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In [1] it is mentioned that the probability flux

\Bj(\Bx, t) = -\frac{i\Hbar}{2 m} \lr{ \psi^\conj \spacegrad \psi – \psi \spacegrad \psi^\conj },

is related to the momentum expectation at a given time by the integral of the flux over all space

\int d^3 x \Bj(\Bx, t) = \frac{\expectation{\Bp}_t}{m}.

That wasn’t obvious to me at a glance, however, this can be seen by recasting the integral in bra-ket form. Let

\psi(\Bx, t) = \braket{\Bx}{\psi(t)},

and note that the momentum portions of the flux can be written as

-i \Hbar \spacegrad \psi(\Bx, t) = \bra{\Bx} \Bp \ket{\psi(t)}.

The current is therefore

\Bj(\Bx, t)
&= \frac{1}{2 m}
\psi^\conj \bra{\Bx} \Bp \ket{\psi(t)}
+\psi {\bra{\Bx} \Bp \ket{\psi(t)} }^\conj
} \\
&= \frac{1}{2 m}
{\braket{\Bx}{\psi(t)}}^\conj \bra{\Bx} \Bp \ket{\psi(t)}
+ \braket{\Bx}{\psi(t)} {\bra{\Bx} \Bp \ket{\psi(t)} }^\conj
} \\
&= \frac{1}{2 m}
\braket{\psi(t)}{\Bx} \bra{\Bx} \Bp \ket{\psi(t)}
\bra{\psi(t)} \Bp \ket{\Bx} \braket{\Bx}{\psi(t)}

Integrating and noting that the spatial identity is \( 1 = \int d^3 x \ket{\Bx}\bra{\Bx} \), we have

\int d^3 x \Bj(\Bx, t)
\bra{\psi(t)} \Bp \ket{\psi(t)},

This is just the expectation of \( \Bp \) with respect to a specific time-instance state.


[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.