is just the formulation of quantum mechanics.

Schrödinger picture

波函数随时间 or any others 演化，其他如算符不变。

• 波函数保持归一化 norm，时间演化算符是unitary $\hat U^\dagger\hat U=\hat I.$ ← 基本假设：概率守恒与范数不变性

  • density matrix：描述量子系统的系综

  • 概率流密度：纯态在特定表象下的概率密度

    

• observable $\hat F$ 不变: $\lang \psi|\hat F|\psi\rang\to\bra{\phi}\hat F\ket \psi$, where $\ket\phi=\hat U(t,t_0)\ket \psi$.

  • 孤立系统（系统的概率守恒）：time evoltion operator, a unitary operator $\hat U_{t_0\to t}(t,t_0)$. 有 $\ket {\psi_t}=\hat U(t,t_0)\ket{\psi_0}.$
  • 时间演化算符

    • 系统哈密顿量不随时间变化 $\partial_t \hat H=0$, 写为 $i\hbar\frac\partial{\partial t}\hat U(t)=\hat H\hat U(t)$的薛定谔方程的解为$U(t)=e^{-iHt/\hbar}$（按算符计算即可）。

    • 系统哈密顿量随时间变化$\partial_t\hat H\ne 0$，时间演化算符是不同时间哈密顿量的积分

      $$ U(t)=\exp\biggl(-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime}) dt^{\prime}\biggr), $$

      $$ U(t)=\mathrm{T}\exp\biggl(-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime}) dt^{\prime}\biggr), $$

the quantum state evolves in time or any other parameter that characterizes the process.

Schrödinger equation:

$$ \mathrm{i}h\frac{\partial|\psi\rangle}{\partial t}=\hat{H}|\psi\rangle $$

The evolution should preserve the norm of the quantum state $\langle\psi|\psi\rangle.$→$\langle\psi_1|\psi_2\rangle.$

$$ \frac{\mathrm{d}\langle\psi_1|\psi_2\rangle}{\mathrm{d}t}=\frac{\mathrm{i}}{\hbar}\left\langle\psi_1\left|(\hat{H}^\dagger-\hat{H})\right|\psi_2\right\rangle=0. $$

and see that the operator $\hat H$ is Hermitian.$\hat H^\dagger=\hat H$

Note: Operators in the Schrödinger picture are time-independent→ operators in the Schrödinger picture do not evolve in time due to the Hamiltonian of the system.[电磁系统位矢有规范不变性，显含时间并非一定引起总能量、系统状态的变化]