The infinite square well

A particle in a box

The infinite square well models a particle confined between two impenetrable walls. Let the well extend from $x=0$ to $x=L$. The potential is

$V(x)=0quad 0<x<L$

and

$V(x)=inftyquad xleq0 ext{or} xgeq L$

The particle cannot exist outside the box, so the wavefunction must vanish at the walls:

$psi(0)=0$

$psi(L)=0$

Schrödinger equation inside the well

Inside the box, the time-independent Schrödinger equation is

-rac{hbar^2}{2m}rac{d^2psi}{dx^2}=Epsi

Rearrange as

rac{d^2psi}{dx^2}+k^2psi=0

where

k^2=rac{2mE}{hbar^2}

The general solution is

$psi(x)=Asin(kx)+Bcos(kx)$

Boundary conditions and quantization

The condition $psi(0)=0$ gives $B=0$. The condition $psi(L)=0$ gives

$Asin(kL)=0$

For a nonzero wavefunction, we need

$kL=npi$

so

k_n=rac{npi}{L}

where

$n=1,2,3,ldots$

The allowed wavelengths satisfy

lambda_n=rac{2L}{n}

Only standing waves that fit in the box are allowed.

Energy levels

Using $E=hbar^2k^2/(2m)$, the allowed energies are

E_n=rac{n^2pi^2hbar^2}{2mL^2}

or equivalently

E_n=rac{n^2h^2}{8mL^2}

The energies are discrete, not continuous. Larger boxes have more closely spaced levels, while smaller boxes have widely separated levels.

Normalized eigenfunctions

The normalized wavefunctions are

ight)$$ for $0<x<L$. These states are orthonormal: $$int_0^Lpsi_m^*(x)psi_n(x)dx=delta_{mn}$$ ## Zero-point energy The lowest state has $n=1$, not $n=0$. Its energy is $$E_1=rac{pi^2hbar^2}{2mL^2}$$ This nonzero ground-state energy is called zero-point energy. A particle confined in a box cannot have zero kinetic energy because that would require definite zero momentum and complete delocalization, conflicting with confinement. ## Nodes and probability The $n$th eigenstate has $n-1$ interior nodes. Higher-energy states oscillate more rapidly and have shorter wavelengths. The probability density is $$|psi_n(x)|^2=rac{2}{L}sin^2left(rac{npi x}{L} ight)$$ A position measurement is more likely where this density is larger. ## The big idea The infinite square well shows how confinement creates quantization. Boundary conditions allow only certain standing waves, producing discrete energies proportional to $n^2$. The model introduces eigenstates, nodes, orthogonality, and zero-point energy in the simplest possible setting.