The finite potential well (also known as the finite square well) is a concept from
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
. It is an extension of the
infinite potential well, in which a particle is confined to a "box", but one which has finite
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
"walls". Unlike the infinite potential well, there is a
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
of the particle is less than the 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
Quantum tunnelling, also known as tunneling ( US) is a quantum mechanical phenomenon whereby a wavefunction can propagate through a potential barrier.
The transmission through the barrier can be finite and depends exponentially on the barrier h ...
).
Particle in a 1-dimensional box
For the 1-dimensional case on the ''x''-axis, the
time-independent Schrödinger equation can be written as:
where
*
is the reduced Planck's constant,
*
is
Planck's constant,
*
is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
of the particle,
*
is the (complex valued)
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ma ...
that we want to find,
*
is a function describing the potential energy at each point ''x'', and
*
is the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
, a real number, sometimes called eigenenergy.
For the case of the particle in a 1-dimensional box of length ''L'', the potential is
outside the box, and zero for ''x'' between
and
. The wavefunction is considered to be made up of different wavefunctions at different ranges of ''x'', depending on whether ''x'' is inside or outside of the box. Therefore, the wavefunction is defined such that:
Inside the box
For the region inside the box, ''V''(''x'') = 0 and Equation 1 reduces to
Letting
the equation becomes
This is a well-studied
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
and
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
problem with a general solution of
Hence,
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,
and equation becomes:
There are two possible families of solutions, depending on whether ''E'' is less than
(the particle is bound in the potential) or ''E'' is greater than
(the particle is free).
For a free particle,
, and letting
produces
with the same solution form as the inside-well case:
This analysis will focus on the bound state, where
. Letting
produces
where the general solution is exponential:
Similarly, for the other region outside the box:
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 condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s on the differential equations previously derived, that is, the matching conditions between the solutions inside and outside the well.
In this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations.
Summarizing the previous sections:
where we found
,
, and
to be:
We see that as
goes to
, the
term goes to infinity. Likewise, as
goes to
, the
term goes to infinity. In order for the wave function to be square integrable, we must set
, and we have:
and
Next, we know that the overall
function must be continuous and differentiable. In other words, the values of the functions and their derivatives must match up at the dividing points:
These equations have two sorts of solutions, symmetric, for which
and
, and antisymmetric, for which
and
. For the symmetric case we get
so taking the ratio gives
Similarly for the antisymmetric case we get
Recall that both
and
depend on the energy. What we have found is that the continuity conditions ''cannot'' be satisfied for an arbitrary value of the energy; because that's a result of the infinite potential well case. Thus, only certain energy values, which are solutions to one or either of these two equations, are allowed. Hence we find that the energy levels of the system below
are discrete; the corresponding eigenfunctions are ''
bound state
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
* ...
s''. (By contrast, for the energy levels above
are continuous.)
The energy equations cannot be solved analytically. Nevertheless, we will see that in the symmetric case, there always exists at least one bound state, even if the well is very shallow.
Graphical or numerical solutions to the energy equations are aided by rewriting them a little. If we introduce the dimensionless variables
and
, and note from the definitions of
and
that
, where
, the master equations read
In the plot to the right, for
, solutions exist where the blue semicircle intersects the purple or grey curves (
and
). Each purple or grey curve represents a possible solution,
within the range
. The total number of solutions,
, (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,
, by the range of each solution
and using the floor or ceiling functions:
In this case there are exactly three solutions, since
.
and
, with the corresponding energies
If we want, we can go back and find the values of the constants
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
):
We note that however small
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,
, the radius of the semicircle gets larger and the roots get closer and closer to the values
, and we recover the case of the
infinite square well.
The other case is that of a very narrow, deep well - specifically the case
and
with
fixed. As
it will tend to zero, and so there will only be one bound state. The approximate solution is then
, and the energy tends to
. But this is just the energy of the bound state of a
Delta function potential of strength
, as it should be.
A simpler graphical solution for the energy levels can be obtained by normalizing the potential and the energy through multiplication by
. The normalized quantities are
giving directly the relation between the allowed couples
as
for the even and odd parity wave functions, respectively. In the previous equations only the positive derivative parts of the functions have to be considered. The chart giving directly the allowed couples
is reported in the figure.
Unbound states
If we solve the time-independent Schrödinger equation for an energy
, the solutions will be oscillatory both inside and outside the well. Thus, the solution is never square integrable; that is, it is always a non-normalizable state. This does not mean, however, that it is impossible for a quantum particle to have energy greater than
, it merely means that the system has continuous spectrum above
. The non-normalizable eigenstates are close enough to being square integrable that they still contribute to the spectrum of the Hamiltonian as an unbounded operator.
Asymmetrical well
Consider a one-dimensional asymmetrical potential well given by the potential
[Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.]