The finite square well

A more realistic well

The infinite square well assumes perfectly impenetrable walls. Real potential wells have finite depth. In a finite square well, a particle can be mostly confined while still having a nonzero probability of being found outside the classically allowed region.

A common symmetric finite well is

$V(x)=-V_0quad |x|<a$

and

$V(x)=0quad |x|geq a$

where $V_0>0$. Bound states have energies

$-V_0<E<0$

The particle is trapped because its total energy is below the outside potential.

Oscillatory inside, exponential outside

Inside the well, $E>V$, so the Schrödinger equation gives oscillatory solutions. Define

k=rac{sqrt{2m(E+V_0)}}{hbar}

Outside the well, $E<0=V$, so the wavefunction decays exponentially. Define

kappa=rac{sqrt{-2mE}}{hbar}

The outside solutions behave like

$psi(x)propto e^{-kappa |x|}$

for large $|x|$. These exponential tails are a major difference from the infinite well.

Boundary matching

The wavefunction must be continuous at the boundaries $x=pm a$. Its derivative must also be continuous if the potential is finite:

$psi_{inside}=psi_{outside}$

rac{dpsi_{inside}}{dx}=rac{dpsi_{outside}}{dx}

These matching conditions determine the allowed energies.

Even and odd states

Because the potential is symmetric, bound states can be chosen with definite parity. Even states satisfy

$psi(-x)=psi(x)$

Odd states satisfy

$psi(-x)=-psi(x)$

Inside the well, even states use cosine-like solutions, while odd states use sine-like solutions. This symmetry simplifies the calculation.

Transcendental equations

For the symmetric finite well, matching conditions lead to equations such as

$k an(ka)=kappa$

for even states and

$-kcot(ka)=kappa$

for odd states, depending on conventions. These equations usually must be solved numerically or graphically.

Unlike the infinite well, there is no simple formula $E_npropto n^2$.

Number of bound states

A finite well supports only a finite number of bound states. A deeper or wider well supports more states. A shallow narrow well may support only one bound state in one dimension.

As energy approaches zero from below, the bound state becomes weakly bound and its wavefunction extends far outside the well.

Penetration into forbidden regions

Classically, a particle with $E<V$ cannot enter a region. Quantum mechanically, the wavefunction can penetrate with exponential decay. The probability density outside the finite well is not zero:

eq0$$ This is not tunneling through a barrier yet, but it is the same mathematical feature: wavefunctions extend into classically forbidden regions. ## Infinite well limit As $V_0 oinfty$, the decay constant $kappa$ becomes very large, the outside tails shrink to zero, and the finite well approaches the infinite square well. Boundary conditions effectively become $psi(pm a)=0$. ## The big idea The finite square well shows that quantum confinement is not all-or-nothing. Bound-state wavefunctions oscillate where classically allowed and decay where classically forbidden. Energies are still quantized, but they are found by matching wavefunctions and derivatives at finite potential boundaries.