this provides the lower bound on the product of standard deviations:

${\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }[A,B]{\bigr \rangle }\right|.}$

Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an ${\displaystyle i/\hbar }$ factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum ${\displaystyle p_{i}}$ is replaced by ${\displaystyle -i\hbar {\frac {\partial }{\partial x}}}$, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times ${\displaystyle -\hbar ^{2}}$.^[24]

Composite systems and entanglement

When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces ${\displaystyle {\mathcal {H}}_{A}}$ and ${\displaystyle {\mathcal {H}}_{B}}$, respectively. The Hilbert space of the composite system is then

${\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}$

If the state for the first system is the vector ${\displaystyle \psi _{A}}$ and the state for the second system is ${\displaystyle \psi _{B}}$, then the state of the composite system is

${\displaystyle \psi _{A}\otimes \psi _{B}.}$

Not all states in the joint Hilbert space ${\displaystyle {\mathcal {H}}_{AB}}$ can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ${\displaystyle \psi _{A}}$ and ${\displaystyle \phi _{A}}$ are both possible states for system ${\displaystyle A}$, and likewise ${\displaystyle \psi _{B}}$ and ${\displaystyle \phi _{B}}$ are both possible states for system ${\displaystyle B}$, then

${\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}$

is a valid joint state that is not separable. States that are not separable are called entangled.^[27][28]

If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define

As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.^[30]

Equivalence between formulations

There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "

The Hamiltonian ${\displaystyle H}$ is known as the generator of time evolution, since it defines a unitary time-evolution operator ${\displaystyle U(t)=e^{-iHt/\hbar }}$ for each value of ${\displaystyle t}$. From this relation between ${\displaystyle U(t)}$ and ${\displaystyle H}$, it follows that any observable ${\displaystyle A}$ that commutes with ${\displaystyle H}$ will be conserved: its expectation value will not change over time.^[7]: 471 This statement generalizes, as mathematically, any Hermitian operator ${\displaystyle A}$ can generate a family of unitary operators parameterized by a variable ${\displaystyle t}$. Under the evolution generated by ${\displaystyle A}$, any observable ${\displaystyle B}$ that commutes with ${\displaystyle A}$ will be conserved. Moreover, if ${\displaystyle B}$ is conserved by evolution under ${\displaystyle A}$, then ${\displaystyle A}$ is conserved under the evolution generated by ${\displaystyle B}$. This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law.

Examples

Free particle

The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:

${\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}$

The general solution of the Schrödinger equation is given by

${\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}$

which is a superposition of all possible plane waves ${\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}$, which are eigenstates of the momentum operator with momentum ${\displaystyle p=\hbar k}$. The coefficients of the superposition are ${\displaystyle {\hat {\psi }}(k,0)}$, which is the Fourier transform of the initial quantum state ${\displaystyle \psi (x,0)}$.

It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.^{[note 1]} Instead, we can consider a Gaussian wave packet:

${\displaystyle \psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}$

which has Fourier transform, and therefore momentum distribution

${\displaystyle {\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}$

We see that as we make ${\displaystyle a}$ smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making ${\displaystyle a}$ larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.

As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.^[33]

Particle in a box

The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region.^{[24]: 77–78} For the one-dimensional case in the ${\displaystyle x}$ direction, the time-independent Schrödinger equation may be written

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}$

With the differential operator defined by

${\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}$

the previous equation is evocative of the classic kinetic energy analogue,

${\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}$

with state ${\displaystyle \psi }$ in this case having energy ${\displaystyle E}$ coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are

${\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}$

or, from Euler's formula,

${\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}$

The infinite potential walls of the box determine the values of ${\displaystyle C,D,}$ and ${\displaystyle k}$ at ${\displaystyle x=0}$ and ${\displaystyle x=L}$ where ${\displaystyle \psi }$ must be zero. Thus, at ${\displaystyle x=0}$,

${\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}$

and ${\displaystyle D=0}$. At ${\displaystyle x=L}$,

${\displaystyle \psi (L)=0=C\sin(kL),}$

in which ${\displaystyle C}$ cannot be zero as this would conflict with the postulate that ${\displaystyle \psi }$ has norm 1. Therefore, since ${\displaystyle \sin(kL)=0}$, ${\displaystyle kL}$ must be an integer multiple of ${\displaystyle \pi }$,

${\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}$

This constraint on ${\displaystyle k}$ implies a constraint on the energy levels, yielding

${\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}$

A

As in the classical case, the potential for the quantum harmonic oscillator is given by[7]^: 234

${\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}$

This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The

${\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }$

${\displaystyle n=0,1,2,\ldots .}$

where H_n are the Hermite polynomials

${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}$

and the corresponding energy levels are

${\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}$

This is another example illustrating the discretization of energy for bound states.

Mach–Zehnder interferometer

The

Both beam splitters are modelled as the unitary matrix ${\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}$, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of ${\displaystyle 1/{\sqrt {2}}}$, or be reflected to the other path with a probability amplitude of ${\displaystyle i/{\sqrt {2}}}$. The phase shifter on the upper arm is modelled as the unitary matrix ${\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}$, which means that if the photon is on the "upper" path it will gain a relative phase of ${\displaystyle \Delta \Phi }$, and it will stay unchanged if it is in the lower path.

A photon that enters the interferometer from the left will then be acted upon with a beam splitter ${\displaystyle B}$, a phase shifter ${\displaystyle P}$, and another beam splitter ${\displaystyle B}$, and so end up in the state

${\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}$

and the probabilities that it will be detected at the right or at the top are given respectively by

${\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}$

${\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}$

One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.

It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by ${\displaystyle p(u)=p(l)=1/2}$, independently of the phase ${\displaystyle \Delta \Phi }$. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.^[36]

Applications

Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by classical methods.^{[note 2]} Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Solid-state physics and materials science are dependent upon quantum mechanics.^[37]

In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the optical amplifier and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy.^[38] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

Relation to other scientific theories

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Classical mechanics

The rules of quantum mechanics assert that the state space of a system is a

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.^[7]: 234

Complications arise with chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.^[40]: 353

Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.^[42]

Special relativity and electrodynamics

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.^[43][44]

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical ${\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}$ Coulomb potential.^[7]: 285 Likewise, in a Stern–Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.^[40]: 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated empirical evidence, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.^[46]

One proposal for doing so is

Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong

The views of Niels Bohr, Werner Heisenberg and other physicists are often grouped together as the "Copenhagen interpretation".^[52][53] According to these views, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but is instead a final renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the complementary nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,^[54] Heisenberg,^[55] Schrödinger,^[56] Feynman,^[2] and Zeilinger^[57] as well as 21st-century researchers in quantum foundations.^[58]

Everett's many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.^[65] This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,^[66][67] with no consensus on whether they have been successful.^[68][69][70]

Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as

During the early 19th century,

This phase is known as the

Explanatory notes

References