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# Improved Notation

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 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

 (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: Two-State System Up: Time-Independent Perturbation Theory Previous: Introduction
Richard Fitzpatrick 2010-07-20