The Finite Well and Wavefunction Penetration
In a finite square well, a bound-state wavefunction extends beyond each edge and decays in the constant exterior potential. Mass and the gap between exterior potential and state energy set its decay length. A well of finite width and depth has finitely many bound states.
What a learner can do afterwards
- Identifies wavefunction value and slope as the quantities that must match at an ordinary finite potential step with constant particle mass
- Describes the exponential decay in the forbidden region and what sets its length
- Recognises that a square well of fixed finite width and depth has finitely many bound states for a particle of fixed finite mass
1 · Read
Recall that an infinite well requires ψ=0 at each wall. Now use a square well of width L with U=0 inside and constant U=U0>0 outside on both sides, extending to infinity. Keep mass m constant across the edges. A bound state has 0<E<U0. Its wavefunction is nonzero at each edge and extends outside in a tail that decays away from the well, giving exterior probability density. A stationary bound state does not escape.
Inside, E>U gives oscillating solutions. Outside, E<U0 gives exponential solutions. A term growing toward either infinity cannot be normalised, so each exterior tail decays away from its edge. At an ordinary finite potential step with constant mass, ψ and its derivative dψ/dx must both match. These conditions select the allowed energies. There are no singular potential spikes in this model.
At outward distance d from an edge, ψ(d)=C exp(-κd), where κ=√[2m(U0-E)]/ℏ. The amplitude decay length is ℓ=1/κ: over ℓ, the magnitude falls by a factor of e, where e≈2.718. Increasing mass or U0-E, with the other fixed, shortens ℓ. Probability density is |C|² exp(-2κd). Its decay length is ℓ/2, and its value at d=ℓ is 1/e² times its edge value.
A well of fixed finite width, depth and mass has finitely many bound states. Higher states need more interior oscillation, but E<U0 limits k=√(2mE)/ℏ. Only finitely many oscillating solutions can fit that width and match both tails. The count depends on width, depth and mass; it can be large. For E>U0, exterior solutions oscillate and the energies form an unbound continuum.
The ground state has no interior nodes. If a second bound state exists, it has one interior node and decaying tails on both sides. Each higher bound state adds one interior node. Compared with an infinite well of the same width, mass and interior potential, each finite-well bound level lies lower than its corresponding infinite-well level. The shifts need not be small. Use the same potential-energy reference for this comparison.
Bound-state waves match in value and slope at finite edges. Their exterior tails decay. Mass and U0-E set tail length; width, depth and mass affect the finite bound-state count.
Lesson and questions written for this dataset. OpenStax University Physics Volume 3, sections 7.3, 7.4 and 7.6, were consulted for physics facts and boundary-matching methods.
Readings and flashcards written for this dataset. OpenStax University Physics Volume 3, sections 7.3, 7.4 and 7.6, were consulted for physics facts. The readings distinguish bound-state penetration from transmission through a finite-width barrier.
More to read
Penetration without escape
Specify the potential before interpreting a tail. In this square-well model, the floor has U=0 and the two exterior regions have the same constant potential U0>0. Those exterior regions extend without end. Bound-state energies satisfy 0<E<U0.
The interior wave oscillates. Outside, the equation permits growing and decaying exponentials. A normalisable bound state cannot grow toward either infinity, so only outward decay remains. The wavefunction has nonzero values at the edges and nonzero probability density outside.
This exterior density is penetration into a region that classical motion at energy E could not enter. It is not a statement that the particle gradually escapes. A stationary bound state's density stays unchanged in time.
Transmission through a finite-width barrier is a different problem. There is an allowed region beyond that barrier where a transmitted wave can oscillate. The semi-infinite exterior regions of this bound-well model do not provide such an escape route.
Matching boundaries and defining tail length
For constant particle mass and ordinary finite potential steps, the wavefunction and its first spatial derivative are continuous at each edge. Matching only the value is insufficient. A bound state must satisfy both conditions at both edges and remain normalisable over all space.
Let d measure distance outward from an edge. Write its tail as ψ(d)=C exp(-κd), where κ=√[2m(U0-E)]/ℏ. The amplitude decay length ℓ=1/κ is the distance over which the wavefunction magnitude falls to 1/e of its starting value.
Larger mass shortens ℓ at a fixed energy gap. A larger gap U0-E also shortens ℓ at fixed mass. Width can affect the state energy E, but width alone does not specify the decay length.
Probability density has the exponential factor exp(-2κd). Its decay length is therefore 1/(2κ). At distance ℓ, density is 1/e² of its edge value. A penetration length must say whether it describes amplitude or density.
The finite list of bound levels
Increasing the bound-state energy increases the interior wave number k=√(2mE)/ℏ. But bound states must stay below U0, so their wave numbers have a finite upper limit. A well of finite width can fit only finitely many bound-state solutions below that limit while satisfying both boundary matches.
The count depends on width, depth and mass. Finite does not mean two or three: a deep or wide well can have many bound levels. Their node ordering starts with no interior nodes in the ground state. The next bound state, if present, has one interior node. Each successive bound state adds another.
Compare finite and infinite wells only after fixing the same width, mass and interior potential reference. Each existing finite-well bound level lies below the corresponding infinite-well level. The changes need not be small.
For E>U0, the exterior equation has oscillating solutions. These unbound states form a continuum rather than continuing the discrete bound-state list.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.