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List of equations in quantum mechanics ：ウィキペディア英語版

List of equations in quantum mechanics

This article summarizes equations in the theory of quantum mechanics.
==Wavefunctions==

A fundamental physical constant occurring in quantum mechanics is the Planck constant, ''h''. A common abbreviation is , also known as the ''reduced Planck constant'' or ''Dirac constant''.
\left(\Psi^
* \nabla \Psi - \Psi \nabla \Psi^
*\right)

= \frac\hbar m \mathrm(\Psi^*\nabla\Psi)=\mathrm(\Psi^* \frac \nabla \Psi)

star
* is complex conjugate
| m⁻² s⁻¹
| ()⁻¹ ()⁻²
|-
|}
The general form of wavefunction for a system of particles, each with position r_''i'' and z-component of spin ''s_{z i}''. Sums are over the discrete variable ''s_z'', integrals over continuous positions r.
For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.
, t \right )
in bra–ket notation:

|\Psi\rangle = \sum_}\cdots\sum_\int_\cdots\int_ \mathrm\mathbf_1\mathrm\mathbf_2\cdots\mathrm\mathbf_N \Psi |\mathbf, \mathbf\rangle

for non-interacting particles:

\Psi  = \prod_^N\Psi \left (\mathbf_n,s_, t \right )

|-
!Position-momentum Fourier transform (1 particle in 3d)
|

*Φ = momentum-space wavefunction
*Ψ = position-space wavefunction

\begin \Phi(\mathbf,s_z,t) & = \frac} e^/\hbar} \Psi(\mathbf, s_z,t)\mathrm^3\mathbf \\ &\upharpoonleft \downharpoonright\\\Psi(\mathbf,s_z,t) & = \frac} e^/\hbar} \Phi(\mathbf,s_z,t)\mathrm^3\mathbf_n \\\end

|-
!General probability distribution
|

*''V_j'' = volume (3d region) particle may occupy,
*''P'' = Probability that particle 1 has position r₁ in volume ''V''₁ with spin ''s''_''z''1 and particle 2 has position r₂ in volume ''V''₂ with spin ''s''_''z''2, etc.

P = \sum_}\sum_\cdots\int_\int_ \left | \Psi \right |^2\mathrm^3\mathbf_1\mathrm^3\mathbf_2\cdots\mathrm^3\mathbf_N\,\!

|-
!General normalization condition
|
|

P = \sum_}\sum_}\cdots\int\limits_} \left | \Psi \right |^2\mathrm^3\mathbf_1\mathrm^3\mathbf_2\cdots\mathrm^3\mathbf_N = 1\,\!

|-
|}

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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