Abstract
So far only properties such as the structure and spectra of quantum physical systems have been discussed, properties for which time development is irrelevant and can thus be ignored. In this chapter properties will be discussed that can be understood only by taking into account time development.
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Notes
- 1.
A pre-Hilbert space is a linear, scalar-product space in which the convergence of infinite sequences ϕ 1, ϕ 2, …ϕ n, ϕ n+1 … is not discussed. Often the Schwartz space, a space in which convergence is defined by an infinite number of norms, is used for Φ.
- 2.
The interaction or Dirac picture is a third commonly used picture in which the Hamiltonian H is split into two parts, H = H 0 + H 1. The observable A evolves in time with H 0, \(A^D(t)=e^{iH_0\,t/\hbar }A^D(0)e^{-iH_0\,t/\hbar }\), and the state ϕ evolves in time with H 1, \(\phi ^D(t)=e^{-iH_1\,t/\hbar }\phi ^D(0)e^{iH_1\,t/\hbar }\). As must be the case, predictions are the same in each picture: \(\mathcal {P}_{\rho ^D(t)}(A^D(t))= \mathcal {P}_{\rho (t)}(A)=\mathcal {P}_\rho (A(t))\).
- 3.
The experimental quantity N n(t)∕N necessarily changes in discrete steps while the theoretical quantity \(\mathcal {P}_{\rho (t)}(A)=\mathcal {P}_\rho (A(t))\) is a continuous function of t. The comparison between experiment and theory is in principle approximate, but as the number of events N becomes larger, the comparison between experiment and theory becomes more accurate.
- 4.
U(t) possesses the following properties:
$$\displaystyle \begin{aligned} &U(0) =I \, , \\ &U^{-1} (t) = U (-t) = U^\dagger (t) \, \\ &U(t_1 + t_2 ) = U(t_1) Ut_2) = U(t_2) U(t_1) \, ; \;\;-\infty < t_1\,, t_2 < +\infty \, . \end{aligned} $$U(t) is a continuous operator function of the parameter t and is called a one-parameter group of unitary operators.
- 5.
The operators |j, j i〉〈j, j i| represent the observables that measure the probability for the angular momentum values j, j i. That is, |〈ϕ(t)|j, j i〉|2 is the probability for the values j, j i in the state ϕ(t).
- 6.
The Larmor frequency ω L for a proton (g = 2.79, e = 1.60 × 10−19 C, m = 1.67 × 10−27 kg) in a typical magnetic field B = 1.0 T is ω L = 1.34 × 108 Hz, which is in the radio frequency range.
- 7.
L. D. Landau and E. M. Lifshitz, Statistical Physics 5 V.1 (3 ed.) Pergamon Press, Oxford, 1980).
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Bohm, A., Kielanowski, P., Mainland, G.B. (2019). Time Evolution of Quantum Systems. In: Quantum Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1760-9_5
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DOI: https://doi.org/10.1007/978-94-024-1760-9_5
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