The uncertainty, \(\Delta A)_\psi\), in a dynamical variable, \(X\) in a state, \(\psi\), is defined by\begin{equation}(\Delta X)^2_\psi = \langle (\hat{X} -\overline{X})^2 \rangle_\psi.\end{equation}Using this definition we derive an uncertainty relation between two non commuting dynamical variables,.
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The question of simultaneous measurement of two dynamical variables is analysed. Starting from the postulates it is argued that two variables can be measured simultaneously if and only of they commute. This result generalizes to simultaneous measurement of several dynamical variables.
Starting with the postulates of quantum mechanics, it shown that the average value of a dynamca variable is given by\begin{equation} \langle A \rangle_\psi = \frac{\langle\psi|\hat{A}|\psi\rangle}{\langle{\psi}|{\psi}\rangle}\end{equation}
We list the postulates of the Hilbert space formulation of quantum mechanics. These are
The third postulate of quantum mechanics is discussed in detail.
Many concepts of classical mechanics have to be revised or given up completely. Also in quantum mechanics many new concepts, such as quantization of dynamical variables, appear. In order to understand the structure of quantum mechanics and its applications well, it si recommended that teaching and learning of quantum mechanics should begin with the postulates.
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