Abstract
As we have seen, the states of quantum mechanics are defined on an (extended) Hilbert space . Changes of these states are caused by operators: This can be, for example, the time evolution of the system itself, or the filtering of certain states out of a general state.
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Notes
- 1.
Further material on operators is found in Appendix I, Vol. 1.
- 2.
The term is not defined in the same way everywhere, and is sometimes rather avoided. The reason for this rejection stems in part from the fact that the name ‘observable’ suggests that without an observer (perhaps even a human), physical quantities cannot become real. We explicitly point out that for us, the term observable does not imply this problem, but is simply a technical term in the above sense.
- 3.
The spin matrices \(s_{i}\) and the Pauli matrices \(\sigma _{i}\) are related by \(s_{i}=\frac{\hbar }{2}\sigma _{i}\).
- 4.
Much in the way that each function can be decomposed into a mirror-symmetric and a point-symmetric part.
- 5.
We note that the term ‘positive operator’ is common but not negative or positive-semidefinite would be more correct. However, one can make the distinction between positive (\(\,\ge 0\)) and strictly positive (\(\,>0\)).
- 6.
One can show that violating the uncertainty principle implies that it is also possible to violate the second law of thermodynamics; see Esther Hänggi & Stephanie Wehner, ‘A violation of the uncertainty principle implies a violation of the second law of thermodynamics’, Nature Communications 4, Article number 1670 (2013), doi:10.1038/ncomms2665.
- 7.
It is not just about the comb, so to speak, but also about the hair that is combed.
- 8.
Quite apart from the literary process, such as with David Foster Wallace in Infinite jest: “The mind says, a box-and-forest-meadows-mind can move with quantum-speed and be anytime anywhere and hear in symphonic sum of the thoughts of the living ... The mind says: It does not really matter whether Gately knows what the term quanta means. By and large, it says there are ghosts ... in a completely different Heisenberg dimension of exchange rates and time courses.”
- 9.
Two subspaces \(\mathcal {H}_{n}\) and \(\mathcal {H}_{m}\) are mutually orthogonal if any vector in \(\mathcal {H}_{n}\) is orthogonal to any vector in \(\mathcal {H}_{m}\).
- 10.
To be exact, there is the second requirement, \(U\alpha \left| \varphi \right\rangle =\alpha U\left| \varphi \right\rangle \). For antiunitary operators \(T\), it holds also that \(TT^{\dag }=T^{\dag }T=1\), but in contrast to the unitary operators, \(T\alpha \left| \varphi \right\rangle =\alpha ^{*}T\left| \varphi \right\rangle \). Anti-unitary operators appear, apart from the complex conjugation, in quantum mechanics only in connection with time reversal (see Chap. 21, Vol. 2). So the equation \(UU^{\dag }=U^{\dag }U=1\) almost always refers to unitary operators.
- 11.
Since it, so to say, impels or propagates the state \(\left| \Psi \right\rangle \) through time, it is also called propagator.
- 12.
Note that \(H\) does not depend on time, which is why we obtain such simple formulations. Propagators for time-dependent Hamiltonians can also be formulated, but this is somewhat more complicated.
- 13.
Here the notation \(p\) has, of course, nothing to do with the momentum, but with \(p\) as \(p\)rojection.
- 14.
To avoid misunderstandings, we repeat the remark that the last equation is an operator equation, i.e. simply two different representations of one operator.
- 15.
We have found it already as an example in an exercise of Chap. 11.
- 16.
We have here a connection to logic (via ‘\(1\hat{=}\) true’ and ‘\(0\hat{=}\) false’). In classical physics, such a statement (the quantity \(A\) has the value \(a_{k}\)) is either true or false; in quantum mechanics or quantum logic, the situation may be more complex.
- 17.
More on this topic in Chap. 27, Vol. 2.
- 18.
Some remarks on terms that arise in connection with ‘measurements’ are given in Appendix S, Vol. 1.
- 19.
We use here the fact that all states \(e^{i\alpha }\left| \Psi \right\rangle \) are physically equivalent for arbitrary real \(\alpha \). See also Chap. 14.
- 20.
The proof is found in Appendix I, Vol. 1.
- 21.
See also Appendix Q, Vol. 1, ‘Schrödinger picture, Heisenberg picture and interaction picture’.
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Pade, J. (2014). Operators. In: Quantum Mechanics for Pedestrians 1: Fundamentals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00798-4_13
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