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

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?

• If you like this question you may also enjoy reading this Phys.SE post. Commented Jun 9, 2020 at 14:47
• Related: QM without complex numbers, and links therein.
– user87745
Commented Jun 9, 2020 at 16:02

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