The Finite Square Well

Core Idea

The finite square well is a region of constant negative potential, V = −V₀ for |x| < a, with V = 0 outside. Bound states have energies between −V₀ and 0; their wavefunctions oscillate inside the well and decay exponentially outside, so the particle has a nonzero chance of being found in the classically forbidden region. Matching ψ and dψ/dx at the walls gives transcendental equations. Their graphical solution shows that the well holds only a finite number of bound states, and in one dimension it always holds at least one. States with E > 0 are scattering states, which pass over the well and are partly reflected.

How It's Best Learned

Solve inside and outside separately, then impose continuity of ψ and ψ' at x = a. Use the even/odd symmetry to split the problem in two. Solve the resulting equations graphically by plotting tan z against √(z₀²/z² − 1), which shows at a glance how many states exist. Compare the results with the infinite well of the same width in both the deep and shallow limits.

Common Misconceptions

• A particle bound in the well is never found outside it. (The wavefunction decays exponentially outside but is not zero, so there is a nonzero probability of finding the particle in the classically forbidden region.)
• A shallow enough well has no bound states. (In one dimension any attractive square well, however shallow or narrow, has at least one bound state. In three dimensions this is not true.)
• The energy levels are the same as for an infinite well of the same width. (The wavefunctions leak past the walls, so the effective width is larger and each level lies lower, measured from the bottom of the well.)

Explainer

The infinite square well is a useful first model, but real potentials have finite depth. Take V = −V₀ inside |x| < a and V = 0 outside, and look first for bound states with −V₀ < E < 0.

Inside the well the kinetic energy E + V₀ is positive, so the Schrödinger equation gives oscillating solutions with wavenumber k = √(2m(E + V₀))/ħ. Outside, the energy is below the potential, and the solutions are real exponentials, e^−κ|x|, with κ = √(−2mE)/ħ. Only the decaying exponential is allowed, because a growing one cannot be normalized. The particle has a real chance of being found outside the well, in a region where classical physics says it could never go.

The potential is symmetric, so the solutions can be chosen even or odd. For even states, ψ = cos kx inside. Requiring both ψ and dψ/dx to be continuous at x = a gives κ = k tan(ka). For odd states, ψ = sin kx inside, and the condition becomes κ = −k cot(ka). These equations cannot be solved in closed form. Introduce z = ka and z₀ = (a/ħ)√(2mV₀). The even condition becomes tan z = √(z₀²/z² − 1). Plotting both sides shows the solutions as crossing points. The right side falls to zero at z = z₀, so only a finite number of crossings exist.

Several results follow from the graph. There is always at least one even state, since the first branch of tan z starts at zero and rises without limit before z = π/2. A new bound state, alternately odd and even, appears each time z₀ passes a multiple of π/2. For a deep, wide well (large z₀), the low-lying levels approach those of an infinite well of width 2a, measured from the bottom. For any finite depth they lie a little lower, because the wavefunction leaks into the walls.

For E > 0 the particle is not bound. It travels over the well and is partly reflected at the edges. The transmission coefficient reaches exactly 1 when the well's width holds a whole number of half-wavelengths, a condition 2a√(2m(E + V₀))/ħ = nπ. This perfect transmission is related to the Ramsauer-Townsend effect, in which slow electrons pass through noble-gas atoms almost without scattering. The finite well also serves as a first model for electrons in quantum-well semiconductor devices and for nucleons in a nucleus.