Numerical Units and Dimensionless Variables

Nondimensionalization rewrites a quantum calculation in terms of explicit reference scales. It keeps numerical magnitudes manageable and reveals which parameter combinations control the physics. Setting $\hbar=1$ is meaningful only after specifying the corresponding time and energy units; it is not permission to mix frequencies, energies, and inverse seconds.

Required background. Array representations explains why wavefunction normalization carries a measure.

Scaling the Schrödinger equation

Choose a length $\ell>0$ and energy $E_*>0$. With

$$x=\ell\xi,\qquad t=\frac{\hbar}{E_*}\tau, \qquad \psi(x,t)=\ell^{-1/2}\phi(\xi,\tau),$$

the one-dimensional equation becomes

$$i\partial_\tau\phi =\left[-\alpha\partial_\xi^2+v(\xi,\tau)\right]\phi, \qquad \alpha=\frac{\hbar^2}{2m\ell^2E_*}, \qquad v=\frac{V}{E_*}.$$

The factor $\ell^{-1/2}$ follows by substituting $dx=\ell\,d\xi$ into the norm. In $d$ spatial dimensions it becomes $\ell^{-d/2}$ for a single common length scale. Coordinate transformations with nonconstant Jacobians require the more general treatment in coordinate transformations.

One may choose $E_*=\hbar^2/(2m\ell^2)$ so that $\alpha=1$. That convention differs from oscillator units, where the kinetic coefficient is $1/2$. Record the actual scaled equation instead of relying on a phrase such as “natural units.”

Oscillator units as a worked example

For $V(x)=m\omega^2x^2/2+\lambda x^4$, take

$$\ell=\sqrt{\frac{\hbar}{m\omega}}, \qquad E_*=\hbar\omega.$$

Then

$$\frac{H}{\hbar\omega} =-\frac12\partial_\xi^2+\frac12\xi^2+g\xi^4, \qquad g=\frac{\lambda\hbar}{m^2\omega^3}.$$

At $g=0$, the dimensionless energies are $n+1/2$. Restoring units multiplies energies by $\hbar\omega$, lengths by $\ell$, and times by $1/\omega$. A parameter sweep in $g$ therefore compares equivalent dimensionless Hamiltonians even when the dimensional masses and trap frequencies differ.

A grid spacing $\Delta\xi=0.02$ is a physical spacing $0.02\ell$. A box $[-8,8]$ means $[-8\ell,8\ell]$, not a universal length in meters. This distinction is essential when reusing a numerical benchmark for a different system.

Frequencies, rates, and observables

An angular frequency $\Omega$ corresponds to energy $\hbar\Omega$; an ordinary frequency $f$ corresponds to $hf=2\pi\hbar f$. For a driven two-level Hamiltonian $H=(\hbar\Omega/2)\sigma_x$, dimensionless evolution uses $(\Omega\hbar/E_*)\sigma_x/2$. The factor of $1/2$ belongs to the Hamiltonian convention, independently of the units.

A decay rate $\gamma$ has units of inverse time. Its dimensionless value is $\widetilde\gamma=\gamma\hbar/E_*$. If a jump operator includes $\sqrt\gamma$, its coefficient scales with the square root of this rate. Scaling $H$ but leaving dissipative rates untouched changes their competition.

For the one-dimensional probability current,

$$j(x,t)=\frac{\hbar}{m}\operatorname{Im}(\psi^*\partial_x\psi) =\frac{\hbar}{m\ell^2} \operatorname{Im}(\phi^*\partial_\xi\phi).$$

The density scales as $\ell^{-1}$, while current scales as inverse time. Check derived quantities individually rather than assuming every output scales like energy.

Tolerances and numerical scaling

An eigenvalue residual $r=Hc-Ec$ has energy units for a normalized coefficient vector. A dimensionless residual uses $\widetilde r=r/E_*$. State whether an acceptance criterion is absolute, relative to $\|H\|$, or relative to a physically chosen energy scale. A tolerance of $10^{-10}$ without units or normalization is incomplete.

Nondimensionalization does not by itself cure ill-conditioning. Multiplying an entire matrix by a scalar leaves its normwise condition number unchanged. It can prevent overflow and make tolerances interpretable; resolving disparate physical scales may still require adaptive meshes, suitable bases, or preconditioning.

Keep conversion at the input/output boundary where practical. Save $\ell,E_*$ and all dimensionless couplings with the result. A second calculation in another consistent unit system should recover the same physical observable after conversion; this tests for missing $\hbar$, $2\pi$, or mass factors.

Exercises

Using $E_*=\hbar^2/(2m\ell^2)$, find the dimensionless energies of an infinite well of width $L=3\ell$.

Solution

$E_n=\hbar^2\pi^2n^2/(2mL^2)$ gives $E_n/E_*=\pi^2n^2/9$. The dimensionless interval has width three and the kinetic operator is $-\partial_\xi^2$.

An energy splitting is quoted as $E/h=5\,\mathrm{GHz}$. What angular frequency belongs in $e^{-iEt/\hbar}$?

Solution

$E/\hbar=2\pi\times5\times10^9\,\mathrm{s}^{-1}$. Inserting $5\times10^9$ directly would lengthen the phase period by a factor of $2\pi$.

References

• Higham, N. J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002, publisher edition — scaling and numerical error.
• Tannor, D. J., Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007 — quantum propagation, physical scales, and oscillator conventions.