Measurement
As with any quantum mechanical observable, in order to discuss measurement, we need to calculate the spectral resolution of Q:
Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let denote the indicator function of B. We see that the projection-valued measure ΩQ is given by
i.e. ΩQ is multiplication by the indicator function of B. Therefore, if the system is prepared in state ψ, then the probability of the measured position of the particle being in a Borel set B is
where μ is the Lebesgue measure. After the measurement, the wave function collapses to either
or
, where is the Hilbert space norm on L2(R).
Read more about this topic: Position Operator
Famous quotes containing the word measurement:
“That’s the great danger of sectarian opinions, they always accept the formulas of past events as useful for the measurement of future events and they never are, if you have high standards of accuracy.”
—John Dos Passos (1896–1970)