// ＜Complete Normalized Orthogonal System＞完全正規直交系(正規直交基底) The j-th wave function is expressed by complete normalized orthogonal system as follows.(j番目の波動関数は完全正規直交系を用いて次のように表される。) \begin{eqnarray}…

// ＜Time-independent Hamiltonian＞ Schrödinger's Wave Equation is described as follows. \begin{eqnarray}i\hbar\frac{\partial}{\partial t}\Psi &=&\hat{H}\Psi\end{eqnarray} ] ] ] ] ] ] ] Schrödinger's Wave Equation is described as follows w…

// ＜Linear Differential Equation part 2＞ Linear differential equation can be solved as below.(線形微分方程式は下記のように解くことができる。) \begin{eqnarray}\prod_{i=0}^k\left(\frac{d}{dx}-\lambda_i\right)y&=&0\notag \\ \left(\frac{d}{d…

// ＜Linear Differential Equation＞ Linear differential equation can be solved as follows.(線形微分方程式は次のようにして解くことができる。) \begin{eqnarray}\left(\frac{d}{dx}-\lambda\right)^{k}y&=&0\notag\\\left(\frac{d}{dx}-\lambda\right…

// ＜Linear Differential Operator＞ A linear combination of differential operators is described as below. \begin{eqnarray}\sum_{i=0}^na_i\left(\frac{d}{dx}\right)^i\end{eqnarray} Then, the next formula is true. \begin{eqnarray}\sum_{i=0}^n…

// ＜Wave Equation＞ Wave equation in 3 dimensions is described as below. \begin{eqnarray}\frac{\partial^2u}{\partial t^2} &=& v^2\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\righ…

// ＜Wave Equation＞ Wave equation in one dimension is described as below. \begin{eqnarray}\frac{\partial^2u}{\partial t^2} &=& v^2\frac{\partial^2u}{\partial x^2} \end{eqnarray} Here, is the displacement of the wave at position in time , …