Dimensional Effects in Low-Dimensional Systems
Surfaces, interfaces, thin films, and quantum wires provide abundant examples of quasi two-dimensional or one-dimensional systems in science and technology. Quantum mechanics in low dimensions has become an important tool for modeling properties of these
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Dimensional Effects in Low-Dimensional Systems
Surfaces, interfaces, thin films, and quantum wires provide abundant examples of quasi two-dimensional or one-dimensional systems in science and technology. Quantum mechanics in low dimensions has become an important tool for modeling properties of these systems. Here we wish to go beyond the simple low-dimensional potential models of Chap. 3 and discuss in particular implications of the dependence of energy-dependent Green’s functions on the number d of spatial dimensions. However, if it is true that the behavior of electrons in certain systems and parameter ranges can be described by low-dimensional quantum mechanics, then there must also exist ranges of parameters for quasi low-dimensional systems where the behavior of electrons exhibits inter-dimensional behavior in the sense that there must exist continuous interpolations e.g. between two-dimensional and three-dimensional behavior. We will see that inter-dimensional (or “dimensionally hybrid”) Green’s functions provide a possible avenue to the identification and discussion of interdimensional behavior in physical systems.
21.1 Quantum Mechanics in d Dimensions Suppose an electron is strictly confined to a two-dimensional quantum well. Unless the material in the quantum well has special dielectric properties, that electron would still “know” that it exists in three spatial dimensions, because it feels the 1/r Coulomb interaction with other charged particles, and this 1/r distance law is characteristic for three dimensions. If the electron would not just be confined to two dimensions, but exist in a genuine two-dimensional world, it would experience a logarithmic distance law for the Coulomb potential. The reason for the 1/r Coulomb law in three dimensions is that the solution of the equation G(r) = − δ(x) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_21
(21.1) 563
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21 Dimensional Effects in Low-Dimensional Systems
in three dimensions is given by G(r) =
1 , 4π r
(21.2)
but in general the solution of Eq. (21.1) depends on the number d of spatial dimensions. Appendix J explains in detail the derivation of the d-dimensional version of G(r) with the result
Gd (r) =
⎧ ⎪ ⎪ ⎪ ⎨
(a − r)/2,
d = 1,
− (2π )−1 ln(r/a), ⎪ ⎪ ⎪ ⎩ d−2 4√π d r d−2 −1 , 2
d = 2,
(21.3)
d ≥ 3.
The most direct application of these results are electrostatic potentials. Equation (21.1) implies that the electrostatic potential of a point charge q in d dimensions is given by d (r) =
q Gd (r). 0
(21.4)
However, from the point of view of non-relativistic quantum mechanics, the Green’s functions (21.3) are the special zero energy values of the energy-dependent free Schrödinger Green’s functions, Gd (r) = Gd (x, E = 0). These energy dependent Green’s functions satisfy Gd (x, E) +
2m h¯ 2
EGd (x, E) = − δ(x).
(21.5)
We have solved this condition in
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