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A couple of remarks on the text of Chapter 4, Probabilities vs. amplitudes.

1. In sections 4.3, Stochastic versus unitary operators, and 4.4, in Infinitesmal stochastic vs. self-adjoint operators, it looks like wherever you mean to write $\ge$, it is showing up in the pdf as $\gt$. For instance on p. 40 you say that a stochastic operator gives a square matrix with non-negative entries, but the formula prints as $U_{i j} \gt 0$. For another example, the second condition for a stochastic operator shows up as $\psi \ge 0 \quad \implies \quad U \psi \ge 0$.

2. On page 36, you gave the corrected value for as $H = r(a^\dagger - 1)$, after showing that your initial guess of $H = r a^\dagger$ failed to map probability distributions to probability distributions. After giving the algebraic explanation that you gave for why the one operator works and the other one doesn't, it could make the reader even happier if you pointed out the meaning of the "fudge factor" -1. The creation operator $r a^\dagger$ was a correct guess for part of the story, which is the contribution of $\Psi_n$ to the positive derivative of $\Psi_{n+1}$ (scaled by the rate constant $r$). But but because the probabilities have to add up to one, they must "flow" among the components. In this case, the rising probability of $\Psi_{n+1}$ is accompanied by an equally falling component of $\Psi_{n}$. This second contribution is the "destruction" of probability in $\Psi_n$ due to the existence of probability in $\Psi_n$. This is the content of the factor $-1$ in the adjusted formula.

A couple of remarks on the text of Chapter 4, Probabilities vs. amplitudes.

1. In sections 4.3, Stochastic versus unitary operators, and 4.4, in Infinitesmal stochastic vs. self-adjoint operators, it looks like wherever you mean to write $\ge$, it is showing up in the pdf as $\gt$. For instance on p. 40 you say that a stochastic operator gives a square matrix with non-negative entries, but the formula prints as $U_{i j} \gt 0$. For another example, the second condition for a stochastic operator shows up as $\psi \ge 0 \quad \implies \quad U \psi \ge 0$.

2. On page 36, you gave the corrected value for as $H = r(a^\dagger - 1)$, after showing that your initial guess of $H = r a^\dagger$ failed to map probability distributions to probability distributions. After giving the algebraic explanation that you gave for why the one operator works and the other one doesn't, it could make the reader even happier if you pointed out the meaning of the "fudge factor" -1. The creation operator $r a^\dagger$ was a correct guess for part of the story, which is the contribution of $\Psi_n$ to the positive derivative of $\Psi_{n+1}$ (scaled by the rate constant $r$). But but because the probabilities have to add up to one, they must "flow" among the components. In this case, the rising probability of $\Psi_{n+1}$ is accompanied by an equally falling component of $\Psi_{n}$. This second contribution is the "destruction" of probability in $\Psi_n$ due to the existence of probability in $\Psi_n$. This is the content of the factor $-1$ in the adjusted formula.