# Finite potential well

The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite potential walls. Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling).

## Particle in a 1-dimensional box

For the 1-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as:

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+V(x)\psi =E\psi \quad (1)$ where

$\hbar ={\frac {h}{2\pi }}$ ,
$h\,$ is Planck's constant,
$m\,$ is the mass of the particle,
$\psi \,$ is the (complex valued) wavefunction that we want to find,
$V\left(x\right)\,$ is a function describing the potential energy at each point x, and
$E\,$ is the energy, a real number, sometimes called eigenenergy.

For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to a value $V_{o}$ at x = -L/2 and x = L/2. The wavefunction is considered to be made up of different wavefuctions at different ranges of x, depending on whether x is inside or outside of the box. Therefore the wavefunction is defined such that:

$\psi ={\begin{cases}\psi _{1},&{\mbox{if }}x<-L/2{\mbox{ (the region outside the box)}}\\\psi _{2},&{\mbox{if }}-L/2L/2{\mbox{ (the region outside the box)}}\end{cases}}$ ### Inside the box

For the region inside the box V(x) = 0 and Equation 1 reduces to

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{2}}{dx^{2}}}=E\psi _{2}.$ Letting

$k={\frac {\sqrt {2mE}}{\hbar }},$ the equation becomes

${\frac {d^{2}\psi _{2}}{dx^{2}}}=-k^{2}\psi _{2}.$ This is a well-studied differential equation and eigenvalue problem with a general solution of

$\psi _{2}=A\sin(kx)+B\cos(kx)\quad .$ Hence,

$E={\frac {k^{2}\hbar ^{2}}{2m}}.$ Here, A and B can be any complex numbers, and k can be any real number.

### Outside the box

For the region outside of the box, since the potential is constant, V(x) = $V_{o}$ and Equation 1 becomes:

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{1}}{dx^{2}}}=(E-V_{o})\psi _{1}$ There are two possible families of solutions, depending on whether E is less than $V_{o}$ (the particle is bound in the potential) or E is greater than $V_{o}$ (the particle is free).

$k'={\frac {\sqrt {2m(E-V_{o})}}{\hbar }}$ produces

${\frac {d^{2}\psi _{1}}{dx^{2}}}=-k'^{2}\psi _{1}$ with the same solution form as the inside-well case:

$\psi _{1}=C\sin(k'x)+D\cos(k'x)\quad$ This analysis will focus on the bound state, where $V_{o}$ > E. Letting

$\alpha ={\frac {\sqrt {2m(V_{o}-E)}}{\hbar }}$ produces

${\frac {d^{2}\psi _{1}}{dx^{2}}}=\alpha ^{2}\psi _{1}$ where the general solution is exponential:

$\psi _{1}=Fe^{-\alpha x}+Ge^{\alpha x}\,\!$ Similarly, for the other region outside the box:

$\psi _{3}=He^{-\alpha x}+Ie^{\alpha x}\,\!$ Now in order to find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A, B, F, G, H and I that satisfy those conditions.

### Finding wavefunctions for the bound state

Solutions to the Schrödinger equation must be continuous, and continuously differentiable. These requirements are boundary conditions on the differential equations previously derived.

In this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations.

Summarizing the previous section:

$\psi ={\begin{cases}\psi _{1},&{\mbox{if }}x<-L/2{\mbox{ (the region outside the box)}}\\\psi _{2},&{\mbox{if }}-L/2L/2{\mbox{ (the region outside the box)}}\end{cases}}$ $\psi _{1}=Fe^{-\alpha x}+Ge^{\alpha x}\,\!$ $\psi _{2}=A\sin(kx)+B\cos(kx)\quad$ $\psi _{3}=He^{-\alpha x}+Ie^{\alpha x}\,\!$ Next, we know that the overall $\psi \,\!$ function must be continuous and differentiable. In other words the values of the functions and their derivatives must match up at the dividing points:

$He^{-\alpha L/2}=B\cos(kL/2)$ $-\alpha He^{-\alpha L/2}=-kB\sin(kL/2)$ so taking the ratio gives

$\alpha =k\tan(kL/2)$ .

Similarly for the antisymmetric case we get

$\alpha =-k\cot(kL/2)$ .

Recall that both $\alpha$ and $k$ depend on the energy. What we have found is that the continuity conditions cannot be satisfied for an arbitrary value of the energy. Only certain energy values, which are solutions to one or other of these two equations, are allowed. Hence we find, as always, the bound-state energies are quantized.

${\sqrt {u_{0}^{2}-v^{2}}}={\begin{cases}v\tan v,&{\mbox{(symmetric case) }}\\-v\cot v,&{\mbox{(antisymmetric case) }}\end{cases}}$ In the plot to the right, for $u_{0}^{2}=20$ , solutions exist where the blue semicircle intersects the purple or grey curves ($v\tan v$ and $-v\cot v$ ). Each purple or grey curve represents a possible solution, $v_{i}$ within the range ${\frac {\pi }{2}}(i-1)\leq v_{i}<{\frac {\pi }{2}}i$ . The total number of solutions, $N$ , (i.e., the number of purple/grey curves that are intersected by the blue circle) is therefore determined by dividing the radius of the blue circle, $u_{0}$ , by the range of each solution $\pi /2$ and then rounding up the result to the nearest integer.

$N=\left\lceil {\frac {2u_{0}}{\pi }}\right\rceil$ In this case there are exactly three solutions, since $N=\lceil 2{\sqrt {20}}/\pi \rceil =\lceil 2.85\rceil =3$ .

$E_{n}={2\hbar ^{2}v_{n}^{2} \over mL^{2}}$ .

If we want, we can go back and find the values of the constants $A,B,G,H$ in the equations now (we also need to impose the normalisation condition). On the right we show the energy levels and wave functions in this case (where $x_{0}\equiv \hbar /{\sqrt {2mV_{0}}}$ ):

We note that however small $u_{0}$ is (however shallow or narrow the well), there is always at least one bound state.

Two special cases are worth noting. As the height of the potential becomes large, $V_{0}\to \infty$ , the radius of the semicircle gets larger and the roots get closer and closer to the values $v_{n}=n\pi /2$ , and we recover the case of the infinite square well.

Note: The above derivation does not consider the possibility that the effective mass of the particle could be different inside the potential well and the region outside the well.

### Spherical cavity

The results above can be used to show that, contrary to the one-dimensional case, there is not always a bound state in a spherical cavity.

The ground state of a spherically symmetric potential will always have zero orbital angular momentum, and the reduced wave function $U(r)\equiv r\psi (r)$ satisfies the equation

$-{\frac {\hbar ^{2}}{2m}}{d^{2}U \over dr^{2}}+V(r)U(r)=EU(r)$ This is identical to the one-dimensional equation, except for the boundary conditions. As before, $U(r)$ and its first derivative must be continuous at the edge of the well $r=R$ . However there is another condition, that $\psi (0)$ must be finite, and that requires $U(0)=0$ .

By comparison with the solutions above, we can see that only the antisymmetric ones have nodes at the origin. Thus only the solutions to $\alpha =-k\cot(kR)$ are allowed. These correspond to the intersection of the semicircle with the grey curves, and so if the cavity is too shallow or small, there will be no bound state.