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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