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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.,

(855) 
Now, we expect the to be orthonormal (see Sect. 4.9).
In one dimension, this implies that

(856) 
In three dimensions (see Cha. 7), the above expression generalizes to

(857) 
Finally, if the are spinors (see Cha. 10) then
we have

(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

(859) 
Here, the term in angle brackets represents the integrals in Eqs. (856)
and (857) in one and threedimensional regular space, respectively,
and the spinor product (858) in spinspace. The advantage of
our new notation is its great generality: i.e., it
can deal with onedimensional wavefunctions, threedimensional wavefunctions,
spinors, etc.
Expanding a general wavefunction, , in terms of the energy
eigenstates, , we obtain

(860) 
In one dimension, the expansion coefficients take the form (see Sect. 4.9)

(861) 
whereas in three dimensions we get

(862) 
Finally, if is a spinor then we have

(863) 
We can represent all of the above possibilities by
writing

(864) 
The expansion (860) thus becomes

(865) 
Incidentally, it follows that

(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)

(867) 
Here, the are unsurprisingly known as the matrix
elements of .
In one dimension, the matrix elements take the form

(868) 
whereas in three dimensions we get

(869) 
Finally, if is a spinor then we have

(870) 
We can represent all of the above possibilities by
writing

(871) 
The expansion (867) thus becomes

(872) 
Incidentally, it follows that [see Eq. (194)]

(873) 
Finally, it is clear from Eq. (872) that

(874) 
where the are a complete set of eigenstates, and 1 is the
identity operator.
Next: TwoState System
Up: TimeIndependent Perturbation Theory
Previous: Introduction
Richard Fitzpatrick
20100720