Quantum mechanics: Difference between revisions

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
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Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics describing the properties of nature on an atomic scale.^[2]

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History

Modern physics
${\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }$ ${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$ Schrödinger and Einstein field equations
Founders Max Planck Albert Einstein Niels Bohr Max Born Werner Heisenberg Erwin Schrödinger Pascual Jordan Wolfgang Pauli Paul Dirac Ernest Rutherford Louis de Broglie Satyendra Nath Bose
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Scientists Witten Röntgen Becquerel Lorentz Planck Curie Wien Skłodowska-Curie Sommerfeld Rutherford Soddy Onnes Einstein Wilczek Born Weyl Bohr Kramers Schrödinger de Broglie Laue Bose Compton Pauli Walton Fermi van der Waals Heisenberg Dyson Zeeman Moseley Hilbert Gödel Jordan Dirac Wigner Hawking P. W. Anderson Lemaître Thomson Poincaré Wheeler Penrose Millikan Nambu von Neumann Higgs Hahn Feynman Yang Lee Lenard Salam 't Hooft Veltman Bell Gell-Mann J. J. Thomson Raman Bragg Bardeen Shockley Chadwick Lawrence Zeilinger Goudsmit Uhlenbeck
Categories Modern physics
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Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as

In 1838, Michael Faraday discovered cathode rays. These studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.^[7] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets) precisely matched the observed patterns of black-body radiation.

In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation,^[8] called Wien's law. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it was valid only at high frequencies and underestimated the radiance at low frequencies.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld, and others. The Copenhagen interpretation of Niels Bohr became widely accepted.

According to Planck, each energy element (E) is proportional to its frequency (ν):

${\displaystyle E=h\nu \ }$,

where h is Planck's constant.

Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the physical reality of the radiation.

In the mid-1920s, quantum mechanics was developed to become the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Due to their particle-like behavior in certain processes and measurements, light quanta came to be called

Quantum mechanics is essential for understanding the behavior of systems at atomic length scales and smaller. If the physical nature of an atom were solely described by classical mechanics, electrons would not orbit the nucleus, since orbiting electrons emit radiation (due to circular motion) and so would quickly lose energy and collide with the nucleus. This framework was unable to explain the stability of atoms. Instead, electrons remain in an uncertain, non-deterministic, smeared, probabilistic wave–particle orbital about the nucleus, defying the traditional assumptions of classical mechanics and electromagnetism.^[26]

Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the spectra of light emitted by different isotopes of the same chemical element, as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.

Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account^[20]:

• certain physical properties
• quantum entanglement
• principle of uncertainty
• wave–particle duality

Mathematical formulations

In the mathematically rigorous formulation of quantum mechanics developed by

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.^[32] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, to arbitrary precision. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown. Contours of constant probability density, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.^[33]

According to one interpretation, as the result of a measurement, the wave function containing the probability information for a system

In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs). Rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates).

Usually, a system will not be in an

For example, the free particle in the previous example will usually have a wave function that is a wave packet centered around some mean position x₀ (neither an eigenstate of position nor of momentum). When one measures the position of the particle, it is impossible to predict with certainty the result.^[34] It is probable, but not certain, that it will be near x₀, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.^[39]

The time evolution of a quantum state is described by the

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.

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. Another method is called "semi-classical equation of motion", which 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. This approach is particularly important in the field of quantum chaos.

Mathematically equivalent formulations

There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).^[45]

Especially since Heisenberg was awarded the

Relation to other scientific theories

The rules of quantum mechanics are fundamental. They 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.

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. 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. 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.

• 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.^[57]
• While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent for extremely small particles or for velocities approaching the speed of light, the laws of classical, often considered "Newtonian", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the velocity of light.^[58]

Copenhagen interpretation of quantum versus classical kinematics

A big difference between classical and quantum mechanics is that they use very different kinematic descriptions.^[59]

In Niels Bohr's mature view, quantum mechanical phenomena are required to be experiments, with complete descriptions of all the devices for the system, preparative, intermediary, and finally measuring. The descriptions are in macroscopic terms, expressed in ordinary language, supplemented with the concepts of classical mechanics.^{[60][61][62][63]} The initial condition and the final condition of the system are respectively described by values in a configuration space, for example a position space, or some equivalent space such as a momentum space. Quantum mechanics does not admit a completely precise description, in terms of both position and momentum, of an initial condition or "state" (in the classical sense of the word) that would support a precisely deterministic and causal prediction of a final condition.^[64][65] In this sense, a quantum phenomenon is a process, a passage from initial to final condition, not an instantaneous "state" in the classical sense of that word.^[66][67] Thus there are two kinds of processes in quantum mechanics: stationary and transitional. For a stationary process, the initial and final condition are the same. For a transition, they are different. Obviously by definition, if only the initial condition is given, the process is not determined.^[64] Given its initial condition, prediction of its final condition is possible, causally but only probabilistically, because the Schrödinger equation is deterministic for wave function evolution, but the wave function describes the system only probabilistically.^[68][69]

For many experiments, it is possible to think of the initial and final conditions of the system as being a particle. In some cases it appears that there are potentially several spatially distinct pathways or trajectories by which a particle might pass from initial to final condition. It is an important feature of the quantum kinematic description that it does not permit a unique definite statement of which of those pathways is actually followed. Only the initial and final conditions are definite, and, as stated in the foregoing paragraph, they are defined only as precisely as allowed by the configuration space description or its equivalent. In every case for which a quantum kinematic description is needed, there is always a compelling reason for this restriction of kinematic precision. An example of such a reason is that for a particle to be experimentally found in a definite position, it must be held motionless; for it to be experimentally found to have a definite momentum, it must have free motion; these two are logically incompatible.^[70][71]

Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces. This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage.

Relation to general relativity

Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence, and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.^[73]

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

Since its inception, the many

The Copenhagen interpretation – due largely to Niels Bohr and Werner Heisenberg – remains most widely accepted some 75 years after its enunciation. According to this interpretation, 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". It also states that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the conjugate nature of evidence obtained under different experimental situations.

Albert Einstein, himself one of the founders of quantum theory, did not accept some of the more philosophical or metaphysical interpretations of quantum mechanics, such as rejection of

Quantum mechanics has had enormous[18] success in explaining many of the features of our universe. 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). Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism).

Quantum mechanics is also critically important for understanding how individual atoms are joined by covalent bonds to form

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.^[87] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.^[88]

Electronics

Many modern electronic devices are designed using quantum mechanics. Examples include the

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to more fully develop quantum cryptography, which will theoretically allow guaranteed secure transmission of information.

An inherent advantage yielded by quantum cryptography when compared to classical

Another active research topic is quantum teleportation, which deals with techniques to transmit quantum information over arbitrary distances.

Macroscale quantum effects

While quantum mechanics primarily applies to the smaller atomic regimes of matter and energy, some systems exhibit quantum mechanical effects on a large scale. Superfluidity, the frictionless flow of a liquid at temperatures near absolute zero, is one well-known example. So is the closely related phenomenon of superconductivity, the frictionless flow of an electron gas in a conducting material (an electric current) at sufficiently low temperatures. The fractional quantum Hall effect is a topological ordered state which corresponds to patterns of long-range quantum entanglement.^[91] States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

Quantum theory

Quantum theory also provides accurate descriptions for many previously unexplained phenomena, such as

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. For the one-dimensional case in the ${\displaystyle x}$ direction, the time-independent Schrödinger equation may be written^[96]

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

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 Born interpretation. Therefore, since ${\displaystyle \sin(kL)=0}$, ${\displaystyle kL}$ must be an integer multiple of ${\displaystyle \pi }$,

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

The quantization of energy levels follows from this constraint on ${\displaystyle k,}$ since

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

The ground state energy of the particles is ${\displaystyle E_{1}}$ for ${\displaystyle n=1.}$

The energy of the particle in the ${\displaystyle n}$th state is ${\displaystyle E_{n}=n^{2}E_{1},\;n=2,3,4,\dots }$

Particle in a box with boundary condition ${\displaystyle V(x)=0,\;-a/2<x<+a/2}$

In this condition the general solution will be same, there will little change to the final result, since the boundary conditions are changed only slightly:

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

At ${\displaystyle x=0,}$ the wave function is not actually zero at all values of ${\displaystyle n.}$

Clearly, from the wave function variation graph we have, At ${\displaystyle n=1,3,5,\dots ,}$ the wave function follows a cosine curve with ${\displaystyle x=0}$ as the origin.

At ${\displaystyle n=2,4,6,\dots ,}$ the wave function follows a sine curve with ${\displaystyle x=0}$ as the origin.

From this observation we can conclude that the wave function is alternatively sine and cosine. So in this case the resultant wave equation is

${\displaystyle \psi _{n}(x)={\begin{cases}A\cos(k_{n}x),&n=1,3,5,\dots \\B\sin(k_{n}x),&n=2,4,6,\dots \end{cases}}}$

Finite potential well

A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth.

The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.

Rectangular potential barrier

This is a model for the

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 quantification of energy for bound states.

Step potential

The potential in this case is given by:

${\displaystyle V(x)={\begin{cases}0,&x<0,\\V_{0},&x\geq 0.\end{cases}}}$

The solutions are superpositions of left- and right-moving waves:

${\displaystyle \psi _{1}(x)={\frac {1}{\sqrt {k_{1}}}}\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\qquad x<0}$

and

${\displaystyle \psi _{2}(x)={\frac {1}{\sqrt {k_{2}}}}\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\qquad x>0}$,

with coefficients A and B determined from the

Notes