齐次保守振荡线性系统的Schroedinger方程表示
The Schroedinger-equation presentation of any oscillatory classical linear system that is homogeneous and conservative
摘要：
哈密顿矩阵与时间无关的含时Schroedinger方程是一个标准型齐次线性振动系统。我们研究是否任何经典系统，本身是线性的，齐次的，振荡的和保守的是保证线性映射到一个薛定谔方程。这样的振荡经典系统可以被分析成它们的简正模，它们是相互独立的、非耦合的简谐振子，并且这样的系统的运动方程线性地映射到薛定谔方程，该薛定谔方程的哈密顿矩阵是对角的，其中h乘以各个简谐振子频率作为其对角项。因此，如果这样一个振荡系统的耦合强度矩阵以对称正定形式表示，则它映射到的薛定谔方程的哈密顿矩阵是该耦合强度矩阵的平方根的h倍。得到了将这类振荡的经典运动方程映射为Schroedinger方程的一般表达式，并将其应用于实值经典Klein-Gordon方程和无源麦克斯韦方程组，得到了严格符合对应原理的相对论性Hamilton算子.一旦这样一个振荡的经典系统被映射到薛定谔方程中，它就自动地处于标准形式，使得薛定谔方程的二次量子化成为一种技术上简单且物理上非常可解释的方法来重新定义原始的经典系统。
英文摘要：
The time-dependent Schroedinger equation with time-independent Hamiltonian matrix is a homogeneous linear oscillatory system in canonical form. We investigate whether any classical system that itself is linear, homogeneous, oscillatory and conservative is guaranteed to linearly map into a Schroedinger equation. Such oscillatory classical systems can be analyzed into their normal modes, which are mutually independent, uncoupled simple harmonic oscillators, and the equation of motion of such a system linearly maps into a Schroedinger equation whose Hamiltonian matrix is diagonal, with h times the individual simple harmonic oscillator frequencies as its diagonal entries. Therefore if the coupling-strength matrix of such an oscillatory system is presented in symmetric, positive-definite form, the Hamiltonian matrix of the Schroedinger equation it maps into is h times the square root of that coupling-strength matrix. We obtain a general expression for mapping this type of oscillatory classical equation of motion into a Schroedinger equation, and apply it to the real-valued classical Klein-Gordon equation and the source-free Maxwell equations, which results in relativistic Hamiltonian operators that are strictly compatible with the correspondence principle. Once such an oscillatory classical system has been mapped into a Schroedinger equation, it is automatically in canonical form, making second quantization of that Schroedinger equation a technically simple as well as a physically very interpretable way to quantize the original classical system.
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