Finite Square Well

Equivalent forms

-k\cot(ka)=\kappa\;(\text{odd})

A finite wall can never fully confine a quantum particle — it always leaks out a little, and that little is why tunneling, alpha decay, and the scanning tunneling microscope exist.

Symbol	Name	SI unit	Dimension	Range
$E$	Bound-state energyoutput Allowed energy below the well depth	J	ML^2T^-2	—
$V_0$	Well depth Height of the potential walls	J	ML^2T^-2	1e-19 – 5e-18
$a$	Half-width Half the well width	m	L	1e-10 – 1e-9
$m$	Particle mass Mass of the trapped particle	kg	M	—
$\hbar$	Reduced Planck constant h/2π, the quantum of action	J*s	ML^2T^-1	—

Where it holds

1D, piecewise-constant potential. Real wells are smooth; this is the idealized limit. A very shallow or narrow well

(V_{0}a^{2}

small) still holds at least one bound state in 1D.

Dimensional analysis

k \tan (ka) \to [L^-1] (\tan

is dimensionless, ka dimensionless)

\kappa \to [L^-1]

k and \kappa

both

\surd

(energy

\cdot

mass)/action

= 1

/length; ka is dimensionless.

\checkmark

Discovery

Erwin Schrödinger (early wave mechanics) · 1926

The finite well is one of the first exactly analyzable models built directly from Schrödinger's 1926 equation. It became the standard textbook demonstration that quantization and barrier penetration follow from wave mechanics, and the direct precursor to Gamow's 1928 tunneling theory of alpha decay.

Research status: stable

Real-world applications

Semiconductor quantum wells in lasers and detectors
Modeling nucleons bound in a nucleus
Quantum dots and the foundation of tunneling devices

Common misconceptions

The wavefunction is nonzero outside the well — it decays, not vanishes (unlike the infinite well)
There are finitely many bound states, not infinitely many
Higher-energy states leak farther out because $\kappa is$ smaller

Experimental verification

Semiconductor quantum wells (GaAs/AlGaAs) realize finite wells whose discrete bound-state energies are measured by optical absorption; nuclear potential wells and quantum dots show the same finite-level structure.

Derivation

Inside the well (|x|<a,

V=0)

solutions are oscillatory

\psi \sim \cos (kx)

or sin(kx) with

k=\sqrt{2mE}/\hbar

.

Outside (|x|>a,

V=V_{0}

>E) they decay

as \psi \sim e^{-\kappa |x|}

with

\kappa =\surd (2m(V_{0}-E))/\hbar

.

Demanding

\psi and \psi ' be

continuous at

x=a

gives, for even states,

k \tan (ka)=\kappa

, and for odd states

-k \cot (ka)=\kappa

.

These transcendental equations have only a finite number of solutions, set by the dimensionless strength

z_{0}=a\sqrt{2mV_{0}}/\hbar

.

What if…

What if

V_{0} \to \infty

?

$\kappa \to \infty$ , the leakage vanishes, and you recover the infinite square well with its infinitely many exactly-sinusoidal levels $E_n \propto n^{2}$ .

What if the well is extremely shallow?

In 1D there is always at least one bound (even) state, no matter how weak the well — a special feature of one dimension.

1

Number of bound states

Given · well strength

z_{0} = a\sqrt{2mV_{0}}/\hbar = 5

Find · how many bound states

Steps

Bound states appear each time $z_{0}$ crosses a multiple $of \pi /2$
$z_{0} = 5 \to 5/(\pi /2) \approx 3.18$
So 4 bound states (even/odd alternating)

Result · Number ≈ ⌈

z_{0}/(\pi /2)

⌉ = ⌈5/1.571⌉

= 4

states

infinite square wellquantum tunnelingTime-Independent Schrödinger Equation→