Let be:
Operator | Eigenvalue | Eigenvector |
\(A\) | \(a_k\) | \(f_k\) (orthonormal) |
The representation of a general quantum mechanical state in the space of the above defined Eigenvectors is:
|
| \begin{equation*} |\psi\rangle = \sum c_k |f_k\rangle \end{equation*} | (2.1) |
For the length we find:
|
| \begin{equation*} \langle \psi|\psi\rangle = \sum c_l \langle f_l| c_k |f_k\rangle = \sum c_{k}^{*} c_k \end{equation*} | (2.2) |
Since \(c_{k}^{*} c_k \) is the fraction of the state \(k\) to the whole system, for a system with one particle we must find:
|
| \begin{equation*} \langle \psi|\psi\rangle = 1 \end{equation*} | (2.3) |
According to axiom 4 we get:
|
| \begin{equation*} \langle a\rangle = \sum a_k c_k^* c_k \end{equation*} | (2.4) |
Easily we find:
|
| \begin{equation*} \langle \psi|A|\psi\rangle = \sum a_k c_l^* \langle f_l| c_k |f_k\rangle = \sum a_k c_k^* c_k = \langle a\rangle \label{avg_def} \end{equation*} | (2.5) |
This is the abstract representation of an expectation value, since it is independent of the choice of the orthonormal basis. If \(\psi\) is not orthonormal, Eq. (2.5) has to be replaced by
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| \begin{equation*} \frac{\langle \psi|A|\psi\rangle }{\langle \psi|\psi\rangle } = \langle a\rangle \end{equation*} | (2.6) |
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© J. Carstensen (Quantum Mech.)