11.2: Improved Notation

Before commencing our investigation, it is helpful to introduce some improved notation. Let the \(\psi_i\) be a complete set of eigenstates of the Hamiltonian, \(H\), corresponding to the eigenvalues \(E_i\): that is, \[H\,\psi_i = E_i\,\psi_i.\] Now, we expect the \(\psi_i\) to be orthonormal. (See Section [seig].) In one dimension, this implies that \[\label{e12.1} \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx = \delta_{ij}.\] In three dimensions (see Chapter [sthree]), the previous expression generalizes to

\[\label{e12.2} \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx\,dy\,dz = \delta_{ij}.\] Finally, if the \(\psi_i\) are spinors (see Chapter [sspin]) then we have

\[\label{e12.3} \psi_i^\dagger\,\psi_j = \delta_{ij}.\] The generalization to the case where \(\psi\) is a product of a regular wavefunction and a spinor is fairly obvious. We can represent all of the previous possibilities by writing \[\langle \psi_i|\psi_j\rangle \equiv \langle i|j\rangle = \delta_{ij}.\] 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, \(\psi_a\), in terms of the energy eigenstates, \(\psi_i\), we obtain

\[\label{e12.7} \psi_a = \sum_i c_i\,\psi_i.\] In one dimension, the expansion coefficients take the form (see Section [seig]) \[c_i = \int_{-\infty}^\infty\psi_i^\ast\,\psi_a\,dx,\] whereas in three dimensions we get \[c_i = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast\,\psi_a\,dx\,dy\,dz.\] Finally, if \(\psi\) is a spinor then we have \[c_i = \psi_i^\dagger\,\psi_a.\] We can represent all of the previous possibilities by writing \[c_i =\langle\psi_i|\psi_a\rangle\equiv \langle i|a\rangle.\] The expansion ([e12.7]) thus becomes \[\label{e12.13a} \psi_a = \sum_i\langle\psi_i|\psi_a\rangle\,\psi_i\equiv \sum_i \langle i|a\rangle\,\psi_i.\] Incidentally, it follows that \[\langle i|a\rangle^\ast=\langle a| i\rangle.\]

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

\[\label{e12.14} \langle A\rangle = \sum_{i,j} c_i^\ast\,c_j\,A_{ij}.\] Here, the \(A_{ij}\) are unsurprisingly known as the matrix elements of \(A\). In one dimension, the matrix elements take the form \[A_{ij} = \int_{-\infty}^\infty\psi_i^\ast\,A\,\psi_j\,dx,\] whereas in three dimensions we get \[A_{ij} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast\,A\,\psi_j\,dx\,dy\,dz.\] Finally, if \(\psi\) is a spinor then we have \[A_{ij}=\psi_i^\dagger\,A\,\psi_j.\] We can represent all of the previous possibilities by writing \[A_{ij}=\langle \psi_i|A|\psi_j\rangle \equiv \langle i|A|j\rangle.\] The expansion ([e12.14]) thus becomes

\[\label{e12.20a} \langle A\rangle \equiv\langle a|A|a\rangle= \sum_{i,j} \langle a|i\rangle \langle i|A|j\rangle \langle j|a\rangle.\] Incidentally, it follows that [see Equation ([e5.48])] \[\langle i|A|j\rangle^\ast=\langle j| A^\dagger|i\rangle.\] Finally, it is clear from Equation ([e12.20a]) that

\[\label{e12.20} \sum_{i} |i\rangle \langle i| \equiv 1,\] where the \(\psi_i\) 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)

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