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Section 8.1 Kronecker Delta
Definition 8.1.1 . The Kronecker Delta.
The Kronecker Delta is a simple switch that compares two values \(i\) and \(j\) in some discrete set. The switch turns on when \(i\) is the same as \(j\) and off when they are different, i.e.,
\begin{equation}
\delta_{ij}
= \begin{cases}0\quad \amp i\not= j\\
1\quad\amp i=j
\end{cases}\tag{8.1.1}
\end{equation}
Example 8.1.2 . Using the Kronecker Delta to Simplify Related Equations.
The most common place that the Kronecker delta appears is as a shorthand to write several related equations as a single equation instead of describing the different cases in words. For example, in the quantum infinite square well (ISW), we use several related integrals to show that the energy eigenstates are orthogonal.
\begin{equation}
\int_{0}^{L} \sin\left(\frac{n\pi x}{L}\right)
\sin\left(\frac{n'\pi x}{L}\right)\, dx
= \begin{cases}0\quad n\not= n'\\
\frac{L}{2}\quad n=n'
\end{cases}\tag{8.1.2}
\end{equation}
We use the Kronecker delta to write these two equations as a a single equation
\begin{equation}
\int_{0}^{L} \sin\left(\frac{n\pi x}{L}\right)
\sin\left(\frac{n'\pi x}{L}\right)\, dx
= \delta_{n,n'}\tag{8.1.3}
\end{equation}
Example 8.1.3 . The Kronecker Delta in a Sum.
The most useful place that the Kronecker delta appears is inside a sum. In this case, it can be used to pull a single term out of a sum. For example:
\begin{align}
\sum_{i=1}^{\infty} a_i\, \delta_{i3}
\amp = a_1 \cancelto{0}{\delta_{13}}
+ a_2 \cancelto{0}{\delta_{23}} +
a_3 \cancelto{1}{\delta_{33}} +\dots\tag{8.1.4}\\
\amp =a_3\tag{8.1.5}
\end{align}
In this example the factor of \(\delta_{i3}\) is zero in every term except the one when \(i=3\text{,}\) so all of those terms are zero and do not contribute to the sum. When \(i=3\text{,}\) the factor of \(\delta_{i3}\) is just one, so that is the only term that contributes.
Example 8.1.4 . Trace of a Matrix with Kronecker Deltas.
If you want to add up all of the elements \(a_{ij}\) of a matrix \(A\) (where, by convention, the first index \(i\) labels the rows and the second index \(j\) labels the columns), this requires a double sum, one for each index.
\begin{equation}
\sum_{i=1}^{3} \sum_{i=j}^{3}a_{ij}\tag{8.1.6}
\end{equation}
If, instead, you want to find the trace of the matrix \(\mathrm{tr} A\text{,}\) then you only want to add up the diagonal elements. A Kronecker delta in the double sum will ensure that you only take the elements for which the row and column labels are the same.
\begin{align}
\mathrm{tr} A\amp\doteq
\sum_{i=1}^{3} \sum_{i=j}^{3}
a_{ij}\delta_{ij}\tag{8.1.7}\\
\amp = \sum_{i=1}^{3} a_{ii}\tag{8.1.8}
\end{align}
Prerequisites 8.1.5 . Try it yourself: The Identity Matrix.
Write the elements of the \(3\times 3\) identity matrix in terms of the Kronecker delta.
Answer .
\begin{equation}
I\doteq \begin{pmatrix}
\delta_{11}\amp \delta_{12}\amp \delta_{13}\\
\delta_{21}\amp \delta_{22}\amp \delta_{23}\\
\delta_{31}\amp \delta_{32}\amp \delta_{33}
\end{pmatrix}\tag{8.1.9}
\end{equation}
