Abstract
So far, we have excluded continuous spectra from the discussion, e.g. by placing our quantum object between infinitely high potential walls, thus discretizing the energy spectrum. The fact that we adopted this limitation had less to do with physical reasons, but rather almost exclusively with mathematical ones.
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Notes
- 1.
Another possibility would be the introduction of periodic boundary conditions.
- 2.
Below the Planck scale (\(\sim \)10\(^{-35}\) m, \(\sim \)10\(^{-44}\) s), neither space nor time may exist, so that ultimately these variables would become ‘grainy’ or discrete. (On this scale, space is thought to be something like a foam bubbling with tiny black holes, continuously popping in and out of existence.) There are attempts to determine whether space is truly grainy, but results have so far been inconclusive. Experimentally, these orders of magnitude are still very far from being directly accessible (if they ever will be); the currently highest-energy accelerator, the LHC at CERN in Geneva, attains a spatial resolution of ‘only’ \(\sim \)10\(^{-19}\) m. Recently, however, indirect methods were proposed; see Jakob D. Bekenstein, ‘Is a tabletop search for Planck scale signals feasible?’, http://arxiv.org/abs/1211.3816 (2012); or Igor Pikovski et al., ‘Probing Planck-scale physics with quantum optics’, Nature Physics 8, 393–397 (2012). For a recent paper see e.g. V. Faraoni, ‘Three new roads to the Planck scale’, American Journal of Physics 85, 865 (2017); https://doi.org/10.1119/1.4994804
- 3.
The factor \(\frac{1}{\sqrt{2\pi }}\) is due to the normalization of the function, see below.
- 4.
It is clear that the delta function cannot be a function. That it is still denoted as one may be due to the often rather nonchalant or easygoing approach of physicists to mathematics. More is given on the delta function in Appendix H, Vol. 1.
- 5.
These states are also called Dirac states.
- 6.
The strict mathematical theory of continuous spectra is somewhat elaborate (keywords e.g. rigged Hilbert space or Gel’fand triple). We content ourselves here with a less rigorous and more heuristic approach.
- 7.
The electron is a point object, but not its wavefunction—that would be in contradiction to the uncertainty principle.
- 8.
Discretizations of continuous variables are used also in other areas, e.g. in lattice gauge theories or in the numerical treatment of differential equations. Moreover, a discrete space is taken as a basis for an alternative derivation/motivation of the SEq (hopping equation; see Appendix J, Vol. 1).
- 9.
Whoever wishes may keep in mind \( \frac{1}{\sqrt{2\pi }}e^{i\lambda x}\) instead of \(\left| \varphi _{\lambda }\right\rangle \equiv \left| \lambda \right\rangle \) — this is not quite correct, but may be helpful here and is preferable to the auxiliary notion of a column vector.
- 10.
Note that we cover the axis completely with non-overlapping intervals \( \Delta \lambda \).
- 11.
A remark on the summation index: the range of values of \(\lambda \) runs through all integral multiples of the grid size \(\Delta \lambda \).
- 12.
This is nothing more than the transition from a sum to an integral, well-known from school mathematics. One lets the length of the subdivisions tend to zero, so that the upper sum and lower sum approach and converge to the integral in the limit, i.e. for infinitesimal interval length. This process is reflected in the integral sign \(\int \)—it is simply a stylized ‘S’, for ‘sum’.
- 13.
We repeat the remark that \(e^{ikx}\) is actually an oscillation in space. But one always refers in this context to a wave, because one keeps the time-dependent factor \(e^{i\omega t}\) in mind, so to speak.
- 14.
We recall that the increment of the sum is 1 (so we have \(\Delta n=1\)). With this, one can emphasize the formal similarity between sums and integrals even more, e.g. in the form \(\sum \left| \varphi _{n}\right\rangle \left\langle \varphi _{n}\right| \Delta n=1\).
- 15.
There are other notations; e.g. Schwabl uses the symbol \(\mathcal {S}\).
- 16.
It is clear that this is an idealized assumption which is not compatible with the uncertainty principle. But we can proceed on this assumption in terms of the above considerations concerning the eigendifferential.
- 17.
We repeat this derivation briefly by writing the first line of (12.28) with bra-kets (remember: \(\int f^{*}g \mathrm{d}x=\left\langle f\right. \left| g\right\rangle \)):
$$\begin{aligned} \int \Psi ^{*}\left( x\right) \Psi \left( x\right) dx&=\left\langle \Psi \right. \left| \Psi \right\rangle =\sum _{n}\left\langle \varphi _{n}\right. \left| \Psi \right\rangle \left\langle \Psi \right. \left| \varphi _{n}\right\rangle \nonumber \\&=\sum _{n}\left\langle \Psi \right. \left| \varphi _{n}\right\rangle \left\langle \varphi _{n}\right. \left| \Psi \right\rangle =\left\langle \Psi \right| \left( \sum _{n}\left| \varphi _{n}\right\rangle \left\langle \varphi _{n}\right| \right) \left| \Psi \right\rangle . \end{aligned}$$ - 18.
We note again that this is an idealized formulation.
- 19.
Again, the above statement on oscillations and waves applies.
- 20.
Because these are two representations of the same ket \(\left| \Psi \right\rangle \), sometimes the same symbol is used for both representations, i.e. \(\Psi (x)\) and \(\Psi (k)\), although these two functions are not the same (as mapping, that is, in the sense that one does not obtain \(\Psi (k)\) by simply replacing x by k in \(\Psi (x)\)). What is precisely meant has to be inferred from the context. To avoid confusion, we use the notation \( \hat{\Psi }(k)\).
- 21.
For an introduction to Fourier transforms, see Appendix H, Vol. 1.
- 22.
Quasi-local operators are defined via the derivative of the delta function:
$$\begin{aligned} \begin{array}{ll} A\left( x,y\right) =a\left( x\right) \delta \left( x-y\right) &{} \text {local operator}\\ B\left( x, y\right) =b\left( x\right) \delta ^{\prime }\left( x-y\right) &{} \text {quasi-local operator} \end{array} \end{aligned}$$ - 23.
“Caminante no hay camino, se hace camino al andar...” Antonio Machado, Spanish poet.
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Pade, J. (2018). Continuous Spectra. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_12
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