One-Dimensional Bound Systems

One-dimensional models show how confinement, matching conditions, and symmetry select energy levels. Hard walls, finite wells, and point interactions each impose a different constraint on a wavefunction. Periodic boundaries provide a useful comparison: their discrete spectrum comes from compact geometry rather than reflecting walls.

Finite walls allow tails

For the symmetric square well $V=-V_0$ inside $|x|<a$ and $V=0$ outside, a bound energy satisfies $-V_0<E<0$. The wavefunction oscillates inside and decays outside, with

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

Matching the wavefunction and its derivative gives $q\tan(qa)=\kappa$ for even states and $-q\cot(qa)=\kappa$ for odd states. The tail is part of the normalized state; setting the wavefunction to zero at a finite wall changes the problem into a hard-wall approximation.

For the usual short-range wells on the line, bound states lie below the continuum threshold and are square integrable. Energy need not be called negative: adding a constant to the potential moves the energies and threshold together. Ring eigenstates instead live on a compact configuration space and need no decay at infinity.

In regular one-dimensional bound problems with separated endpoint conditions, energy levels are nondegenerate and node counts order them. The ring’s opposite-momentum degeneracy shows why the domain assumptions matter.

Reading sequence

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?

Compare one model at a time

Begin with the infinite well, then the finite well and parity. An asymmetric well tests what changes when parity is unavailable. Delta potentials replace derivative continuity by a jump; double wells connect level splitting to coherent transfer between localized states.

Exercises

A finite-well state approaches threshold from below. What happens to its exterior decay length, and why must threshold itself be checked separately?

Solution

The decay length $1/\kappa=\hbar/\sqrt{-2mE}$ diverges as $E\to0^-$. At zero energy the exterior solution is linear rather than exponentially decaying, so the ordinary finite square well has no nonzero square-integrable threshold eigenfunction. A threshold root must not be counted as an additional bound state.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• S. Flügge, Practical Quantum Mechanics, Springer, 1999.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.