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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Madelung equations ： ウィキペディア英語版 Madelung equations The Madelung equations are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation. ==Equations== The Madelung equations are quantum Euler equations: :$\backslash partial\_t\; \backslash rho+\backslash nabla\backslash cdot(\backslash rho\backslash bold\; u)=0,$ :$\backslash partial\_t\backslash bold\; u+\backslash bold\; u\backslash cdot\backslash nabla\backslash bold\; u=-\backslash frac\backslash nabla\backslash left(\backslash frac\backslash right)=-\backslash frac\backslash nabla\backslash cdot\backslash bold\; p\_Q-\backslash frac\backslash nabla\; U,$ where $\backslash bold\; u$ is the flow velocity in the quantum probability space with mass density $\backslash bold\; \backslash rho=m|\backslash psi|^2$. The circulation of the flow velocity field along any closed path obeys the auxiliary condition . The term in the brackets represents a quantum chemical potential $\backslash mu$ in vacuum. The kinetic energy operator from the Hamiltonian $\backslash hat\; H$ results in a non-local quantum pressure tensor :$\backslash bold\; p\_Q=-(\backslash hbar/2m)^2\; \backslash rho\; \backslash nabla\backslash otimes\backslash nabla\; \backslash ln\; \backslash rho$ which is related to the Bohm quantum potential $Q$. While the latter is the icon of the de Broglie–Bohm theory, $\backslash bold\; p\_Q$ is the quantum symbol of the Madelung hydrodynamics.〔 〕 The integral energy stored in the quantum pressure tensor is proportional to the Fisher information, which accounts for the quality of measurements. Thus, according to the Cramér–Rao bound, the Heisenberg Uncertainty principle is equivalent to a standard inequality for the efficiency (statistics) of measurements. The thermodynamic definition of the quantum chemical potential $\backslash mu\; =Q+U$ follows from the hydrostatic force balance above $\backslash \; \backslash nabla\; \backslash mu\; =\; (m/\backslash rho)\backslash nabla\; \backslash cdot\; \backslash bold\; p\_Q\; +\; \backslash nabla\; U$. According to thermodynamics, at equilibrium the chemical potential is constant everywhere, which corresponds straightforward to the stationary Schrödinger equation. Therefore, the eigenvalues of the Schrödinger equation are free energies, which differ from the internal energies of the system. The particle internal energy is calculated via $\backslash \; \backslash epsilon\; =\; \backslash mu\; -\; tr(\backslash bold\; p\_Q)(m\; /\; \backslash rho)\; =\; -\backslash hbar^2(\backslash nabla\backslash ln\; \backslash rho)^2/8m\; +\; U$ and it is related to the local Carl Friedrich von Weizsäcker correction. In the case of a quantum harmonic oscillator, for instance, one can easily show that the zero point energy is the value of the oscillator chemical potential, while the oscillator internal energy is zero in the ground state, $\backslash \; \backslash epsilon\; =\; 0$. Hence, the zero point energy represents the energy to place a static oscillator in vacuum, which shows again that the vacuum fluctuations are the reason for quantum mechanics. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Madelung equations」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.