# Discussion: how to represent the time evolution operator as a countable Markov process?

$$\hat{H} = H(\hat{X},\hat{P})$$
- H: hamiltonian(energy operator), H is a hermition matrix in a given Hilbert space
- X: position operator, P: momentum operator (both hermite)
- Represent the eigenvalue problem of H and corresponding eigenvectors/eigenvalues by Dirac-notation, as following form:
$$\hat{H}|E> = E|E>$$
- E: eigen-energy, |E> is the corresponding basis state vector
- Similarly,
$$\hat{X}|x>=x|x>,\\\hat{P}|p>=p|p>$$

- Time evolution operator $$e^{-i\hat{H}t}$$

- where t: time,t>=0, and satisfies:

$$|x_t> = e^{-i\hat{H}t}|x_0>$$

- The **transition probability** from x0 to xt is:
$$Pr(x_0\rightarrow x_t)=<x_t|x_0>^{*}<x_t|x_0>$$

- Eg. Consider the free particle case, the simplest case, H can be rewriten as:
$$H(\hat{P}) = \frac{\hat{P}^2}{2m} $$
- which satisfies:
$$ \frac{\hat{P}^2}{2m}|x> = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}|x>$$
- hbar is reduced planck constant