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First, let's say I have a classical system involving throwing a fair coin. There are two possible events $\{\text{head},\text{tails}\}$. Their respective probabilities are:

$$ P(\text{head})=\frac{1}{2}\\ P(\text{tails})=\frac{1}{2}\\ P(\text{head})+P(\text{tails})=1 $$

In a quantum system scenario, the probabilities are replaced by a complex amplitude. The square modulus of the amplitude is a "probability density" evaluated at a point in phase space. For instance:

$$ A(\text{head})=c_1e^{i\theta_1}\\ A(\text{tails})=c_2e^{i\theta_2}\\ A(\text{head})+A(\text{tails})=c_1e^{i\theta_1}+c_2e^{i\theta_2}\\ I=(c_1e^{i\theta_1}+c_2e^{i\theta_2})(c_1e^{-i\theta_1}+c_2e^{-i\theta_2})=c_1^2+c_2^2 + 2c_1c_2 \cos (\theta_2-\theta_2) $$

And the sum of probabilities is given by the integral over all of phase space:

$$ \int_{-\infty}^\infty I(c_1[x],c_2[x],\theta_1[x],\theta_2[x])dx=1 $$

where $x$ is a prametrization for $c_1,c_2,\theta_1,\theta_2$.

Thus the quantum analog of

$$ P(\text{head})+P(\text{tails})=1 $$

is

$$ \int_{-\infty}^\infty I(c_1[x],c_2[x],\theta_1[x],\theta_2[x])dx=1 $$

Now, say I want to show that classical probability is a special case of quantum probability. I can set, as a restriction, the complex amplitude to be real, then the usual sum of probabilities is obtained without interference:

$$ |\Re[A_1]+\Re[A_2]|^2=|A_1|^2+|A_2|^2+2|A_1||A_2| $$

But I do not understand the procedure to get rid of the integral?

$$ \int_{-\infty}^{\infty}( |A_1|^2+|A_2|^2+2|A_1||A_2|) dx=1 $$

It seems that, at best, the integral may reduce to a continuous probability distribution in the classical case, such as a gaussian integral - is that the case?

Qmechanic
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Anon21
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1 Answers1

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Quantum mechanics relies on the usual probability theory. Specifically, the modulus square of a wave function is conventional probability density: $$w(x) = \psi^*(x)\psi(x).$$ The difference between quantum theory and the probabilistic description of classical systems is that the latter operates with the probability density (e.g., via Fokker-Plank equation or rate equations), whereas the former formulates equations for wave function or density matrix (which is a generalization of the wave function).

To summarize: quantum probability theory is not a spacial case, but the same probability theory.

Update
Note that the classical random processes mentioned above are not the classical limit of quantum fluctuations. Indeed, the classical limit of the quantum probability is deterministic classical mechanics. When processes in classical physics are random it is for different reasons - e.g., Brownian motion of a particle interacting with an environment. In the quantum domain one then has two sources of randomness - due to the quantum nature of particles and due to the other random effects, like collisions with other particles, etc. In fact, this is precisely the case where the wave function is not sufficient and one has to use the density matrix. Such processes are described using quantum kinetic equations or *master equation.

Roger V.
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