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11.2: Improved Notation

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Before commencing our investigation, it is helpful to introduce some improved notation. Let the ψi be a complete set of eigenstates of the Hamiltonian, H, corresponding to the eigenvalues Ei: that is, Hψi=Eiψi.

Now, we expect the ψi to be orthonormal. (See Section [seig].) In one dimension, this implies that ψiψjdx=δij.
In three dimensions (see Chapter [sthree]), the previous expression generalizes to

ψiψjdxdydz=δij.

Finally, if the ψi are spinors (see Chapter [sspin]) then we have

ψiψj=δij.

The generalization to the case where ψ is a product of a regular wavefunction and a spinor is fairly obvious. We can represent all of the previous possibilities by writing ψi|ψji|j=δij.
Here, the term in angle brackets represents the integrals appearing in Equations ([e12.1]) and ([e12.2]) in one- and three-dimensional regular space, respectively, and the spinor product appearing in Equation ([e12.3]) in spin-space. The advantage of our new notation is its great generality: that is, it can deal with one-dimensional wavefunctions, three-dimensional wavefunctions, spinors, et cetera.

Expanding a general wavefunction, ψa, in terms of the energy eigenstates, ψi, we obtain

ψa=iciψi.

In one dimension, the expansion coefficients take the form (see Section [seig]) ci=ψiψadx,
whereas in three dimensions we get ci=ψiψadxdydz.
Finally, if ψ is a spinor then we have ci=ψiψa.
We can represent all of the previous possibilities by writing ci=ψi|ψai|a.
The expansion ([e12.7]) thus becomes ψa=iψi|ψaψiii|aψi.
Incidentally, it follows that i|a=a|i.

Finally, if A is a general operator, and the wavefunction ψa is expanded in the manner shown in Equation ([e12.7]), then the expectation value of A is written (see Section [seig])

A=i,jcicjAij.

Here, the Aij are unsurprisingly known as the matrix elements of A. In one dimension, the matrix elements take the form Aij=ψiAψjdx,
whereas in three dimensions we get Aij=ψiAψjdxdydz.
Finally, if ψ is a spinor then we have Aij=ψiAψj.
We can represent all of the previous possibilities by writing Aij=ψi|A|ψji|A|j.
The expansion ([e12.14]) thus becomes

Aa|A|a=i,ja|ii|A|jj|a.

Incidentally, it follows that [see Equation ([e5.48])] i|A|j=j|A|i.
Finally, it is clear from Equation ([e12.20a]) that

i|ii|1,

where the ψi are a complete set of eigenstates, and 1 is the identity operator.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 11.2: Improved Notation is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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