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Xingdong Zuo
Xingdong Zuo
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Proof of the solution of von Neumann equation will never stabilizefluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$

If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will never be stabilizedfluctuate forever. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to show that it will endlessly oscillate and never be damped.)

Mathematically, $H$ is an Hermitian matrix and $\rho(t)$ is a square matrix with trace one, positiveness and self-adjoint.

We need a mathematical proof that particularly by using the fact that $[H, \rho(0)] \neq 0$, i.e. $H\rho(0)\neq\rho(0)H$

Proof of the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$

If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will never be stabilized. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to show that it will endlessly oscillate and never be damped.)

Mathematically, $H$ is an Hermitian matrix and $\rho(t)$ is a square matrix with trace one, positiveness and self-adjoint.

We need a mathematical proof that particularly by using the fact that $[H, \rho(0)] \neq 0$, i.e. $H\rho(0)\neq\rho(0)H$

Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$

If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate forever. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to show that it will endlessly oscillate and never be damped.)

Mathematically, $H$ is an Hermitian matrix and $\rho(t)$ is a square matrix with trace one, positiveness and self-adjoint.

We need a mathematical proof that particularly by using the fact that $[H, \rho(0)] \neq 0$, i.e. $H\rho(0)\neq\rho(0)H$

Source Link
Xingdong Zuo
Xingdong Zuo
  • 325
  • 1
  • 9

Proof of the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$

If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will never be stabilized. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to show that it will endlessly oscillate and never be damped.)

Mathematically, $H$ is an Hermitian matrix and $\rho(t)$ is a square matrix with trace one, positiveness and self-adjoint.

We need a mathematical proof that particularly by using the fact that $[H, \rho(0)] \neq 0$, i.e. $H\rho(0)\neq\rho(0)H$