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Before commencing our investigation, it is helpful to introduce some
improved notation. Let the
be a complete set of eigenstates
of the Hamiltonian,
, corresponding to the eigenvalues
:
i.e.,
![\begin{displaymath}
H \psi_i = E_i \psi_i.
\end{displaymath}](img2099.png) |
(855) |
Now, we expect the
to be orthonormal (see Sect. 4.9).
In one dimension, this implies that
![\begin{displaymath}
\int_{-\infty}^\infty \psi_i^\ast \psi_j dx = \delta_{ij}.
\end{displaymath}](img681.png) |
(856) |
In three dimensions (see Cha. 7), the above expression generalizes to
![\begin{displaymath}
\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \psi_i^\ast \psi_j dx dy dz = \delta_{ij}.
\end{displaymath}](img2100.png) |
(857) |
Finally, if the
are spinors (see Cha. 10) then
we have
![\begin{displaymath}
\psi_i^\dag \psi_j = \delta_{ij}.
\end{displaymath}](img2101.png) |
(858) |
The generalization to the case where
is a product of a regular
wavefunction and a spinor is fairly obvious. We can represent all
of the above possibilities by writing
![\begin{displaymath}
\langle \psi_i\vert\psi_j\rangle \equiv \langle i\vert j\rangle = \delta_{ij}.
\end{displaymath}](img2102.png) |
(859) |
Here, the term in angle brackets represents the integrals in Eqs. (856)
and (857) in one- and three-dimensional regular space, respectively,
and the spinor product (858) in spin-space. The advantage of
our new notation is its great generality: i.e., it
can deal with one-dimensional wavefunctions, three-dimensional wavefunctions,
spinors, etc.
Expanding a general wavefunction,
, in terms of the energy
eigenstates,
, we obtain
![\begin{displaymath}
\psi_a = \sum_i c_i \psi_i.
\end{displaymath}](img2103.png) |
(860) |
In one dimension, the expansion coefficients take the form (see Sect. 4.9)
![\begin{displaymath}
c_i = \int_{-\infty}^\infty\psi_i^\ast \psi_a dx,
\end{displaymath}](img2104.png) |
(861) |
whereas in three dimensions we get
![\begin{displaymath}
c_i = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast \psi_a dx dy dz.
\end{displaymath}](img2105.png) |
(862) |
Finally, if
is a spinor then we have
![\begin{displaymath}
c_i = \psi_i^\dag \psi_a.
\end{displaymath}](img2106.png) |
(863) |
We can represent all of the above possibilities by
writing
![\begin{displaymath}
c_i =\langle\psi_i\vert\psi_a\rangle\equiv \langle i\vert a\rangle.
\end{displaymath}](img2107.png) |
(864) |
The expansion (860) thus becomes
![\begin{displaymath}
\psi_a = \sum_i\langle\psi_i\vert\psi_a\rangle \psi_i\equiv \sum_i \langle i\vert a\rangle \psi_i.
\end{displaymath}](img2108.png) |
(865) |
Incidentally, it follows that
![\begin{displaymath}
\langle i\vert a\rangle^\ast=\langle a\vert i\rangle.
\end{displaymath}](img2109.png) |
(866) |
Finally, if
is a general operator, and the wavefunction
is expanded in the manner shown in Eq. (860), then the expectation value of
is written (see Sect. 4.9)
![\begin{displaymath}
\langle A\rangle = \sum_{i,j} c_i^\ast c_j A_{ij}.
\end{displaymath}](img2110.png) |
(867) |
Here, the
are unsurprisingly known as the matrix
elements of
.
In one dimension, the matrix elements take the form
![\begin{displaymath}
A_{ij} = \int_{-\infty}^\infty\psi_i^\ast A \psi_j dx,
\end{displaymath}](img2112.png) |
(868) |
whereas in three dimensions we get
![\begin{displaymath}
A_{ij} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast A \psi_j dx dy dz.
\end{displaymath}](img2113.png) |
(869) |
Finally, if
is a spinor then we have
![\begin{displaymath}
A_{ij}=\psi_i^\dag A \psi_j.
\end{displaymath}](img2114.png) |
(870) |
We can represent all of the above possibilities by
writing
![\begin{displaymath}
A_{ij}=\langle \psi_i\vert A\vert\psi_j\rangle \equiv \langle i\vert A\vert j\rangle.
\end{displaymath}](img2115.png) |
(871) |
The expansion (867) thus becomes
![\begin{displaymath}
\langle A\rangle \equiv\langle a\vert A\vert a\rangle= \sum_...
...rangle
\langle i\vert A\vert j\rangle \langle j\vert a\rangle.
\end{displaymath}](img2116.png) |
(872) |
Incidentally, it follows that [see Eq. (194)]
![\begin{displaymath}
\langle i\vert A\vert j\rangle^\ast=\langle j\vert A^\dag \vert i\rangle.
\end{displaymath}](img2117.png) |
(873) |
Finally, it is clear from Eq. (872) that
![\begin{displaymath}
\sum_{i} \vert i\rangle \langle i\vert \equiv 1,
\end{displaymath}](img2118.png) |
(874) |
where the
are a complete set of eigenstates, and 1 is the
identity operator.
Next: Two-State System
Up: Time-Independent Perturbation Theory
Previous: Introduction
Richard Fitzpatrick
2010-07-20