Why do we describe probability amplitude rather than probability itself in quantum mechanics?

Question

In the quantum mechanics, the dynamics of quantum system are described in terms of probability amplitude. However, we want to calculate the probability in the end which can be measured. Why don't we develop quantum mechanics directly describing the probability instead of probability amplitude? Wouldn't this make the quantum mechanics more interpretable and simple?

Answer 1

To make sense of interference fringes without probability amplitudes, you would need a messy ad-hoc theoretical framework like the Bohmian pilot wave. Probability amplitudes are used simply because it explains interference in a simple way

$$P_{A+B} = (\langle \Psi_A | + \langle \Psi_B | )(| \Psi_A \rangle + | \Psi_B \rangle) = \langle \Psi_A | \Psi_A \rangle + \langle \Psi_B | \Psi_B \rangle + 2 \langle \Psi_A | \Psi_B \rangle $$

If the $| \Psi_A \rangle$ are eigenstates of energy expressible as $e^{-i\omega_A t}| \phi(x)_A \rangle$, then the above expression becomes:

$$ \langle \phi_A(x) | \phi_A(x) \rangle + \langle \phi_B(x) | \phi_B(x) \rangle + 2 \cos((\omega_A - \omega_B)t) \langle \phi_A(x) | \phi_B(x) \rangle = {P_A + P_B + 2 \cos((\omega_A - \omega_B)t) \langle \phi_A(x) | \phi_B(x) \rangle}, $$

where $P_A$ and $P_B$ are the separated probabilities of each state. Even if you were able to express everything without amplitudes, you would probably need a lot of mathematical contortions to get that last interference term. In fact it is possible since Bohmian pilot wave theory exists and it is equivalent to Quantum Mechanics, but it is not straightforward for calculation purposes

Answer 2

You are right. In proper treatments of the mathematical foundations of quantum mechanics, following von Neumann, the probability amplitude is simply defined from the probability using the Born rule and satisfying Hilbert space. This does indeed make quantum mechanics much easier to understand, and being rigorous, it actually follows that interpretation is a mathematically solve problem. I have a published paper on the topic, The Hilbert space of conditional clauses.