In mechanics, a constant of motion is quantity that is conserved throughout the motion, imposing 构成 in effect a (mathematical )constraint on the motion.
example: energy, linear momentum, angular momentum, Laplace-Runge-Lenz vector.etc.
Constant of motion allow properties of the motion to be derived without solving the equations of motion.
Hamilton-Jacobi equation
Noether’s theorem
A quantity $A$ is a constant of the motion if its time derivative is zero. $0=\frac{dA}{dt}=\frac{\partial A}{\partial t}+\{A,H\}.$
Poisson’s theorem in Hamiltonian mechanics. $\frac{dA}{dt}=\frac{dB}{dt}=0\to \frac{d}{dt}\{A,B\}=0$. This implies that if $F$ and $G$ are invariants, then the Poisson brackets$\{F,G\}$ is an invariant
An observable quantity $Q=Q(p,q,t)$ will be a constant of motion if it commutes with the Hamiltonian, and it does not iteself depend explicitly on time.
there is a wave function which obeys Schrödinger's equation
$$ \begin{aligned}i\hbar \frac{d\left | \psi \right \rangle }{dt} & = H\left | \psi \right \rangle \\-i\hbar \frac{d\left \langle \psi \right | }{dt} & = \left \langle \psi \right | H\end{aligned} $$
so finally
$$ \begin{aligned}\frac d{dt}\langle\psi|Q|\psi\rangle &=\frac{d\left \langle Q\right \rangle}{dt }\\ &=\frac{d\left \langle \psi \right | }{dt} Q\left | \psi \right \rangle +\left \langle \psi \right | \frac{dQ}{dt} \left | \psi \right \rangle +\left \langle \psi \right | Q\frac{d\left | \psi \right \rangle }{dt} \\& = -\frac1{i\boldsymbol{\hbar}}\left\langle\psi\right|\left[H,Q\right]\left|\psi\right\rangle+\left\langle\psi\right|\frac{dQ}{dt}\left|\psi\right\rangle \end{aligned} $$
For an arbitary state of a quantum mechanical system
if $[H,Q]=0$, and $\frac{\partial Q}{\partial t}=0$ (Q is not explicitly dependent on time), {then $\{H,Q\}=0\to \frac{d Q}{dt}=0$}, then $\frac{d\left \langle Q\right \rangle}{dt }=0$
if $\psi$ is an eigenfunction of Hamiltonian$H|\psi\rangle=E|\psi\rangle$ and $\frac{\partial Q}{\partial t}=0$, then even if $[H,Q]\ne 0$, it is still the case that Q is independent of time$\frac{d\left \langle Q\right \rangle}{dt }=0$.
$$ \begin{aligned}\frac d{dt}\langle Q\rangle & = -\frac1{i\hbar}\langle\psi|\left[H,Q\right]|\psi\rangle +\left\langle\psi\right|\frac{dQ}{dt}\left|\psi\right\rangle\\&=-\frac1{i\boldsymbol{\hbar}}\langle\psi|\left(HQ-QH\right)|\psi\rangle +\left \langle \psi \right | \frac{\partial Q}{\partial t}+\{Q,H\}\left | \psi \right \rangle \\&=-\frac1{i\boldsymbol{\hbar}}\left(E\langle\psi|Q|\psi\rangle-E\langle\psi|Q|\psi\rangle\right)+\left \langle \psi \right | \lambda [Q,H]\left | \psi \right \rangle \\&=0\end{aligned} $$