- #1
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1. Given a Markov state density function:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ##
##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of the state at ##\textbf{r}_{n}## transitioning to itself.
2. A probability density function can be decomposed into probability amplitudes, where the probability density function ##P## is the product of a probability amplitude ##\psi_{\textbf{r}_{n-1}} ## and the complex modulus of second probability amplitude ##\psi^*_{\textbf{r}_{n}} ## where:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) = \psi^*_{\textbf{r}_{n}} \cdot \psi_{\textbf{r}_{n-1}} ##
Again, if ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) = \psi^*_{\textbf{r}_{n}} \cdot \psi_{\textbf{r}_{n}} = |\psi_{\textbf{r}_{n}}|^2##
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Unfortunately, I have not been able to find a derivation for #2. Does anyone know where I can find such a derivation?
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ##
##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of the state at ##\textbf{r}_{n}## transitioning to itself.
2. A probability density function can be decomposed into probability amplitudes, where the probability density function ##P## is the product of a probability amplitude ##\psi_{\textbf{r}_{n-1}} ## and the complex modulus of second probability amplitude ##\psi^*_{\textbf{r}_{n}} ## where:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) = \psi^*_{\textbf{r}_{n}} \cdot \psi_{\textbf{r}_{n-1}} ##
Again, if ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) = \psi^*_{\textbf{r}_{n}} \cdot \psi_{\textbf{r}_{n}} = |\psi_{\textbf{r}_{n}}|^2##
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Unfortunately, I have not been able to find a derivation for #2. Does anyone know where I can find such a derivation?