ABOUT QUANTUM WELLS AND QUANTUM BARRIERS

Flavio Aristone

Institute of Physics – Federal University of South Mato Grosso, Brazil

79070-900 Campo Grande, MS – Brazil

ABSTRACT

A particular discussion of the ground state energy level and the corresponding wave

function shape is discussed for quantum well systems aiming to provide a better

comprehension of such confinement effects for quantum mechanics beginners. The results

have been numerically obtained from the solution of one dimensional time independent

Schrödinger equation for a confined space inside which either a barrier or a quantum well is

grown. The wave function and energy level for the ground state is extensively analyzed in

both scenarios, when the inserted barrier height increases considerably and when the

inserted quantum well becomes extremely thin and deeper simulating a delta-doped

system. The physics interpretations of these results are particularly rich and may become

useful to undergraduate students not only of physics courses but also for other domains

where structural concepts of quantum theory are necessary, required or wanted.

KEYWORDS: Quantum well, quantum barrier, wave function, Eigen state of energy,

confinement effects, delta-doped potential.

I. INTRODUCTION

Quantum wells and quantum barriers are important examples frequently used to

introduce quantum effects before getting into the postulates and the specific mathematics

tolls necessary to develop a more formal discussion of quantum mechanics. Actually,

quantum wells and quantum barriers are routinely discussed during classes of Modern

Physics even if the Schrödinger equation has to be imposed without further explanations.

Students of physics courses normally don’t have enough time to completely absorb the

first ideas introduced by teachers of quantum mechanics or modern physics when they are

presented to quantum effects. Normally quantum mechanics is presented as something

describing effects that are not allowed to occur in the classical world as, e.g. the tunneling

effect. Another startling feature appearing already at the first contact and associated with

quantum effects is the separation of the energy values in discrete levels that a particle is

allowed to assume, separated by inaccessible ranges of values that the particle cannot

access.

Quantum wells and quantum barriers are particularly interesting since the Schrödinger

equation can be analytically solved at every coordinate point due to the flat regions where

the potential energy is constant. Some special careful calculations must be taken at the

interfaces where the potential energy changes abruptly though, implying that the continuity

of the wave function and of its derivative have to be preserved.

The particular interest discussed in this paper is to start with a confined region leading

to confined states, actually a quantum well and slowly grow in the middle of this region

another potential energy, either an attracting well or a scattering barrier, to analyzed the

effect of such structures on the original wave function and ground state energy. The

consequence is clearly to identify the tunneling process occurring through the barrier in the

first case and notice the confinement effect that happens due to the quantum well in the

second case.

II. SCHRÖDINGER EQUATION FOR A CONFINED REGION

The Schrödinger equation may be presented in different ways. Actually the most formal

introduction presents it as the 6th postulate of quantum mechanics theory [1]. However, this

process doesn’t carry any especial understanding for this paper and we prefer to avoid such

a description in order to discuss a more direct approach directly applicable to our interests.

The system we will study is time independent as the involved potentials, the quantum

barrier and/or the quantum well don’t change as a function of time. Therefore the specific

Schrödinger equation is simply:

(1)

where are the Eigen states of the system; are the Eigen energies. is the Hamiltonian

operator describing the total energy of the system given by the sum of kinetic ( ) plus

potential ( ) energies, or:

(2)

Equation (1) looks very similar to the Hamilton-Jacobi formalism of Classical

Mechanics excepted by the presence of the new term described as the quantum state of

the particle / system. The “state” of the system is not discussed in classical theory as

classical particles are simply particles without any further discussion about it. They may be

little spheres, cubes, or simply dots having all necessary physical characteristics treated as

parameters. Equation (1) may therefore be seen as a supplementary discussion wherein the

particle, from now on called quantum system, must be somehow described before any

calculations to consider other mechanical property.

The very first conclusion is that equation (1) is completely equivalent to a problem of

matrix diagonalization since is an operator and is a number. This interpretation is

absolutely correct and is called Heisenberg formalism. Nevertheless it is also possible to

expand equation (1) as a second order differential equation, in which case the process is

called Schrödinger formalism.

It is beyond the scope of this paper to describe how equation (1) transforms into the

usual and most common form of Schrödinger equation represented in equation (3) for a one

dimensional system:

(3)

where is the Planck constant divided by and is the mass of

the particle.

Up to this point everything is plausible even if not entirely justifiable as some critical

mathematical steps have been avoided. Actually, equation (3) is frequently exhibited in

textbooks prior to the introduction of the quantum mechanics theory postulates. The first

postulate determines the meaning of , defining such a function as the mathematical

description of a quantum state. An easier physical picture of is to understand it as a

function whose modulus square represents the density of probability of finding the particle

(or the quantum state). Therefore, given the value of is the probability of

finding the particle around position within the interval .

Equation (3) is easily solved for , i.e., a constant value. An ideal quantum

well is defined by a region of constant potential surrounded by barriers of infinity height

where the wave function must not exist. Realistic quantum wells are limited by barriers of

elevated but finite values wherein the wave function may penetrate even though it must

vanish rapidly. It is this condition of confinement in both cases that imposes the

discretization of the energy levels accessible to the quantum particle.

It is not extremely complicated to solve numerically equation (3). Some typical

results are exhibited and discussed in this paper as examples. Initially we compare the

ground state energy and wave function for both cases: when the barriers of potential

energy of confinement are infinite and when they are limited to fixed values. In the second

case the wave function penetrates the forbidden region and the ground state of quantized

energy is smaller than the values found out previously.

The results expressed in this paper have been obtained through the solution of

Schrödinger equation treated generally. An indisputably confirmed numerical method, the

fourth order Runge-Kutta method [2], is applied to solve the Schrödinger equation of a

confined system limited between 0 (zero) and . Outside these limits the potential energy is

assumed to be infinite, where as a result the wave function is null. Actually, this first analysis

defines exactly a typical infinite quantum well whose width is . Conversely, when the

barriers are not infinite, the wave function is not zero at the edges even though it must

vanish. To find out the Eigen state analytically it is necessary to match the wave function

and its derivative in both edges. Numerically this is not a real concern as the function and its

derivative are continuous by construction.

The numerical processes are not trivial but neither they are of extreme complexity.

However, we opted for presenting the numerical solutions as the results may be more

interesting, easily repeatable and even implemented by any person with minimal computer

skills [3]. Our computer program has been written in FORTRAN 95 and it is accessible to

anyone on demand.

The computational details employed to obtain all present data are not discussed in

this article as the complications involved are quite limited. The main interest is to discuss

the physics interpretation of wave functions and density of probability arising from the

proposed systems. Further questions about the numerical and computational specificities

are encouraged to be sent directly to the author who shall gladly provide all the information

demanded.

To make equation (3) more appropriate to work numerically with, it is convenient to

rewrite it entirely in a units free system. Two parameters of scale are necessary and we opt

to rewrite equation (3) in terms of a dimensionless position variable that is proportional to

, i.e., . The second parameter suitable to be treated in terms of a

dimensionless value is the energy that we write as being proportional to , i.e.,

. Consequently the potential energy ought also to be written as for sake

of direct comparison. Actually, equation (3) becomes entirely units free when it is rewrote in

terms of the dimensionless variable and the dimensionless parameter . Therefore, the

two parametrical values connecting laboratory units and the results presented here are the

system or particle mass and the frequency that is determined once is fixed.

The energy levels of ideal quantum wells for this free of units system units described

previously are:

(4)

The complete values are immediately regained simply taking into consideration that the real

size of the quantum well in units of length is .

0 1 2 3 40,00

0,15

0,30

0,45

0,60

0,75

0()

|0()|²

0=0.30845

Figure 1. Wave function in black and density of probability in red for the ground state of

energy of an ideal quantum well, i.e., when the limiting barriers of confinement are

described as assuming infinite values, which obligates the vanishing of at the edges.

The marked area in the figure represents the infinite energy potential barriers.

In Figure 1 it is represented the normalized wave function and the normalized

density of probability for the lowest energy level, also called the ground state level.

For both and the barrier of potential energy is infinite and therefore the wave

function must be null out there. The discontinuities of

occurring at and are

“authorized” only because the potential is assumed to be infinite for these values. The

energy found in our system of units is indicated in the Figure and corresponds exactly to the

theoretical value obtained in equation (4).

-1 0 1 2 3 4 50,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

V0(

)

0(

)

0()

|0()|²

0=0.22947

Figure 2. Wave function and density of probability for the ground state of energy of a real

quantum well, i.e., when the limiting barriers of confinement are limited, allowing then the

penetration of the function into the normally forbidden region, represented by the shadow

areas of the figure.

In figure (2) we represent the same ground state: wave function, density of probability

and energy level for a system whose confinement is given by high but finite barriers. These

barriers are considered to exist from to and from to . The Eigen value of energy

decreases when compared to the precedent ideal case. The energy level is considerably

lower than the previous one. The penetration of the wave function is responsible for such

an effect. The infinite barriers raise the energy level by approximately 34% in this particular

example. In this case the wave function must vanish for infinite values, positive and

negative, of , i.e. of . The energy gaps between Eigen levels also decrease and for energies

higher than 5.0 units the spectrum is continuum again, since there is no more confinement.

The limited length represented in the figure from -1.0 to +5.0 provide only a reference as it

can be seen from the fact that the wave function is not yet zero for those values.

III. CONFINEMENT WITH AN INSIDE BARRIER

The first specific case to be discussed is that of a quantum barrier inside a quantum well.

Actually, the “external” quantum well is useful to provide the initial confinement of the

particle physical states resulting in quantized energy levels. Let us give some numbers to the

system in order to get quantitative rather than only qualitative arguments.

The barrier inside the quantum well is located at the center of the well and its length is

measured as a percentage of the well width. Its height is represented by as described in

the precedent paragraph. Two limit situations regarding are easier to analyze, it is the

case when and . When the condition is trivial and simply reports to

the isolated quantum well energy levels and Eigen functions. When the solution is

also trivial as it reports to the situation when there are two separated quantum wells

instead of only one within a barrier inside. The two quantum wells have widths equal the

quantum well width minus the barrier width. As the barrier is initially symmetrically located

at the center of the well, the new quantum wells are also symmetric. The energy levels of

each well are identical and the wave functions get obligatory null values at the both

borders. The energy level values are exactly the same as the values calculated for a single

quantum well having the same width given by the region where the energy is zero.

The interesting analysis arises then when the situation of the introduced barrier is

intermediary. In order to make the results clearer the study ought to be separated in two

parts, the first one considering a constant width barrier and variable potential height. The

second part takes into account a fixed barrier height while its width is modified. It is very

didactic to spend time investigating the energy levels and wave functions that arise from

these calculations.

For sake of simplicity we will keep the width of this new barrier constant and equal to

of the initial quantum well width, which means in our calculation that the inside

barrier width is equal . Nevertheless the barrier height will be augmented from to

a limit considered much higher than the first value of the ground state confined energy,

.

0 1 2 3 40,00

0,15

0,30

0,45

0,60

0,75

V0=0.0

V0=2.0

V0=4.0

V0=8.0

V0=16.0

V0=32.0

V0=56.0

0(

)

0 1 2 3 40

10

20

30

40

50

60

V0(

)

Figure 3. Wave functions of the ground state for different values of the barrier energy height

(a). In part (b) the different barriers are represented in the center of the well

Obviously nothing changes while the new potential energy is null. Conversely, both the

wave function and the energy level are modified as the new barrier height increases. In

figure 3 it is plotted two families of curves, representing the wave functions and the

inserted barrier. The dot-dashed lines are the wave functions and the continuum lines

represent the barriers of potential energy.

The first effect to notice is the deformation of wave functions as the barrier height

increases, reducing the probability of presence of the particle in the barrier region. The

Eigen values of energy increases as a function of the barrier height and the ground state

energy is smaller than all barriers represented in figure 3, which means that for all situations

the particle may be found in the barrier region that is classically forbidden. However, it is

evident that the barrier acts like a scattering center since the probability of finding the

particle is clearly moving away at both left and right directions.

Actually, it becomes also clear another limiting situation, when the new barrier is

extremely high. The wave function in this case almost splits completely in two quasi-

separated parts, each one centered exactly in the middle of the new free regions at left and

right. In fact, the “old” quantum well has been divided in two new quantum wells, limited by

the former infinite barriers and by the new inserted barrier.

The Eigen energy obtained for the highest barrier coincides exactly with the expect value

calculated for an infinite quantum well of width equal to the half of the previous one

diminished by 5%, which is half of the inserted barrier width. In figure 4 we represent the

Eigen energy for the ground state as function of the potential barrier height. The straight

line is the analytical Eigen energy for a quantum well of width limited by infinite

barriers.

0 10 20 30 40 50 600,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

0

V0

Figure 4. Ground state of energy for a quantum well limited by infinite barriers and width =

4.0 within an inside centered barrier of height = V0 and width = 0.4. The blue line represents

the theoretical value of the ground state energy for a quantum well limited by infinite

barriers and width = 1.8.

IV. A QUANTUM WELL INSIDE A QUANTUM WELL

Our second consideration is to introduce a different quantum well inside the first one

instead of a barrier as in the previous session. This second well is also centered at and

its presence naturally modifies the ground state of energy and the corresponding wave

function that are analyzed as a function of this inserted quantum well deepness and width.

Even though those analyses bring interesting pieces of information of quantum systems, it is

much more interesting to study the effects induced by the presence of the second quantum

well as a function of the product of the well deepness multiplied by the well width

maintained constant. For very small widths this inserted quantum well shall behave like a

delta-doped potential energy. For sake of definition these different wells can be called as

quantum well of confinement and inserted quantum well respectively, which eventually

becomes a delta doped-like system.

Holding a constant value for the inserted quantum well width and increasing the

deepness of the potential energy value the Eigen function for the ground state becomes

more concentrate in the new well region. The ground state of energy decreases and

eventually becomes negative. Therefore, it is like both the energy values and the wave

function enter the new quantum well, even though the probability to find the particle in the

region of the old quantum is still non-negligible.

An equivalent analysis arises when the potential energy of the new quantum well is kept

constant, e.g. , and its width is modified, from to . Both the energy level and the

wave function will end inside the new quantum well at some point. When the system

is absolutely identical to the pervious one and when too except that the bottom line

of energy is no longer but instead. The wave functions and density of probability are

exactly the same for these two cases.

In spite of these transitions described above and all the adjustment suffered by the

wave function, the comprehension of the process is not extremely elaborated.

Nevertheless, the situation changes a bit when a new condition is practiced. Instead of

keeping either the well width constant or its potential energy constant, we investigate what

happens to the ground state of energy and its corresponding wave function when the

product well width multiplied by the potential energy intensity is kept constant, while

varying them both. The different relationship connecting energy and width can be visualized

in figure (5).

0 1 2 3 4

-800

-700

-600

-500

-400

-300

-200

-100

0

V0

(A)

1,95 1,96 1,97 1,98 1,99 2,00 2,01 2,02 2,03 2,04 2,05

V0=-800

V0=-400

V0=-200

V0=-100

(B)

Figure 5. Schematic representation of the potential energy considered to solve the

Schrödinger equation. The system is limited by infinite barriers for and as usual

and it is free in between except these additional quantum wells. The product is kept

constant and equal 4 in our work.

In the part (A) of figure (5) we see all the different quantum wells at once and it is clear

then that the pattern converges to a delta doped – like system. In part (B) it is represented a

magnification of the central part of the well to make clear the meaning of description.

An important analysis arises when is kept small and constant, typically 2%, and gets

deeper until the ground state is located inside this tiny confinement, meaning that the Eigen

energy for is negative. The corresponding wave function is represented in figure (5) and

shows a much localized form. Actually, it is possible to obtain the analytical function when

the potential energy is described as a -function obeying its natural mathematical

properties. The resulting wave function is given by decaying exponential centered at the

origin of the and the energy of confinement is given by:

(5)

where is the parameter characterizing the function, or more appropriately saying, the

distribution. In the system of units assumed here to solve Schrödinger equation the energy

is therefore equal to

.

In order to exhibit some numerical results a quantum well composed of the following

characteristics: energy potential depth and width = has been considered. The

product is kept constant and equal 4.0. Making very small implies to increase

considerably the intensity , which is one particular form of definition of -function.

0 1 2 3 4

0,00

0,25

0,50

0,75

1,00

1,25

1,50

0

1

2

0(x

)|²

Numerical data

Analytical result

0(x

)

x

Figure 6. Wave function and density of probability (in the small frame) for the confined state

of a -doped system. The red line represents the analytical solution of an ideal system while

the dots represent the numerical results for our simulation of a potential.

In figure (6) it is represented the wave function for the ground state of an analytical -

doped potential energy as the straight line. The numerical solution of Schrödinger equation

for a square well potential extremely thin and deep: and respectively

is represented by the sequence of dots. The coincidence of both curves is remarkable, even

though is noticeable the little deviation close to both edges. The numerical solution is

constructed with the obligation that the wave function vanishes at and while

the analytical solution is described by a pure decaying exponential, which may assume very

small values but not zero.

-800 -600 -400 -200 0

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

0

V0

Figure (7). Variation of the ground state level of energy as a function of the potential energy

depth for a quantum well whose width is also variable but the product is kept

constant and equal 4, simulating a -doped system. The system is composed of an external

quantum well limited by infinite barriers of potential at 0 and 4 inside which another

quantum well is centered at the middle distance.

In figure (7) the dots represent the ground state of energy as a function of the

potential depth maintaining the product constant and equal 4, which classifies

the -function. The bottom straight line is the level of energy for an analytical -doped

potential while the top straight line is the level of energy for the free quantum well system

limited by infinite barriers at and .

Finally, in figure (8) it is exhibited three different Eigen functions obtained while

calculating the points of figure (7) to exemplify its behavior. The shape of the wave function

changes drastically from the free quantum well to the -doped like system. An intermediary

condition is also shown. The system goes from completely delocalized to very localized as

the potential depth increases.

0 1 2 3 4

0,0

0,5

1,0

1,5

2,0

0,0

0,4

0,8

1,2

1,6

2,0

2,4

2,8

3,2

3,6

4,0

0(x

)|²

V0=0.0

V0=-20.0

V0=-800.0

Figure 8. Three different Eigen functions for different values of the potential depth placed as

a quantum well at the center of the system.

In figure (9) it is shown that the Eigen values are correctly calculated for any value of

parameter that classifies de -doped function, expressed in equation (5).

0 1 2 3 4 5 6

-20

-15

-10

-5

0

0

Numerical

Analytical

Figure 9. Ground state of energy or energy of confinement for a single -doped quantum

well characterized by the parameter . The dependence is expressed in equation (5) and

verified numerically as indicated in the figure by the dots following the straight line.

V. CONCLUSION

In this paper the Schrödinger equation independent of time has been numerically solved

inside a quantum well delimited by confining barriers of infinite potential height. The

common situation is defined by a constant potential inside the well whose analytical

solutions are easily found. The ground state energy has been obtained and the

corresponding wave function and density of probability are exhibited. It has also been

shown the effects of changing the barriers from infinite to limited values. The wave function

then penetrates the forbidden region and the energy level is lowered.

The study is more interesting when a perturbation potential is placed inside the well.

This situation is divided in two different cases. The first study is the analysis of a quantum

barrier placed symmetrically at the center of the precedent well width. The ground state

energy is studied as a function of this new barrier potential energy height. The evolution of

these functions as the potential energy for the inside barriers increases is actually

interesting for a didactical point of view. It is clear visible that the new barrier acts as a new

confining condition, separating the previous wave function in two distinct parts. At the limit

of a very high inside barrier, the ground state converges to the expected value of a single

quantum well for which the width is defined by the distance between the previous infinite

barrier and the inside confinement barrier. For a larger inside barrier the ground state

converges quicker as the tunneling effect is reduced.

The second case studied in this paper is the situation of a confining potential placed

inside the initial quantum well. This confining potential is described as a quantum well

centralized in the middle of the previous well, which is always limited by infinite height

barriers. For this specific case a particular control is observed, the product of the new

quantum well energy depth multiplied by its width is maintained constant. This is an initial

simulation of the so-called delta-doped quantum well, which is an electronic device made,

e.g., of Gallium-Arsenide within a single monolayer of doped Silicon. A potential energy in

the form of a function presents a confined level of energy easily calculated from the

Schrödinger equation. However, the procedure presented in this paper is highly educational

as it is possible to actually “see” the wave function getting inside the new confining

potential, as if it is falling down inside the attractive shape of the potential energy.

The results presented in this paper are not exactly new from any research point of view

but they have been used to teach introductory quantum mechanics to physics students at

two different courses: computational physics 2 and quantum mechanics 1. Apparently the

students appreciate seeing the modification of the wave function as the potential inside the

initial confined quantum well is customized. The simplicity and reliability of the numerical

methods are powerful instruments to facilitate the comprehension of the results.

VI. ACKNOWLEDGMENTS

This work has been financially supported by CNPq under contract 407840/2013-3. CAPES

and Fundect/MS must also be acknowledged. It is very important to thank Ms. Victoria Tangi

for careful English editing and text revision.

VII. REFERENCES

[1]. Claude Cohen-Tannoudji, Bernard Diu and Franck Lalöe. Quantum Mechanics Vol. 1,

1977.

[2]. Milton Abramowitz and Irene Stegun. Handbook of mathematical functions, 1972.

[3]. William H. Press, Saul A. Teukolsky, William T. Vettering and Brian P. Flannery.

Numerical Recipes in Fortran, 1992.