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Without this grouping of the RHS, the argument of cos was rendered on the next line, which is really confusing
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Ruslan
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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:

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

Wherewhere $P_A$ 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

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

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

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lurscher
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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