Each operator can be defined in an abstract way by,
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| \begin{equation*} y = A x \qquad , \end{equation*} | (1.32) |
e.g.
\(A\): Rotation of an angle of \(45^o\) around the \(x\)-axis
\(A\): Time developing operator of a quantum mechanical wave function
For using this operator in calculations, we have to define, how this operator acts on each vector \(x\). Taking the component representation of \(x\) and \(y\)
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| \begin{equation*} x = \sum_i x_i |e_i\rangle \qquad \mbox{and} \qquad y = \sum_i y_i |e_i\rangle \end{equation*} | (1.33) |
we find
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| \begin{equation*} y = \sum_i y_i |e_i\rangle = A x = \sum_i x_i A |e_i\rangle \qquad . \end{equation*} | (1.34) |
Multiplication with the Bra-vector leads to
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| \begin{equation*} y_j = \sum_i \langle e_j| y_i |e_i\rangle \sum_i \langle e_j| A |e_i\rangle x_i \qquad . \end{equation*} | (1.35) |
The components \(m_{i,j} = \langle e_j| A |e_i\rangle \) thus define the operator \(A\).
The matrix \(M\) consisting of the components \(m_{j,i}\) is thus equivalent to the operator \(A\).
We call \(M\) a representation of the operator \(A\).
Every linear function can be represented by a matrix.
Example:
Choosing the vector space of square integrable functions,
the set of functions \(e^{ikx}\) serves as an orthonormal basis. Every vector (function) \(f\)
may be represented as components of the functions \(e^{ikx}\). This is the Fourier-Analysis (Fourier-Transformation)
of the function \(f\). The result of an operator \(A\) on a square integrable function is thus
defined, when the components \(m_{i,j} = \langle e_j| A |e_i\rangle \) are known.
All
results, proofed here for matrixes, are valid for linear operators as well.
REMARKS:
The functions \(e^{ikx}\) are not exactly square integrable.
The ”indices” \(i\), \(j\) for plane waves \(e^{ikx}\) are continuous. The sum over \(i\) has therefore to be replaced by an integration over \(k\).
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© J. Carstensen (Quantum Mech.)