Dimensionless Variables and Scaling

Dimensionless variables turn a physical Schrödinger problem into an equation whose essential parameters are visible. They remove irrelevant units, reveal natural length and energy scales, improve numerical conditioning, and make limiting regimes easier to recognize.

The goal is not to hide constants. The goal is to identify which combinations of constants actually control the physics.

Why Dimensionless Variables Matter

A formula such as

$$e^{ikx}$$

is meaningful because $kx$ is dimensionless. Similarly, the Schrödinger phase

$$e^{-iEt/\hbar}$$

is meaningful because $Et/\hbar$ is dimensionless. Dimensionless scaling applies the same discipline to whole differential equations.

Scaling helps in four common ways:

• it shows which parameters are physically independent;
• it exposes small or large dimensionless quantities for approximation methods;
• it makes eigenvalue equations cleaner;
• it keeps numerical variables and matrix entries near natural order-one scales.

Changing units does not change the physics. Choosing natural dimensionless variables changes how clearly the physics is displayed.

Basic Scaling Template

Consider a one-dimensional stationary Schrödinger equation

$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x) \right]\psi(x) =E\psi(x).$$

Choose a length scale $L$ and define

$$\xi=\frac{x}{L}.$$

Then

$$\frac{d}{dx} = \frac{1}{L}\frac{d}{d\xi}, \qquad \frac{d^2}{dx^2} = \frac{1}{L^2}\frac{d^2}{d\xi^2}.$$

The kinetic term introduces the natural energy associated with length $L$:

$$E_L = \frac{\hbar^2}{2mL^2}.$$

If the potential has a characteristic size $V_0$ and shape $U(\xi)$,

$$V(x)=V_0U(\xi),$$

then the dimensionless eigenvalue equation is

$$\left[ -\frac{d^2}{d\xi^2} +\lambda U(\xi) \right]\varphi(\xi) = \varepsilon\varphi(\xi),$$

where

$$\lambda=\frac{V_0}{E_L}, \qquad \varepsilon=\frac{E}{E_L}.$$

The dimensionless parameter $\lambda$ is often the real control knob. It compares potential energy to the kinetic energy forced by localization over the scale $L$.

Wavefunction Scaling

The wavefunction must also be scaled if normalization is to keep its meaning. In one dimension, define

$$\psi(x)=\frac{1}{\sqrt L}\varphi(\xi), \qquad \xi=\frac{x}{L}.$$

Then

$$\int_{-\infty}^{\infty} \lvert\psi(x)\rvert^2\,dx = \int_{-\infty}^{\infty} \lvert\varphi(\xi)\rvert^2\,d\xi.$$

In three dimensions,

$$\psi(\mathbf r) = \frac{1}{L^{3/2}}\Phi(\boldsymbol\xi), \qquad \boldsymbol\xi=\frac{\mathbf r}{L},$$

so that

$$\int \lvert\psi(\mathbf r)\rvert^2\,d^3r = \int \lvert\Phi(\boldsymbol\xi)\rvert^2\,d^3\xi.$$

For radial central-potential problems, the scaling depends on which radial function is used:

$$R(r)=L^{-3/2}\mathcal R(s), \qquad u(r)=L^{-1/2}w(s), \qquad s=\frac{r}{L}.$$

These powers are not optional decoration; they preserve the normalization convention.

Time Scaling

If the natural energy scale is $E_0$, the corresponding time scale is

$$t_0=\frac{\hbar}{E_0}.$$

With

$$\tau=\frac{t}{t_0},$$

the phase $Et/\hbar$ becomes

$$\frac{Et}{\hbar} = \frac{E}{E_0}\tau.$$

Thus the dimensionless energy $\varepsilon=E/E_0$ also controls the rate of phase evolution in dimensionless time.

Square Well Scale

For a particle in an infinite square well of width $L$, the natural kinetic scale is

$$E_L=\frac{\hbar^2}{2mL^2}.$$

Using $\xi=x/L$ on $0<\xi<1$, the stationary equation becomes

$$-\frac{d^2\varphi}{d\xi^2} = \varepsilon\varphi, \qquad \varphi(0)=\varphi(1)=0.$$

The dimensionless eigenvalues are

$$\varepsilon_n=n^2\pi^2, \qquad n=1,2,\ldots.$$

Restoring units gives

$$E_n = \frac{\hbar^2\pi^2n^2}{2mL^2}.$$

The scaling makes the main dependence immediate: all energies scale as $1/L^2$.

Harmonic Oscillator Scale

For the harmonic oscillator,

$$V(x)=\frac12m\omega^2x^2.$$

The natural length is found by balancing kinetic localization energy with oscillator potential energy:

$$\frac{\hbar^2}{2m\ell^2} \sim \frac12m\omega^2\ell^2.$$

This gives the oscillator length

$$\ell=\sqrt{\frac{\hbar}{m\omega}}.$$

With $\xi=x/\ell$ and energy in units of $\hbar\omega$, the stationary equation becomes

$$\left[ -\frac12\frac{d^2}{d\xi^2} +\frac12\xi^2 \right]\varphi = \varepsilon\varphi, \qquad \varepsilon=\frac{E}{\hbar\omega}.$$

Some derivations multiply this equation by $2$ and use $2E/(\hbar\omega)$ as the dimensionless eigenvalue. Both conventions are equivalent. The important scales are $\ell$ and $\hbar\omega$.

Coulomb Scale

For the hydrogenic Coulomb problem with reduced mass $\mu$ and charge magnitude $e$, the Hamiltonian is

$$\hat H = -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{4\pi\epsilon_0r}.$$

The natural length is the reduced-mass Bohr radius

$$a_0 = \frac{4\pi\epsilon_0\hbar^2}{\mu e^2}.$$

The corresponding Hartree-scale energy is

$$E_{\mathrm H} = \frac{\hbar^2}{\mu a_0^2} = \frac{\mu e^4}{(4\pi\epsilon_0)^2\hbar^2}.$$

The bound-state energies are then simply

$$E_n = -\frac{E_{\mathrm H}}{2n^2}.$$

The Coulomb scaling explains why changing reduced mass or nuclear charge changes atomic sizes and binding energies in systematic ways. The canonical hydrogen solution is developed in Hydrogen Atom.

Magnetic Length

For a charged particle in a uniform magnetic field, the natural length is the magnetic length

$$\ell_B = \sqrt{\frac{\hbar}{\lvert q\rvert B}}.$$

The corresponding frequency scale is the cyclotron frequency

$$\omega_c = \frac{\lvert q\rvert B}{m},$$

and the energy scale is

$$\hbar\omega_c.$$

The magnetic length measures the spatial scale on which magnetic flux and quantum phase compete. It is the natural scale for Landau levels and many quantum Hall calculations.

Numerical Conditioning

Dimensionless variables are not only aesthetic. They make numerical calculations more stable and easier to audit.

For example, a discretized Hamiltonian with entries near $10^{-34}$ in SI units may be mathematically correct but numerically awkward. After scaling by a natural energy, the same matrix may have entries of order $1$ to $10^3$, with residuals and tolerances that can be interpreted physically.

A numerical eigenpair residual such as

$$r=H\mathbf v-E\mathbf v$$

should be compared with a scale:

$$\frac{\lVert r\rVert}{E_0\lVert\mathbf v\rVert}.$$

Without $E_0$, a statement such as “the residual is $10^{-9}$” is incomplete, because the units and physical scale of the residual have not been specified.

Scaling also helps identify stiffness in time-dependent problems. If the largest energy scale in a Hamiltonian is $E_{\max}$, then the shortest phase-evolution time scale is roughly

$$\frac{\hbar}{E_{\max}}.$$

Time-stepping choices must resolve that scale if those high-energy components matter.

Common Mistakes

• Setting $\hbar=1$ without saying which length, mass, charge, or energy scales are being held fixed.
• Scaling the coordinate but forgetting the wavefunction normalization factor.
• Treating a dimensionful quantity inside an exponential, logarithm, or trigonometric function as meaningful.
• Confusing a unit convention with a physical approximation.
• Hiding the true small parameter by choosing variables that make it look like $1$.
• Reporting numerical residuals or tolerances without a reference scale.
• Mixing two scaling conventions in the same derivation.

Workflow

When scaling a Schrödinger problem:

1. identify the physical domain and boundary conditions;
2. choose a natural length $L$ from geometry, balance of terms, or external fields;
3. define a kinetic energy scale $\hbar^2/(2mL^2)$ or another natural $E_0$;
4. rescale the wavefunction so normalization is preserved;
5. divide the equation by $E_0$;
6. name the remaining dimensionless parameters;
7. solve or approximate the dimensionless problem;
8. restore units only after the dimensionless result is understood.

The best scaling is the one that makes the important physics hard to miss.

Exercises

1. Rescale the infinite square well on $0<x<L$ using $\xi=x/L$ and show that the dimensionless energies are $n^2\pi^2$ when $E_L=\hbar^2/(2mL^2)$.

Solution

The stationary equation is

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi.$$

With $\xi=x/L$,

$$\frac{d^2}{dx^2} = \frac{1}{L^2}\frac{d^2}{d\xi^2}.$$

Dividing by $E_L=\hbar^2/(2mL^2)$ gives

$$-\frac{d^2\varphi}{d\xi^2} = \varepsilon\varphi, \qquad \varepsilon=\frac{E}{E_L}.$$

The boundary conditions are $\varphi(0)=\varphi(1)=0$, so

$$\varphi_n(\xi)=\sqrt2\sin(n\pi\xi), \qquad \varepsilon_n=n^2\pi^2.$$

2. Balance the kinetic localization scale $\hbar^2/(2m\ell^2)$ with the oscillator potential scale $m\omega^2\ell^2/2$ and solve for $\ell$.

Solution

Set

$$\frac{\hbar^2}{2m\ell^2} = \frac12m\omega^2\ell^2.$$

Multiplying by $2m\ell^2$ gives

$$\hbar^2=m^2\omega^2\ell^4.$$

Therefore

$$\ell^2=\frac{\hbar}{m\omega}, \qquad \ell=\sqrt{\frac{\hbar}{m\omega}}.$$

3. In one dimension, show that $\psi(x)=L^{-1/2}\varphi(x/L)$ preserves normalization.

Solution

Let $\xi=x/L$, so $dx=L\,d\xi$. Then

$$\int \lvert\psi(x)\rvert^2\,dx = \int \frac{1}{L} \lvert\varphi(\xi)\rvert^2 L\,d\xi = \int \lvert\varphi(\xi)\rvert^2\,d\xi.$$

Thus the scaled wavefunction has the same dimensionless norm.

Where This Is Used

• Dimensionless Parameters Table gives the quick lookup version of the scales and control parameters derived here.
• Energy Scales in One Dimension specializes the scaling logic to one-dimensional wells, barriers, delta binding, and oscillator approximations.
• Infinite Square Well uses the geometric length $L$ to set the energy scale.
• Quantum Harmonic Oscillator uses the oscillator length and $\hbar\omega$.
• Hydrogen Atom uses the Bohr radius and Hartree-scale energy.
• Floating-Point Arithmetic explains why scale matters in computation.
• Semiclassical Limit uses dimensionless action ratios such as $S_0/\hbar$.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.