the paradigm of a classical harmonic oscillator

$$ F=-kx=m\frac{d^2x}{dt^2} $$

$$ x(t)=A\sin(\omega t)+B\cos(\omega t),where,\omega\equiv\sqrt{\frac km} $$

$\omega$ is the (angular) frequncy of osillation. The potential energy is

$$ V(x)=\frac12kx^2; $$

Virtually, any osillatoty motion is approximately simple harmonic, as long as the amplitude is small 微振动

$$ \begin{aligned}V(x)&=V(x_0)+V'(x_0)(x-x_0)+\frac{1}{2}V''(x_0)(x-x_0)^2+\cdots \\ &\cong\frac12V^{\prime\prime}(x_0)(x-x_0)^2, \end{aligned} $$

The quantum problem

$$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+(\frac12m\omega^2x^2)\psi=E\psi. $$

代数法|对易子Commuator|算符|二次量子化

time-independent annihilation and creation operator.

$$ \frac1{2m}[p^2+(m\omega x)^2]\psi=E\psi. $$

The basic idea is to factor the Hamiltonian. Operators do not, in general, commute.

$$ H=\frac1{2m}[p^2+(m\omega x)^2]. $$

the commuator of operator A and B:

$$ [A,B]\equiv AB-BA. $$

Operators are notoriously出了名的 slippery to work with in the abstract, and you are bound to make mistakes unless you give them a “test function” $f(x)$, to act on.

$$ [x,p]f(x)=\left[x\frac{\hbar}{i}\frac{d}{dx}(f)-\frac{\hbar}{i}\frac{d}{dx}(xf)\right]=\frac{\hbar}{i}\left(x\frac{df}{dx}-x\frac{df}{dx}-f\right)=i\hbar f(x). $$

We get the canonical coummutation relation.正则对易关系。坐标和动量（算符）不对易the position and momentum do not coummute.

取分解

$$ \boxed{\quad a_{\pm}\equiv\frac1{\sqrt{2\hbar m\omega}}\left(\mp ip+m\omega x\right)} $$