$$\require{cancel}$$

# 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.
