quantum physics equations


A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant. m E {\displaystyle ={\frac {\hbar }{m}}\mathrm {Im} (\Psi ^{*}\nabla \Psi )=\mathrm {Re} (\Psi ^{*}{\frac {\hbar }{im}}\nabla \Psi )}. ∈ 2 r − L = /

{\displaystyle {\begin{aligned}&\ell \in \{0\cdots n-1\}\\&m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\\\end{aligned}}\,\! d d

i 2 s i m ( Ψ ⋯

List of equations in nuclear and particle physics List of equations in wave theory List of photonics equations List of relativistic P.M . 2 2

{\displaystyle \sigma (n)\sigma (\phi )\geq {\frac {\hbar }{2}}\,\!

m



1 しかしながら、同じく20世紀の初頭に建設されていった量子力学は、次第に非決定論的な性格を帯びたものであることが知られるようになった。量子力学が非決定論的であることが知られるにつれ、量子力学が真に非決定論であるか、あるいは量子力学に変わる決定論的な理論が存在し得るかなどといった議論が生じ、量子力学の理論形式の解釈をめぐり論争が展開された[21]。, たとえば量子力学が形成される初期において、従来のニュートン力学や相対性理論と異なり、物体が時空上に定まった軌道をとらないが、実験においてはウィルソンの霧箱などを利用することで粒子の軌跡を知ることができ、見かけ上は古典的な運動が実現されていることが指摘された[22]。この粒子の飛跡を説明する過程で、ハイゼンベルクにより不確定性原理が発見され、粒子の飛跡の問題について正当性のある物理的解釈が得られるようになった。不確定性原理によれば、物体の位置と運動量の両方を定めることができず、位置を精度よく定めるほど、運動量を正確には決定できなくなる[23]。しかしながら位置と運動量の不確定性の積は、プランク定数程度の大きさになるので、霧箱の実験においては位置と運動量を充分な精度で測定することができ、粒子が連続的に運動しているように見えることについて説明付けられる。, ハイゼンベルクによって示された不確定性関係の解釈や適用範囲についてもまた、ハイゼンベルクによる提案から現在に至るまで議論が続けられている。

j ∏ DOI:10.1093/acprof:oso/9780199213900.003.03, Part I Probability in classical and quantum physics, Part III Stochastic processes in Hilbert space, 1 Classical Probability Theory and Stochastic Processes, 5 Probability Distributions on Hilbert Space, 8 Applications to Quantum Optical Systems, 10 Non-Markovian Dynamics in Physical Systems, 11 Measurements in Relativistic Quantum Mechanics, Theoretical, Computational, and Statistical Physics. 1 {\displaystyle \|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,}, Ψ

/ t ∑ V

( = 1 − {\displaystyle |\mathbf {J} |=\hbar {\sqrt {j(j+1)}}\,\! ( )

s e }, σ

where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is, ∇ Published to Oxford Scholarship Online: February 2010, DOI: 10.1093/acprof:oso/9780199213900.001.0001, PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). s

+ ℓ }, Non-relativistic time-independent Schrödinger equation, Non-relativistic time-dependent Schrödinger equation, Last edited on 21 September 2020, at 15:25, List of equations in nuclear and particle physics, https://en.wikipedia.org/w/index.php?title=List_of_equations_in_quantum_mechanics&oldid=979584829, Creative Commons Attribution-ShareAlike License, varies with situation and number of particles, This page was last edited on 21 September 2020, at 15:25. { λ ∗ ) = , 2 1

{\displaystyle S_{z}=m_{s}\hbar \,\! | . /



N A ℓ , H

n

) S }, | This article summarizes equations in the theory of quantum mechanics.GEo MATICS. m Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. i d



s

{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }, Time-independent case: p =

j z V ⟩

2 As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. Ψ

m 2 {\displaystyle {\begin{aligned}&\Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)}}\,\hbar \\&m_{s}\in \{-s,-s+1\cdots s-1,s\}\\\end{aligned}}\,\!

≥ ≥ ‖ } , = = |

∂ V ⟨ 1 ∇ }, Energy-time

.

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