Quantum mechanics (QM; also known as quantum physics or quantum
theory), including quantum field theory, is a fundamental theory in
physics which describes nature at the smallest scales of energy levels
of atoms and subatomic particles.[2]
Classical physics
Classical physics (the physics existing before quantum mechanics) is a set of fundamental theories which describes nature at ordinary (macroscopic) scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.[3] Quantum mechanics differs from classical
physics in that: energy, momentum and other quantities of a system may
be restricted to discrete values (quantization), objects have
characteristics of both particles and waves (wave-particle duality),
and there are limits to the precision with which quantities can be
known (uncertainty principle).[note 1]
Quantum mechanics gradually arose from theories to explain
observations which could not be reconciled with classical physics,
such as Max Planck's solution in 1900 to the black-body radiation
problem, and from the correspondence between energy and frequency in
Albert Einstein's 1905 paper which explained the photoelectric effect.
Early quantum theory was profoundly re-conceived in the mid-1920s by
Erwin Schrödinger, Werner Heisenberg,
Max Born
Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Important applications of quantum theory[5] include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.[6] Contents 1 History
2
4.1
5
6.1 Electronics
6.2 Cryptography
6.3
7 Examples 7.1 Free particle
7.2
8 See also 9 Notes 10 References 11 Further reading 12 External links History[edit] Modern physics H ^
ψ n ( t ) ⟩ = i ℏ ∂ ∂ t
ψ n ( t ) ⟩ displaystyle hat H psi _ n (t)rangle =ihbar frac partial partial t psi _ n (t)rangle 1 c 2 ∂ 2 ϕ n ∂ t 2 − ∇ 2 ϕ n + ( m c ℏ ) 2 ϕ n = 0 displaystyle frac 1 c ^ 2 frac partial ^ 2 phi _ n partial t ^ 2 - nabla ^ 2 phi _ n + left( frac mc hbar right) ^ 2 phi _ n =0 Manifold dynamics: Schrödinger and Klein–Gordon equations Founders
Concepts Topology · space · time · energy · matter · work randomness · information · entropy · mind light · particle · wave Branches
Applied · Experimental · Theoretical
Scientists
Röntgen · Becquerel · Lorentz · Planck ·
Curie · Wien · Skłodowska-Curie ·
Sommerfeld · Rutherford · Soddy · Onnes ·
Einstein · Wilczek · Born · Weyl ·
Bohr · 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 ·
Lee · Lenard · Salam · 't Hooft ·
Bell · Gell-Mann ·
v t e Main article: History of quantum mechanics
Scientific inquiry into the wave nature of light began in the 17th and
18th centuries, when scientists such as Robert Hooke, Christiaan
Huygens and
E = h ν displaystyle E=hnu ,
where h is Planck's constant.
Planck cautiously insisted that this was simply an aspect of the
processes of absorption and emission of radiation and had nothing to
do with the physical reality of the radiation itself.[12] In fact, he
considered his quantum hypothesis a mathematical trick to get the
right answer rather than a sizable discovery.[13] However, in 1905
The 1927
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
quantization of certain physical properties quantum entanglement principle of uncertainty wave–particle duality
Fig. 1:
Some wave functions produce probability distributions that are
constant, or independent of time—such as when in a stationary state
of constant energy, time vanishes in the absolute square of the wave
function. Many systems that are treated dynamically in classical
mechanics are described by such "static" wave functions. For example,
a single electron in an unexcited atom is pictured classically as a
particle moving in a circular trajectory around the atomic nucleus,
whereas in quantum mechanics it is described by a static, spherically
symmetric wave function surrounding the nucleus (Fig. 1) (note,
however, that only the lowest angular momentum states, labeled s, are
spherically symmetric).[40]
The
Unsolved problem in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "superposition of states" and "wave function collapse", give rise to the reality we perceive? (more unsolved problems in physics) 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
− e 2 / ( 4 π ϵ 0 r ) displaystyle scriptstyle -e^ 2 /(4pi 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.[52] While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with particles of extremely small size or 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.[53]
A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through potential barriers. (Left: band diagram; Center: transmission coefficient; Right: current-voltage characteristics) As shown in the band diagram(left), although there are two barriers, electrons still tunnel through via the confined states between two barriers(center), conducting current. Many electronic devices operate under effect of quantum tunneling. It
even exists in the simple light switch. The switch would not work if
electrons could not quantum tunnel through the layer of oxidation on
the metal contact surfaces.
Examples[edit]
Free particle[edit]
For example, consider a free particle. In quantum mechanics, a free
matter is described by a wave function. The particle properties of the
matter become apparent when we measure its position and velocity. The
wave properties of the matter become apparent when we measure its wave
properties like interference. The wave–particle duality feature is
incorporated in the relations of coordinates and operators in the
formulation of quantum mechanics. Since the matter is free (not
subject to any interactions), its quantum state can be represented as
a wave of arbitrary shape and extending over space as a wave function.
The position and momentum of the particle are observables. The
1-dimensional potential energy box (or infinite potential well) Main article:
x displaystyle x direction, the time-independent
− ℏ 2 2 m d 2 ψ d x 2 = E ψ . displaystyle - frac hbar ^ 2 2m frac d^ 2 psi dx^ 2 =Epsi . With the differential operator defined by p ^ x = − i ℏ d d x displaystyle hat p _ x =-ihbar frac d dx the previous equation is evocative of the classic kinetic energy analogue, 1 2 m p ^ x 2 = E , displaystyle frac 1 2m hat p _ x ^ 2 =E, with state ψ displaystyle psi in this case having energy E displaystyle E coincident with the kinetic energy of the particle.
The general solutions of the
ψ ( x ) = A e i k x + B e − i k x E = ℏ 2 k 2 2 m displaystyle psi (x)=Ae^ ikx +Be^ -ikx qquad qquad E= frac hbar ^ 2 k^ 2 2m or, from Euler's formula, ψ ( x ) = C sin k x + D cos k x . displaystyle psi (x)=Csin kx+Dcos kx.! The infinite potential walls of the box determine the values of C, D, and k at x = 0 and x = L where ψ must be zero. Thus, at x = 0, ψ ( 0 ) = 0 = C sin 0 + D cos 0 = D displaystyle psi (0)=0=Csin 0+Dcos 0=D! and D = 0. At x = L, ψ ( L ) = 0 = C sin k L . displaystyle psi (L)=0=Csin kL.! in which C cannot be zero as this would conflict with the Born interpretation. Therefore, since sin(kL) = 0, kL must be an integer multiple of π, k = n π L n = 1 , 2 , 3 , … . displaystyle k= frac npi L qquad qquad n=1,2,3,ldots . The quantization of energy levels follows from this constraint on k, since E = ℏ 2 π 2 n 2 2 m L 2 = n 2 h 2 8 m L 2 . displaystyle E= frac hbar ^ 2 pi ^ 2 n^ 2 2mL^ 2 = frac n^ 2 h^ 2 8mL^ 2 . Finite potential well[edit]
Main article: 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[edit]
Main article: Rectangular potential barrier
This is a model for the quantum tunneling effect which plays an
important role in the performance of modern technologies such as flash
memory and scanning tunneling microscopy.
Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy. As in the classical case, the potential for the quantum harmonic oscillator is given by V ( x ) = 1 2 m ω 2 x 2 . displaystyle V(x)= frac 1 2 momega ^ 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 eigenstates are given by ψ n ( x ) = 1 2 n n ! ⋅ ( m ω π ℏ ) 1 / 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n ( m ω ℏ x ) , displaystyle psi _ n (x)= sqrt frac 1 2^ n ,n! cdot left( frac momega pi hbar right)^ 1/4 cdot e^ - frac momega x^ 2 2hbar cdot H_ n left( sqrt frac momega hbar xright),qquad n = 0 , 1 , 2 , … . displaystyle n=0,1,2,ldots . where Hn are the Hermite polynomials H n ( x ) = ( − 1 ) n e x 2 d n d x n ( e − x 2 ) , 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 E n = ℏ ω ( n + 1 2 ) . 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[edit]
Main article: Solution of
Scattering at a finite potential step of height V0, shown in green. The amplitudes and direction of left- and right-moving waves are indicated. Yellow is the incident wave, blue are reflected and transmitted waves, red does not occur. E > V0 for this figure. The potential in this case is given by: V ( x ) = 0 , x < 0 , V 0 , x ≥ 0. displaystyle V(x)= begin cases 0,&x<0,\V_ 0 ,&xgeq 0.end cases The solutions are superpositions of left- and right-moving waves: ψ 1 ( x ) = 1 k 1 ( A → e i k 1 x + A ← e − i k 1 x ) x < 0 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 ψ 2 ( x ) = 1 k 2 ( B → e i k 2 x + B ← e − i k 2 x ) x > 0 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 boundary conditions and by imposing a continuous derivative on the solution, and where the wave vectors are related to the energy via k 1 = 2 m E / ℏ 2 displaystyle k_ 1 = sqrt 2mE/hbar ^ 2 and k 2 = 2 m ( E − V 0 ) / ℏ 2 displaystyle k_ 2 = sqrt 2m(E-V_ 0 )/hbar ^ 2 . Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. Notably, in contrast to classical mechanics, incident particles with energies greater than the potential step are partially reflected. See also[edit]
Notes[edit] ^ Born, M. (1926). "Zur Quantenmechanik der Stoßvorgänge".
Zeitschrift für Physik. 37 (12): 863–867.
Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. Retrieved 16
December 2008.
^ Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). The
^ N.B. on precision: If δ x displaystyle delta x and δ p displaystyle delta p are the precisions of position and momentum obtained in an individual measurement and σ x displaystyle sigma _ x , σ p displaystyle sigma _ p their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements δ x displaystyle delta x and δ p displaystyle delta p , but the standard deviations will always satisfy σ x σ p ≥ ℏ / 2 displaystyle sigma _ x sigma _ p geq hbar /2 ".[4] References[edit] The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. Chester, Marvin (1987) Primer of
More technical: Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds
Interpretation of
Further reading[edit] Bernstein, Jeremy (2009).
On Wikibooks This
External links[edit] Find more about
Definitions from Wiktionary Media from Wikimedia Commons News from Wikinews Quotations from Wikiquote Texts from Wikisource Textbooks from Wikibooks Learning resources from Wikiversity 3D animations, applications and research for basic quantum effects
(animations also available in commons.wikimedia.org (Université paris
Sud))
Course material A collection of lectures on
FAQs Many-worlds or relative-state interpretation.
Measurement in
Media PHYS 201: Fundamentals of
Philosophy Ismael, Jenann. "
v t e
Background Introduction History timeline Glossary Classical mechanics Old quantum theory Fundamentals Bra–ket notation
Casimir effect
Complementarity
Density matrix
ground state
excited state
degenerate levels
Vacuum state
Zero-point energy
Hamiltonian
Operator
Qubit
Qutrit
Observable
Formulations Formulations Heisenberg Interaction Matrix mechanics Schrödinger Path integral formulation Phase space Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations Interpretations
Bayesian
Consistent histories
Copenhagen
de Broglie–Bohm
Ensemble
Hidden variables
Many-worlds
Objective collapse
Experiments Afshar
Bell's inequality
Cold
Science
Technology
Timeline
Extensions
Axiomatic quantum field theory
Category Portal:Physics Commons v t e Branches of physics Divisions Applied Experimental Theoretical Energy Motion Thermodynamics Mechanics Classical Ballistics Lagrangian Hamiltonian Continuum Celestial Statistical Solid Fluid Quantum Waves Fields Gravitation Electromagnetism Optics Geometrical Physical Nonlinear Quantum
Special General By speciality Accelerator Acoustics Astrophysics Nuclear Stellar Heliophysics Solar Space Astroparticle Atomic–molecular–optical (AMO) Communication Computational Condensed matter Mesoscopic Solid-state Soft Digital Engineering Material Mathematical Molecular Nuclear Particle Phenomenology Plasma Polymer Statistical
Biophysics Virophysics Biomechanics Medical physics Cardiophysics
Health physics
Agrophysics Soil Atmospheric Cloud Chemical Econophysics Geophysics Physical chemistry Authority control LCCN: sh85109469 GND: 4047989-4 SUDOC: 02731569X BNF: cb11938463d (data) N |