A particle bound in a potential well cannot carry an arbitrary energy. Normalizability and continuity of the wave function select a discrete set. The infinite well is the case you can solve in closed form; the finite well keeps the same mechanism and a harder matching equation.
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Lesson close · Quantum Computing Fundamentals
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Continue to Barriers & Tunneling, the next prerequisite step in this course.
with . A bound state requires below the potential at infinity so that decays rather than oscillating out to .
Infinite well. on an interval of width and outside, so at the endpoints and outside. Finite well. for and outside. Bound states need . Both and are continuous at .
Inside the infinite well set , so . The general solution is . On an interval the condition kills the cosine. Then forces , hence for ( is the zero function). Therefore
The ground state has no interior zero. Each step in adds one node. The probability density is peaked, not flat: for it is largest at and zero at the walls.
For a finite well the same is used inside, while outside gives real exponentials . Continuity of and at each wall produces a transcendental equation in . Solutions exist only at discrete energies, still ordered by node count, and is small but not identically zero in the classically forbidden region. Sending pushes those tails to zero and recovers the infinite-well spectrum (up to how you label the width).
A qubit is what remains if you keep two isolated well levels and ignore the rest of the spectrum. The discretization step — boundary conditions, not an extra postulate — is why that two-level truncation has a definite gap.
Interactive · infinite-well Eₙ ∝ n²
relative energy units E∝1
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