Abstract
\(D \equiv \frac{d}{dx}\), and also \(\hat{x}\) (which multiplies by x), are examples of linear operators \(\hat{O}\): \(\hat{O}(\varPhi +\varPsi )=\hat{O}\varPhi + \hat{O}\varPsi \). Coordinates and momenta of Classical Mechanics wear a hat and become quantum operators, and we shall meet more. The scalar product of \( \hat{Q} |\varPsi \rangle \) with \( |\varPhi \rangle \), which is \(\langle \varPhi |\hat{Q} |\varPsi \rangle , \) may be regarded as the element \(M_{ij}\) of some matrix M with i and j replaced by the indices \(\varPhi \) and \(\varPsi ,\) which are functions \(\in L^{2}\); it is called a matrix element of \(\hat{Q}\); the only real difference with a conventional matrix \(M_{ij}\) is that the indices are often continuous and the matrix most often has an infinity of rows and columns. Werner Heisenberg initially formulated the theory in terms of matrices.
Keywords
- General Uncertainty Principle
- Expectation valueExpectation Value
- Rigid Planar Rotations
- Hyperdifferential Operators
- Classical Canonical Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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- 1.
\(\hat{p}\) fails to be Hermitean on the space P(N) of polynomials of degree N, but this does not matter because all polynomials are outside \(L^{2}\).
- 2.
For a body with density \(\rho (\overrightarrow{x}),\) the moment of inertia relative to the z axis is \(I = \int d^{3}x \rho (\overrightarrow{x})(x^{2}+y^{2})\).
- 3.
This is the orbital angular momentum; the spin has another nature and will be discussed later.
- 4.
If one picks \(\psi =\) eigenstate of \(\hat{A},\) say, then \(\sigma _{A}= 0;\) in this limiting case \(\sigma _{B}\) blows up.
- 5.
If they fail to commute, it can still happen that the r.h.s. of (13.22) vanishes for some \(\psi \).
- 6.
Indeed, \(\langle \hat{A}-\langle \hat{A} \rangle \rangle =0 \) and \(\sigma ^{2}_{\hat{A}-\langle \hat{A}\rangle }=\sigma ^{2}_{\hat{A}}\); moreover, the commutator of \(\hat{A}\) and \(\hat{B}\) is not changed if we subtract the mean values from the operators.
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Cini, M. (2018). Postulate 2. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_13
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DOI: https://doi.org/10.1007/978-3-319-71330-4_13
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