Coordinate Transformations

A coordinate transformation can concentrate grid points near a singularity, boundary layer, or localized state. It also changes derivatives, integration weights, and the matrix adjoint. A mapped grid is reliable only when all three changes are handled together; substituting a nonuniform list of positions into a uniform-grid second-difference formula does not perform the transformation.

Required background. Representations as arrays distinguishes samples from normalized coordinates. Boundary implementation specifies domains and discrete inner products.

The Jacobian fixes the inner product

Let $x=f(q)$ be smooth and strictly increasing with $J(q)=f'(q)>0$ on a finite computational interval. Set $\widetilde\psi(q)=\psi(f(q))$. Then

$$\int |\psi(x)|^2\,dx =\int J(q)|\widetilde\psi(q)|^2\,dq.$$

One may keep this weighted representation, or define $\chi(q)=\sqrt{J(q)}\,\widetilde\psi(q)$ so that $\int|\chi|^2dq$ is the norm. These are unitarily equivalent continuum descriptions. The Jacobian is positive by the stated orientation; for a reversed map the integration measure uses its absolute value and derivative signs must be handled consistently.

Position probabilities in an interval must be computed with the corresponding measure. An unweighted sum of $|\widetilde\psi|^2$ counts computational grid points, which are deliberately distributed unevenly in physical space.

Transforming the kinetic energy

For constant physical mass, write $T=-\alpha\partial_x^2$, $\alpha=\hbar^2/(2m)$. The chain rule gives

$$\widetilde T=-\alpha\frac1J\partial_q\frac1J\partial_q.$$

It is self-adjoint in the $J\,dq$ inner product with the appropriately transformed domain. In the flat-measure representation,

$$T_\chi=-\alpha J^{-1/2}\partial_q J^{-1}\partial_q J^{-1/2}.$$

The placement of every Jacobian matters. Dropping the derivatives of $J$ changes the operator rather than merely changing its accuracy. Multiplicative potentials remain $V(f(q))$ in either representation.

A robust discretization can start from the quadratic form. For homogeneous Dirichlet data,

$$\langle\widetilde\psi,\widetilde T\widetilde\psi\rangle_J =\alpha\int\frac{|\partial_q\widetilde\psi|^2}{J}\,dq.$$

Discretize this positive form with compatible quadrature to obtain a stiffness matrix $K$ and mass matrix $M$. Solve $Kc=EMc$, including the potential form where necessary. This construction exposes the metric and is often safer than manually expanding several Jacobian derivatives. See finite elements for assembly.

Smooth stretching and an affine check

The map $x=\ell\sinh q$ has $J=\ell\cosh q$. A uniform $q$ grid has physical spacing approximately $\ell\cosh q\,\Delta q$: fine near the origin and coarse far away. It suits a localized central state only while the coarser exterior still resolves the wavefunction and any outgoing wavelength. A mapping cannot remove the need for resolution.

For the simpler map $x=a+s q$, $s>0$, all Jacobian derivatives vanish and $T_\chi=-\alpha s^{-2}\partial_q^2$. On $q\in[0,1]$ with Dirichlet boundaries, the eigenvalues are $\alpha n^2\pi^2/s^2$. This immediately detects a missing scale factor or a mistaken $s^{-1}$ kinetic coefficient.

Change mapping parameters and grid size separately in a convergence study. If an eigenvalue changes when the point distribution changes at fixed high resolution, the purported convergence in a single grid sequence may have hidden a poorly resolved region.

Radial reductions and singular endpoints

A radial wavefunction $R_\ell(r)$ has norm $\int_0^\infty r^2|R_\ell|^2dr$. With $u_\ell=rR_\ell$, the norm is $\int|u_\ell|^2dr$ and the usual central-potential equation becomes

$$-\alpha u_\ell''+ \left[V(r)+\frac{\alpha\ell(\ell+1)}{r^2}\right]u_\ell=Eu_\ell.$$

This reduction has already absorbed the radial measure. If one then maps $r=f(q)$, transform $u_\ell$ with $\sqrt{dr/dq}$; do not insert another $r^2$ norm weight. For a regular central problem, the regular solution behaves as $u_\ell\sim r^{\ell+1}$ near the origin. Singular interactions can require different domain analysis.

For example, $r=r_0e^q$ places $r=0$ at $q=-\infty$. A finite $q$ interval therefore introduces an inner cutoff as well as an outer boundary. Both must be varied or matched to analytic endpoint behavior. Calling the coordinate logarithmic does not impose regularity automatically.

Exercises

If $x=2q$ and $\psi(x)=\sqrt{2/L}\sin(\pi x/L)$ on $0<x<L$, write the flat-measure transformed state and verify its normalization.

Solution

$\chi(q)=\sqrt{4/L}\sin(2\pi q/L)$ on $0<q<L/2$. Its squared integral is $(4/L)(L/4)=1$. The transformed kinetic operator is $-\alpha\partial_q^2/4$, giving the original energy $\alpha\pi^2/L^2$.

Show how zero normal derivative at an endpoint changes in the flat-measure representation.

Solution

$\partial_x\psi=0$ means $\partial_q\widetilde\psi=0$ for a finite nonzero $J$. Since $\widetilde\psi=J^{-1/2}\chi$, it becomes $\chi'-(J'/2J)\chi=0$. Imposing $\chi'=0$ instead is generally a different boundary condition.

References

• Arfken, George B., Hans J. Weber, and Frank E. Harris. Mathematical Methods for Physicists. 7th ed. Academic Press, 2013 — coordinate metrics and differential operators.
• Brenner, Susanne C., and L. Ridgway Scott. The Mathematical Theory of Finite Element Methods. 3rd ed. Springer, 2008. DOI — weak forms and transformed approximation spaces.
• Tannor, David J. Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, 2007 — quantum representations and grid methods.