Abstract
This chapter deals with the solutions of the SEq for two simple but important one-dimensional systems. First, we consider the infinite potential well as a simple model of a bounded system, then force-free unlimited motion as a simple model of an unbounded system. Here, ‘bounded motion’ means basically that the system is confined to a finite region, in contrast to unlimited motion.
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Notes
- 1.
See also the exercises for Chap. 3.
- 2.
Indeed, the quantum-mechanical ping-pong ball is quite a peculiar ball, namely an object described by a standing wave.
- 3.
A conclusive argument for these boundary conditions is given in Chap. 15, Vol. 2. For now, one might think (in an intuitive analogy to the wavefunction) of a rope which is clamped at both ends (although the question remains open as to what a rope has to do with this quantum-mechanical situation). Alternatively, one might consider a continuity requirement for the wavefunction at the walls to be plausible.
- 4.
We have \(k\ne 0\), since for \(k=0\), only the trivial solution is obtained.
- 5.
For \(A=0\), we would obtain the trivial solution.
- 6.
We recall that \(\sin x=\frac{e^{ix}-e^{-ix}}{2i}\).
- 7.
This special choice will be justified below.
- 8.
We recall that we are using a shorthand notation for the summation \(\sum _{n}\). The range of values of n must be clear from the context. Here, it would be \(n=1,\ldots .\infty \) or \( {\frac{x^{2}}{2}}\sum _{n=1}^{\infty }\).
- 9.
- 10.
See the exercises for this chapter.
- 11.
- 12.
Square-integrable (or quadratically integrable) over the interval \(\left[ a, b \right] \) are those functions f(x) for which \(\int \nolimits _{a}^{b} \left| f\left( x\right) \right| ^{2}\mathrm{d}x<\infty \) holds. The short notation reads \(f(x)\in L^{2}\left[ a, b\right] \). For \(a=-\infty \) and \( b=\infty \), the notation \(L^{2}\left[ \mathbb {R}\right] \) is common.
- 13.
Hence the often undifferentiated use of the terms eigenfunction and eigenvector.
- 14.
This use of the term vector has of course nothing to do with arrows or with the properties of transformation behavior (polar and axial vectors).
- 15.
At first sight, it may seem strange that one can compute infinitely many complex numbers \(c_{n}\) from one initial condition \(\Psi (x, 0)\). But in fact, with \(\Psi (x, 0)\) we have uncountably many values.
- 16.
“Physicists usually have a nonchalant attitude when the number of dimensions is extended to infinity. Optimism is the rule, and every infinite sequence is presumed to be convergent, unless proven guilty.” A. Peres, Quantum Theory, p. 79.
- 17.
This equation is very similar to the heat equation \(\overset{.}{f}=\lambda { \nabla }^2 f\)—apart from i in the SEq. As is well known, this ‘small difference’ is the mother of all worlds.
- 18.
Integral and not sum, because k is a continuous ‘index’. The integration variable k may of course also assume negative values here.
- 19.
Some basics on Fourier transformation can be found in Appendix H, Vol. 1.
- 20.
The quite specific form of the coefficients is due to the normalization.
- 21.
Occasionally, this width is referred to as the halfwidth, although the function has dropped not to 1 / 2, but to 1 / e of its maximum value.
- 22.
A slightly more detailed analysis can be found in Appendix D, Vol. 2 (wave packets).
- 23.
In fact, it may also be the case that discrete and continuous spectra overlap, or that discrete levels are embedded in the continuum, as we shall see using the example of the helium atom in Chap. 23, Vol. 2.
- 24.
Moreover, it follows from (5.49) for instance that
$$\begin{aligned} B=-\frac{\varphi _{1}(kL)}{\varphi _{2}(kL)}A \end{aligned}$$and so
$$\begin{aligned} \Phi (x)=A\left[ \varphi _{1}(kx)-\frac{\varphi _{1}(kL)}{\varphi _{2}(kL)} \varphi _{2}(kx)\right] \!, \end{aligned}$$leaving only one remaining free constant (and one must remain because of the linearity of the SEq).
- 25.
When a writer like Terry Pratchett couples the idea of rolled-up dimensions with other physical paradigms, it reads like this: “..and people stopped patiently building their little houses of rational sticks in the chaos of the universe and started getting interested in the chaos itself—partly because it was a lot easier to be an expert on chaos, but mostly because it made really good patterns that you could put on a T-shirt.
And instead of getting on with proper science, scientists suddenly went around saying how impossible it was to know anything, and that there wasn’t really anything you could call reality to know anything about, and how all this was tremendously exciting, and incidentally did you know there were possibly all these little universes all over the place but no-one can see them because they are all curved in on themselves? Incidentally, don’t you think this is a rather good T-shirt?” Terry Pratchett, in Witches Abroad, A Discworld Novel.
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Pade, J. (2018). Two Simple Solutions of the Schrödinger Equation. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_5
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