A linear operator \(A\) is called Hermitian, if:
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| \begin{equation*} \langle x|Ay\rangle = \langle Ax|y\rangle = \langle x|A|y\rangle \qquad \mbox{(Bra ..... Ket)} \end{equation*} | (1.11) |
The third notation is used to point out the symmetry of the first equality.
Examples:
Hermitian matrices:
Applying the above definition to a matrix, we find:
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| \begin{equation*} |Ab\rangle _i = \sum_j A_{i,j}b_j \qquad \mbox{and} \qquad |Aa\rangle _i = \sum_j A_{i,j}a_j \end{equation*} | (1.12) |
With
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| \begin{equation*} \langle Aa|_i = \sum_j A_{i,j}^*a_j^* \end{equation*} | (1.13) |
follows
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| \begin{equation*} \langle a|Ab\rangle = \sum_{i,j} a_i^*A_{i,j}b_j \qquad \mbox{and} \qquad \langle Aa|b\rangle = \sum_{i,j} a_j^*A_{i,j}^*b_j \label{sym_A} \end{equation*} | (1.14) |
For Hermitian matrices we find according to Eq. (1.14):
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| \begin{equation*} A_{i,j} =A_{j,i}^* \end{equation*} | (1.15) |
Multiplication with x
In this case
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| \begin{equation*} \langle f|Ag\rangle = \langle Af|g\rangle = \langle f|A|g\rangle \end{equation*} | (1.16) |
is written as
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| \begin{equation*} \int_{-\infty}^{+\infty} f^*(x)xg(x) dx = \int_{-\infty}^{+\infty} x^*f^*(x)g(x) dx \end{equation*} | (1.17) |
Since \(x\) is a real value, the multiplication with \(x\) is a Hermitian operator.
The differential operator
As proofed by the following calculation, the differential operator is not Hermitian. We thus choose the
correct operator
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| \begin{equation*} Af= i \frac{d}{dx}f \qquad \mbox{$i$: imaginary unit} \end{equation*} | (1.18) |
We find:
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© J. Carstensen (Quantum Mech.)