11.1: 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,

\[H\,\psi_i = E_i\,\psi_i. \nonumber \]

Now, we expect the \(\psi_i\) to be orthonormal. (See Section [seig].) In one dimension, this implies that

\[ \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx = \delta_{ij}. \label{e12.1} \]

In three dimensions (see Chapter [sthree]), the previous expression generalizes to

\[ \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx\,dy\,dz = \delta_{ij}. \label{e12.2} \]

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

\[ \psi_i^\daggerger\,\psi_j = \delta_{ij}. \label{e12.3} \]

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

\[\langle \psi_i|\psi_j\rangle \equiv \langle i|j\rangle = \delta_{ij}. \nonumber \]

Here, the term in angle brackets represents the integrals appearing in Equations \ref{e12.1} and \ref{e12.2} in one- and three-dimensional regular space, respectively, and the spinor product appearing in Equation \ref{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

\[ \psi_a = \sum_i c_i\,\psi_i. \label{e12.7} \]

In one dimension, the expansion coefficients take the form (see Section [seig])

\[c_i = \int_{-\infty}^\infty\psi_i^\ast\,\psi_a\,dx, \nonumber \]

whereas in three dimensions we get

\[c_i = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast\,\psi_a\,dx\,dy\,dz. \nonumber \]

Finally, if \(\psi\) is a spinor then we have

\[c_i = \psi_i^\daggerger\,\psi_a. \nonumber \]

We can represent all of the previous possibilities by writing

\[c_i =\langle\psi_i|\psi_a\rangle\equiv \langle i|a\rangle. \nonumber \]

The expansion \ref{e12.7} thus becomes

\[ \psi_a = \sum_i\langle\psi_i|\psi_a\rangle\,\psi_i\equiv \sum_i \langle i|a\rangle\,\psi_i. \label{e12.13a} \]

Incidentally, it follows that

\[\langle i|a\rangle^\ast=\langle a| i\rangle. \nonumber \]

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

\[ \langle A\rangle = \sum_{i,j} c_i^\ast\,c_j\,A_{ij}. \label{e12.14} \]

Here, the \(A_{ij}\) are unsurprisingly known as the matrix elements of \(A\). In one dimension, the matrix elements take the form

\[A_{ij} = \int_{-\infty}^\infty\psi_i^\ast\,A\,\psi_j\,dx, \nonumber \]

whereas in three dimensions we get

\[A_{ij} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast\,A\,\psi_j\,dx\,dy\,dz. \nonumber \]

Finally, if \(\psi\) is a spinor then we have

\[A_{ij}=\psi_i^\daggerger\,A\,\psi_j. \nonumber \]

We can represent all of the previous possibilities by writing

\[A_{ij}=\langle \psi_i|A|\psi_j\rangle \equiv \langle i|A|j\rangle. \nonumber \]

The expansion \ref{e12.14} thus becomes

\[ \langle A\rangle \equiv\langle a|A|a\rangle= \sum_{i,j} \langle a|i\rangle \langle i|A|j\rangle \langle j|a\rangle. \label{e12.20a} \]

Incidentally, it follows that [see Equation \ref{e5.48}]

\[\langle i|A|j\rangle^\ast=\langle j| A^\daggerger|i\rangle. \nonumber \]

Finally, it is clear from Equation \ref{e12.20a} that

\[ \sum_{i} |i\rangle \langle i| \equiv 1, \label{e12.20} \]

where the \(\psi_i\) are a complete set of eigenstates, and 1 is the identity operator.

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