Abstract
One-dimensional models and models with piecewise constant potentials have been used as simple model systems for quantum behavior ever since the inception of Schrödinger’s equation. These models vary in their levels of sophistication, but their generic strength is the clear demonstration of important general quantum effects and effects of dimensionality of a quantum system at very little expense in terms of effort or computation. Simple model systems are therefore more than just pedagogical tools for teaching quantum mechanics. They also serve as work horses for the modeling of important quantum effects in nanoscience and technology, see e.g. [3, 18].
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Notes
- 1.
The normalization condition (1.20) implies that the function ψ(x, E) does not exist in the sense of classical Fourier theory. We will therefore see in Section 5.2 that ψ(x, E) is rather a series of δ-functions of the energy. This difficulty is usually avoided by using an exponential ansatz \(\psi (x,t) = \psi (x,E)\exp \!\left (-\mathrm{i}Et/\hslash \right )\) instead of a full Fourier transformation. However, if one accepts the δ-function and corresponding extensions of classical Fourier theory, the transition to the time-independent Schrödinger equation through a formal Fourier transformation to the energy axis is logically more satisfactory.
- 2.
E. Schrödinger, Annalen Phys. 384, 361 (1926). Schrödinger found the time-indepedent equation first and published the time-dependent equation (1.2) five months later.
- 3.
The time-dependent Schrödinger equation permits discontinuous wave functions ψ(x, t) even for smooth potentials, because there can be a trade-off between the derivative terms, see e.g. Problem 10.
- 4.
Magnetic tunnel junctions provide yet another beautiful example of the interplay of two quantum effects – tunneling and exchange interactions. Exchange interactions will be discussed in Chapter 17.
- 5.
The propagator is commonly denoted as K(x, t). However, we prefer the notation U(x, t) because the propagator is nothing but the x representation of the time evolution operator U(t) introduced in Chapter 13.
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Dick, R. (2012). Simple Model Systems. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8077-9_3
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DOI: https://doi.org/10.1007/978-1-4419-8077-9_3
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