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Consider a dynamical system consisting of a single non-relativistic particle of mass
moving along the
-axis in some real potential
. In quantum mechanics, the instantaneous state of the system is represented by a complex wavefunction
. This wavefunction evolves in time
according to Schrödinger's equation:
![\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2 m}\frac{\partial^2\psi}{\partial x^2} + V(x) \psi.
\end{displaymath}](img442.png) |
(137) |
The wavefunction is interpreted as follows:
is
the probability density of a measurement of the particle's
displacement yielding the value
. Thus, the probability of
a measurement of the displacement giving a result
between
and
(where
) is
![\begin{displaymath}
P_{x \in a:b}(t) = \int_{a}^{b}\vert\psi(x,t)\vert^{ 2} dx.
\end{displaymath}](img458.png) |
(138) |
Note that this quantity is real and positive definite.
Richard Fitzpatrick
2010-07-20