Consider a Hamiltonian $H$ with discrete eigenvalues $\{E_n\}_{n=1}^\infty$ and eigenstates $\{\psi_n\}_{n=1}^\infty$.
Suppose I prepare a state $\psi=c_1\psi_1+c_2\psi_2$ (normalized) and make a measurement of the "energy". What will I find? The Born rule tells me that I will measure the energy to be one of two values:
\begin{align}
\mathcal E=\cases{E_1\quad \text{with probability} |c_1|^2\\
E_2\quad \text{with probability} |c_2|^2 \tag{1}
}
\end{align}
But there is another possibility that seems natural to me: the measured energy could be the expectation value: $\langle \psi|H |\psi\rangle=E_1 |c_1|^2+E_2|c_2|^2$.
So my question is: which of these possibilities is realised? Will I measure the energy to be $\langle H \rangle$ with certainty, or will I measure the energy according to (1)?
Best Answer
Born's rule is correct. Your measurement result will be $E_1$ or $E_2$, but not $E_\text{average}$.
It is like rolling a die. You get a $1$, $2$, $3$, $4$, $5$ or $6$. But you never get a $3\frac 12$.