Energy Scales in One Dimension

Energy scales are the fastest way to understand a one-dimensional quantum problem before solving it. They tell you whether confinement is strong, whether a bound state should be shallow or deep, how long a forbidden-region tail should be, and which parameters actually control the spectrum.

The central rule is simple:

A length scale $L$ implies a kinetic-energy scale $\hbar^2/(2mL^2)$.

Most one-dimensional estimates are refinements of that statement.

Core Scale Table

Situation	Natural scale	Meaning
Confinement over length $L$	$E_L=\hbar^2/(2mL^2)$	Kinetic cost of localization
Infinite well of width $L$	$E_1=\pi^2\hbar^2/(2mL^2)$	Exact ground-state confinement energy
Finite well of half-width $a$ and depth $V_0$	$z_0=a\sqrt{2mV_0}/\hbar$	Dimensionless depth-width parameter
Attractive delta well $-\alpha\delta(x)$	$E_\delta=-m\alpha^2/(2\hbar^2)$	Zero-range binding energy
Forbidden region $V\gt E$	$\ell_{\text{tail}}=\hbar/\sqrt{2m(V-E)}$	Evanescent decay length
Harmonic minimum	$E_{\mathrm{osc}}=\hbar\omega$	Level-spacing scale near a stable minimum
Smooth tunneling barrier	$\Delta E\propto e^{-S/\hbar}$	Splitting or transmission controlled by action

The numerical constants differ from model to model. The scaling with $m$, length, depth, and $\hbar$ is the durable part.

Confinement Scale

A wavefunction localized over length $L$ must contain momenta of order

$$\Delta p\sim\frac{\hbar}{L}.$$

The corresponding kinetic energy is

$$E_L \sim \frac{(\Delta p)^2}{2m} \sim \frac{\hbar^2}{2mL^2}.$$

This estimate becomes exact up to a numerical factor in the infinite square well:

$$E_1 = \frac{\pi^2\hbar^2}{2mL^2}.$$

The $1/L^2$ dependence is why squeezing a one-dimensional bound state rapidly raises its energy. Doubling the length lowers this scale by a factor of four.

Finite-Well Strength

For the symmetric finite square well

$$V(x)= \begin{cases} -V_0, & \lvert x\rvert\lt a,\\ 0, & \lvert x\rvert\ge a, \end{cases}$$

the natural kinetic scale associated with the half-width $a$ is

$$E_a=\frac{\hbar^2}{2ma^2}.$$

The dimensionless well strength is

$$\frac{V_0}{E_a} = \frac{2ma^2V_0}{\hbar^2} =z_0^2,$$

where

$$z_0=a\frac{\sqrt{2mV_0}}{\hbar}.$$

Large $z_0$ means many oscillations can fit inside the attractive region before the continuum threshold. The approximate number of bound states is controlled by

$$N\sim\frac{2z_0}{\pi}.$$

The exact count still requires threshold care, as explained in Bound-State Counting.

Delta-Well Scale

The attractive delta potential

$$V(x)=-\alpha\delta(x), \qquad \alpha\gt0,$$

has a coupling $\alpha$ with units of energy times length. The only length that can be built from $m$, $\alpha$, and $\hbar$ is

$$\ell_\delta = \frac{\hbar^2}{m\alpha}.$$

The exact bound-state wavefunction decays on this length:

$$\psi(x)\propto e^{-\lvert x\rvert/\ell_\delta}.$$

The binding energy is

$$E_\delta = -\frac{m\alpha^2}{2\hbar^2}.$$

Thus a stronger zero-range attraction makes the state smaller and more deeply bound. Unlike a fixed-width well, the delta problem has no separate geometric length; the interaction strength creates the length scale.

Forbidden-Region Decay

In a region where the potential is locally constant and larger than the energy,

$$V\gt E,$$

the stationary Schrödinger equation gives exponential behavior. Define

$$\kappa = \frac{\sqrt{2m(V-E)}}{\hbar}.$$

The tail length is

$$\ell_{\text{tail}} = \frac{1}{\kappa} = \frac{\hbar}{\sqrt{2m(V-E)}}.$$

This formula appears in finite wells, rectangular barriers, tunneling estimates, and near-threshold bound states. Larger mass, higher barrier excess $V-E$, or smaller $\hbar$ make the tail shorter. As $E$ approaches the threshold from below, $\kappa\to0$ and the tail becomes long.

Harmonic Scale Near A Minimum

Near a stable minimum $x_0$, a smooth potential has the expansion

$$V(x) \approx V(x_0) +\frac12V''(x_0)(x-x_0)^2.$$

Identify

$$m\omega^2=V''(x_0).$$

The natural energy scale for small oscillations is

$$E_{\mathrm{osc}}=\hbar\omega,$$

and the natural length is

$$\ell_{\mathrm{osc}} = \sqrt{\frac{\hbar}{m\omega}}.$$

Low-lying states in a smooth well often resemble oscillator states near the minimum:

$$E_n \approx V(x_0) +\hbar\omega \left( n+\frac12 \right).$$

This approximation is local. It fails for highly excited states that explore anharmonic parts of the well or approach a finite barrier top.

Tunneling And Splitting Scales

Tunneling introduces an energy scale that can be much smaller than the local level spacing. In a symmetric double well, each well may have a local oscillator scale $\hbar\omega_0$, but the splitting between the even and odd combinations can be exponentially small:

$$\Delta E \sim A e^{-S/\hbar}.$$

Here $S$ is a barrier action of the rough form

$$S \sim \int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx,$$

over the classically forbidden region. The prefactor $A$ is model dependent, but the exponential sensitivity is the key scale lesson. Small changes in barrier width, height, or mass can produce large changes in tunneling splittings.

Detailed WKB prefactors belong to the semiclassical volume. In this chapter, the main use is diagnostic: tunneling splittings are often far smaller than ordinary confinement or oscillator spacings.

Mass Scaling

Mass appears in nearly every one-dimensional scale:

$$E_L\propto\frac{1}{m}, \qquad \ell_{\text{tail}}\propto\frac{1}{\sqrt m}, \qquad E_\delta\propto m, \qquad \ell_{\mathrm{osc}}\propto\frac{1}{\sqrt m}$$

when the external potential parameters are held fixed.

This mix of dependencies is not contradictory. The length and strength being held fixed matter. A heavier particle has a lower confinement kinetic energy for a fixed box, a shorter evanescent tail for a fixed barrier excess, and a deeper delta binding energy for a fixed integrated attraction.

Scale-Estimation Workflow

Before solving a one-dimensional model:

1. Identify the relevant length: box width, well half-width, oscillator length, decay length, or barrier width.
2. Form the kinetic scale $\hbar^2/(2mL^2)$.
3. Compare potential depths or barriers to that kinetic scale.
4. Convert the comparison into a dimensionless parameter such as $z_0$.
5. Estimate the number of states, tail lengths, or splitting sizes.
6. Only then solve the exact matching equations or numerical eigenvalue problem.

This workflow catches many wrong answers. A computed state with a tail much longer than the numerical box, or an energy spacing wildly inconsistent with $\hbar^2/(2mL^2)$, deserves suspicion.

Common Mistakes

• Comparing two energies without checking that they are measured from the same zero.
• Forgetting the square on $L$ in the confinement scale.
• Treating the finite-well depth $V_0$ alone as the state-counting parameter; width and mass matter too.
• Applying the harmonic scale to states far from the minimum.
• Ignoring a long near-threshold tail in a finite numerical box.
• Treating an exponentially small tunneling splitting as comparable to the local oscillator spacing.
• Dropping $\hbar$ or $m$ during dimensional estimates and then restoring them inconsistently.

Where This Is Used

• Dimensionless Variables and Scaling gives the general method behind these estimates.
• Infinite Square Well realizes the confinement scale exactly.
• Finite Square Well uses $z_0$ and the forbidden-region tail length.
• Delta-Function Potential gives the exact zero-range binding scale.
• Bound-State Counting turns scales into state-count estimates.
• Quantum Harmonic Oscillator develops $\ell_{\mathrm{osc}}$ and $\hbar\omega$.
• Oscillator as a Universal Local Model explains why curvature near a stable minimum sets the local oscillator approximation.
• Rectangular Barrier Tunneling uses the same $\kappa$ scale for barrier penetration.

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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. A box length is doubled while the mass is fixed. How does the confinement scale change?

Solution

The confinement scale is

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

Replacing $L$ by $2L$ gives

$$E_{2L} = \frac{\hbar^2}{2m(2L)^2} = \frac{E_L}{4}.$$

The scale decreases by a factor of four.

2. Use dimensional reasoning to construct the delta-well length scale from $m$, $\alpha$, and $\hbar$.

Solution

For $V(x)=-\alpha\delta(x)$, the strength $\alpha$ has units of energy times length. The combination

$$\ell_\delta = \frac{\hbar^2}{m\alpha}$$

has units of length. The exact solution confirms that this is the decay length:

$$\psi(x)\propto e^{-\lvert x\rvert/\ell_\delta}.$$

3. A forbidden region has constant $V-E=U\gt0$. Find the decay length.

Solution

The decay constant is

$$\kappa = \frac{\sqrt{2mU}}{\hbar}.$$

Therefore

$$\ell_{\text{tail}} = \frac{1}{\kappa} = \frac{\hbar}{\sqrt{2mU}}.$$

4. Near a smooth minimum, $V''(x_0)=K$. What oscillator frequency and energy scale should be used?

Solution

Compare

$$V(x)\approx V(x_0)+\frac12K(x-x_0)^2$$

with the oscillator form

$$\frac12m\omega^2(x-x_0)^2.$$

Thus

$$\omega=\sqrt{\frac{K}{m}}.$$

The corresponding level-spacing scale is

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