Chemical Education Today
Letters Incorrect Mathematical Operators in a Two-Dimensional Quantum Problem In a recent article in this Journal, Ellison (1) has observed that quantum chemistry, even though it is central to an understanding of modern chemistry, “is a difficult subject to teach and one that students have great difficulty grasping”. In order to make the subject interesting to students and to make connection to real physical systems, Ellison has described the quantum mechanics of a particle in a corral, in relation to an STM image of a ring of iron atoms on a copper surface. While I agree with the observation of the author about the difficulties in the teaching–learning process of quantum mechanics, I discuss below some basic problems in the quantum mechanical treatment as presented in that article. Firstly, the Hamiltonian operator used in Ellison’s article (1) for a particle in a 2-dimensional quantum corral is given as −
2
h 2μ
1 ∂ r 2 ∂r
r2
∂ ∂r
2
1 ∂ r 2 ∂ϕ2
+
This form is wrong. It was presumably adapted from the Laplacian operator in 3-dimensional spherical polar coordinates ∇2 =
1 r 2 sin θ
∂ ∂r
r 2 sin θ
∂ ∂r
+
∂ ∂ sin θ ∂θ ∂θ
+
1 ∂2 sin θ ∂ϕ2
in which a constant value of π∙2 was assigned to the azimuthal angle θ. That this approach is wrong can be easily seen by examining the volume element dτ = r2 sinθ drdθdϕ. The unit of dτ in this expression is (length)3, as it should be. Setting θ = π∙2, the purported dτ in the 2-dimension case becomes r2 drdϕ, which also has a unit of (length)3, in contrast to the expected unit of (length)2. The correct Laplacian operator for a 2-dimensional problem in polar coordinates can be derived from its expression in Cartesian coordinates and the relations x = rcosϕ and y = rsinϕ or obtained from expressions given for generalized orthogonal coordinates (2). This expression is given as ∇2 =
∂ 1 ∂ r ∂r r ∂r
+
1 ∂2 r 2 ∂ϕ2
with dτ = rdrdϕ. Consequently, the correct Schrödinger equation for the particle within a corral is
−
2
∂ h 1 ∂ r ∂r 2 μ r ∂r
+
1 ∂2 Ψ = EΨ r 2 ∂ϕ2
with the potential energy V = 0 for 0 ≤ r
