11.2: Improved Notation
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 ∫∞−∞ψ∗iψjdx=δij.
In three dimensions (see Chapter [sthree]), the previous expression generalizes to
Finally, if the ψi are spinors (see Chapter [sspin]) then we have
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|ψj⟩≡⟨i|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
In one dimension, the expansion coefficients take the form (see Section [seig]) ci=∫∞−∞ψ∗iψadx,
whereas in three dimensions we get ci=∫∞−∞∫∞−∞∫∞−∞ψ∗iψadxdydz.
Finally, if ψ is a spinor then we have ci=ψ†iψa.
We can represent all of the previous possibilities by writing ci=⟨ψi|ψa⟩≡⟨i|a⟩.
The expansion ([e12.7]) thus becomes ψa=∑i⟨ψi|ψa⟩ψi≡∑i⟨i|a⟩ψi.
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])
Here, the Aij are unsurprisingly known as the matrix elements of A. In one dimension, the matrix elements take the form Aij=∫∞−∞ψ∗iAψjdx,
whereas in three dimensions we get Aij=∫∞−∞∫∞−∞∫∞−∞ψ∗iAψjdxdydz.
Finally, if ψ is a spinor then we have Aij=ψ†iAψj.
We can represent all of the previous possibilities by writing Aij=⟨ψi|A|ψj⟩≡⟨i|A|j⟩.
The expansion ([e12.14]) thus becomes
⟨A⟩≡⟨a|A|a⟩=∑i,j⟨a|i⟩⟨i|A|j⟩⟨j|a⟩.
Incidentally, it follows that [see Equation ([e5.48])] ⟨i|A|j⟩∗=⟨j|A†|i⟩.
Finally, it is clear from Equation ([e12.20a]) that
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)