Abstract
We have to go through a few more formalities before we can resume our discussion of quantum effects in physics. In particular, we need to address minimal uncertainties of observables in quantum mechanics, and we have to discuss transformation and solution properties of differential operators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
W. Heisenberg, Z. Phys. 43, 172 (1927).
- 2.
Frequency-time Fourier transformation for Green’s functions and potentials, which generically will depend on two arguments, will often appear asymmetric due to translation invariance \(G(t,t') = G(t - t') \leftrightarrow G(\omega,\omega ') = G(\omega )\delta (\omega -\omega ')\):
$$\displaystyle{ G(t) = \frac{1} {2\pi }\int \!d\omega \,G(\omega )\exp \!\left (-\mathrm{i}\omega t\right ),\quad G(\omega ) =\int \! dt\,G(t)\exp \!\left (\mathrm{i}\omega t\right ). }$$(5.11) - 3.
Free states with initial conditions \(\langle \boldsymbol{x}\vert \psi \rangle \equiv \langle \boldsymbol{ x}\vert \psi (t = 0)\rangle\) yield frequency representations \(\langle \boldsymbol{x}\vert \psi (\omega )\rangle\) in terms of convolutions of \(\langle \boldsymbol{x}\vert \psi \rangle\) with Bessel functions. This is explained in Appendix J, especially equations (J.32)and (J.34–J.36). However, if you began only recently to learn quantum mechanics, don’t let yourself become distracted by the technicalities of Appendix J. Save it for later.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dick, R. (2016). Formal Developments. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-25675-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25674-0
Online ISBN: 978-3-319-25675-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)