================================================== What is core concept of quantum physics www.youtube.com/watch?v=obepXcvPYnE ================================================== Square matrix is used as operator in quantum physics ================================================== Ket vector $$$\begin{pmatrix} x \\ y \end{pmatrix}$$$ x and y are complex numbers It's notated as $$$|\psi\big>$$$ ================================================== Bra vector $$$\begin{pmatrix} x && y \end{pmatrix}$$$ x and y are pair of complex number It's notated by $$$\big<\phi|$$$ ================================================== Eigenvalue equation $$$\begin{pmatrix} 2&&0\\1&&1 \end{pmatrix} \begin{pmatrix} 1\\1 \end{pmatrix}= 2 \begin{pmatrix} 1\\1 \end{pmatrix} $$$ Goal is find number like 2 $$$\begin{pmatrix} 1\\1 \end{pmatrix}$$$ should be kept on both sides And it's called eigenvalue $$$\begin{pmatrix} 2&&0\\1&&1 \end{pmatrix}$$$ is given square matrix ================================================== This is core equation of quantum physics $$$\hat{H} |\psi\big> = E |\psi\big>$$$ $$$\hat{H}$$$ is squaree matrix as operator $$$E$$$ is energy as eigenvalue from Meaning is that $$$E$$$ is emerged as eigenvalue from $$$\hat{H}$$$ ================================================== Multiply bra vector $$$\big<\phi|$$$ $$$\big<\phi| \times \hat{H} |\psi\big> = E \big<\phi| \times |\psi\big>$$$ E is constant value Divide $$$E=\dfrac{\big<\phi| \times \hat{H} |\psi\big>}{\big<\phi| \times |\psi\big>}$$$ Meaning from above equation is that energy $$$E$$$ is expectation value from $$$\hat{H}$$$ ================================================== Bra vector $$$\times$$$ ket vector $$$\big<\phi| \times |\psi\big> = a^* a + b^*b$$$ Let's suppose situations - If $$$\big<\phi| \times |\psi\big> = 1$$$, calculation will be simple Adjusting equation to make $$$\big<\phi| \times |\psi\big>$$$ as 1 is called as Normalization Then, equation will become $$$E = \big<\phi| \times \hat{H} \times |\psi\big> $$$ $$$\hat{H}$$$ is called Hamiltonian Energy is expectation value from Hamiltonian operator