One-Dimensional Bound Systems

One-dimensional bound systems are the first complete laboratory for spectral quantization. They show how a differential expression becomes a quantum Hamiltonian only after its domain and boundary or matching conditions are specified. Hard walls, finite barriers, singular wells, symmetry, and topology then produce distinct spectra even when the local kinetic operator is the same.

Detailed scattering through barriers begins with Quantum Tunneling. WKB counting and tunneling estimates belong in Approximation and Semiclassical Methods. This chapter owns the exact and qualitative wave mechanics of confined one-dimensional models.

What makes a state bound

For

$$\hat H = - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x),$$

a bound state is a square-integrable eigenfunction satisfying the operator’s domain conditions:

$$\hat H\psi_n = E_n\psi_n, \qquad \int \lvert\psi_n(x)\rvert^2\,dx \lt \infty.$$

When $V(x)\to0$ at spatial infinity, the continuum threshold is $E=0$, so bound states have $E_n\lt0$. For other asymptotic conventions the threshold shifts with the asymptotic potential. “Negative energy” is therefore not an absolute definition; normalizability below the continuum threshold is.

The kinetic scale associated with confinement over length $L$ is

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

This scale predicts how spectra respond to mass and width before any exact solution is attempted.

Hard-wall quantization

For an infinite square well on $0\lt x\lt L$, the wavefunction vanishes at both walls. The normalized eigenfunctions and energies are

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right), \qquad n=1,2,\ldots,$$

and

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

The spectrum is discrete because only certain wavelengths satisfy both endpoint conditions. Infinite Square Well owns the full derivation, normalization, expectation values, parity forms, and time-dependent superpositions.

Finite confinement and matching

A symmetric finite well may be written

$$V(x) = \begin{cases} -V_0, & \lvert x\rvert\lt a,\\ 0, & \lvert x\rvert\ge a, \end{cases} \qquad V_0\gt0.$$

Bound states lie in $-V_0\lt E\lt0$. They oscillate inside the well and decay outside with

$$q = \frac{\sqrt{2m(E+V_0)}}{\hbar}, \qquad \kappa = \frac{\sqrt{-2mE}}{\hbar}.$$

Continuity of $\psi$ and $\psi'$ produces the parity-separated equations

$$q\tan(qa)=\kappa$$

for even states and

$$-q\cot(qa)=\kappa$$

for odd states. Their roots, not a closed polynomial formula, determine the energies.

Finite Square Well owns this calculation and its infinite-wall limit. Asymmetric Square Well removes parity symmetry and shows how full left-and-right matching replaces the even/odd shortcut.

Singular wells and jump conditions

The attractive delta well

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

is free everywhere except at one singular point. The wavefunction is continuous, but integrating the Schrödinger equation through the origin gives

$$\psi'(0^+)-\psi'(0^-) = - \frac{2m\alpha}{\hbar^2} \psi(0).$$

The model has exactly one bound state:

$$\psi(x) = \sqrt{\kappa} e^{-\kappa\lvert x\rvert}, \qquad \kappa = \frac{m\alpha}{\hbar^2},$$

with

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

Delta-Function Potential owns the limiting construction, derivative jump, normalization, and scattering preview. Double Delta Potential turns the same matching rule into an analytically controlled model of even/odd splitting and bonding/antibonding structure.

Parity, nodes, and nondegeneracy

If $V(-x)=V(x)$, bound eigenstates can be chosen even or odd. Smooth even states obey $\psi'(0)=0$; odd states obey $\psi(0)=0$. These conditions let a symmetric problem be solved on half the line.

For ordinary one-dimensional bound problems:

• bound-state energies are nondegenerate;
• the ground state can be chosen strictly positive and has no interior node;
• ordering states from the ground state as $j=0,1,2,\ldots$, the $j$th state has $j$ interior nodes;
• in a symmetric well, parity alternates with the level ordering.

Parity and Nodes owns the Wronskian nondegeneracy argument, node ordering, parity classification, and limitations of these statements.

Double wells and two-state structure

Two separated attractive regions produce nearly localized left and right states. For a symmetric double well, the energy eigenstates are approximately

$$\lvert+\rangle \simeq \frac{ \lvert L\rangle+\lvert R\rangle }{\sqrt2}, \qquad \lvert-\rangle \simeq \frac{ \lvert L\rangle-\lvert R\rangle }{\sqrt2}.$$

A two-state effective Hamiltonian

$$H_{\mathrm{eff}} = \begin{pmatrix} E_0&-J\\ -J&E_0 \end{pmatrix}$$

has energies $E_0-J$ and $E_0+J$. The splitting $2J$ controls coherent tunneling between localized combinations. Exact splitting depends on the full barrier and belongs to the detailed model or semiclassical treatment.

Double-Well Potential owns the qualitative smooth-well physics. The double-delta page provides the exact singular comparison. Coupled-state dynamics continues in Coupled Wells and Avoided Crossings.

Periodic topology is not a hard box

Periodic boundary conditions identify $x$ and $x+L$:

$$\psi(x+L)=\psi(x), \qquad \psi'(x+L)=\psi'(x).$$

For a free particle, they select

$$k_n = \frac{2\pi n}{L}, \qquad n\in\mathbb Z.$$

There is no reflection from an endpoint and no wall-imposed node. The interval represents a circle, and the modes form a Fourier-series basis. Periodic Boundary Conditions owns the regulator, density-of-states, and self-adjointness viewpoints.

Particle on a Ring: First Encounter translates periodicity into angular momentum and magnetic-flux language. The full canonical ring treatment remains in Particle on a Ring.

Counting before solving

The dimensionless finite-well strength

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

measures how many half-wavelengths can fit inside a symmetric well of half-width $a$. Deeper, wider, or heavier systems generally support more bound states. Exact threshold cases require care because a state at the continuum edge may fail to be square integrable.

Bound-State Counting combines graphical finite-well roots, nodes, thresholds, and semiclassical estimates. Energy Scales in One Dimension collects the confinement, delta binding, forbidden-region decay, oscillator, and tunneling scales used to check detailed calculations.

Reading route

1. Infinite Square Well
2. Finite Square Well
3. Parity and Nodes
4. Asymmetric Square Well
5. Delta-Function Potential
6. Double Delta Potential
7. Double-Well Potential
8. Bound-State Counting
9. Energy Scales in One Dimension
10. Periodic Boundary Conditions
11. Particle on a Ring: First Encounter

Page map

Page	Central question
Infinite Square Well	How do hard-wall boundary conditions quantize energy?
Finite Square Well	How do finite barriers produce tails and transcendental levels?
Asymmetric Square Well	How is matching organized without parity symmetry?
Delta-Function Potential	How does a singular well replace derivative continuity by a jump?
Double Delta Potential	How do two singular wells create parity splitting?
Double-Well Potential	How do parity eigenstates and localized states encode tunneling?
Periodic Boundary Conditions	How does endpoint identification quantize momentum without walls?
Particle on a Ring: First Encounter	How does periodic one-dimensional motion become angular motion?
Bound-State Counting	How many levels should exist before their energies are solved?
Parity and Nodes	How do symmetry and node order constrain the spectrum?
Energy Scales in One Dimension	Which combinations of mass, length, and potential set the answer’s scale?

Common mistakes

Mistake	Correction
Solving the differential equation without stating its domain	specify the interval, asymptotics, and boundary or matching conditions
Setting a finite-well wavefunction to zero at the barrier	match it to an evanescent exterior tail
Mixing even and odd finite-well equations	solve each parity sector separately
Requiring derivative continuity across a delta interaction	integrate through the singularity and impose the jump condition
Treating left/right localized double-well states as exact energy eigenstates	form the symmetric and antisymmetric stationary combinations
Inferring degeneracy from visual similarity in one dimension	apply the one-dimensional nondegeneracy result and inspect the domain
Treating periodic boundaries as hard walls	remember that endpoints are identified and carry circulating current
Counting a threshold solution as a normalizable bound state	check its asymptotic decay and normalization explicitly

Exercises

1. Confinement scaling

The length of an infinite square well is doubled while the mass is fixed. How does every energy eigenvalue change?

Solution

Since

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

replacing $L$ by $2L$ gives

$$E_n(2L) = \frac14E_n(L).$$

The dimensionless level pattern $n^2$ is unchanged; only the confinement scale changes.

2. Delta-well jump condition

Integrate the stationary Schrödinger equation for $V(x)=-\alpha\delta(x)$ across a small interval around the origin and derive the derivative jump.

Solution

Integrating from $-\epsilon$ to $\epsilon$ gives

$$- \frac{\hbar^2}{2m} \left[ \psi'(\epsilon)-\psi'(-\epsilon) \right] - \alpha\psi(0) = E \int_{-\epsilon}^{\epsilon} \psi(x)\,dx.$$

For a finite continuous wavefunction, the right-hand integral vanishes as $\epsilon\to0$. Therefore

$$\psi'(0^+)-\psi'(0^-) = - \frac{2m\alpha}{\hbar^2} \psi(0).$$

3. Parity at the origin

Show that a differentiable even wavefunction obeys $\psi'(0)=0$ and an odd wavefunction obeys $\psi(0)=0$.

Solution

Evenness gives $\psi(-x)=\psi(x)$. Differentiating yields $-\psi'(-x)=\psi'(x)$, so at $x=0$, $\psi'(0)=-\psi'(0)$ and therefore $\psi'(0)=0$.

Oddness gives $\psi(-x)=-\psi(x)$. Setting $x=0$ gives $\psi(0)=-\psi(0)$, hence $\psi(0)=0$.

4. Momentum on a periodic interval

Derive the allowed wavenumbers for $\psi(x)=Ae^{ikx}$ under $\psi(x+L)=\psi(x)$.

Solution

Periodicity requires

$$e^{ikL}=1.$$

Thus $kL=2\pi n$ for $n\in\mathbb Z$, and

$$k_n = \frac{2\pi n}{L}.$$

The signs of $n$ distinguish opposite momenta; $n$ and $-n$ have the same free-particle energy.

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.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• S. Flügge, Practical Quantum Mechanics, Springer, 1999.