Particle
In A Box

by
Kristen Adams

**Discussion
of Concept**

Part A: Why do we
care about examining a particle in a box?

A particle in a box resembles an electron in a stable orbit around a
nucleus. Such an electron exhibits a standing wave pattern much like the
standing wave pattern that can be produced on a string that is fixed at both
ends.

Figure
1 Diagram of deBroglie matter waves of an electron in a stable orbit.

Source of figure: http://online.cclt.org/physicslab/content/PhyAPB/lessonnotes/dualnature/deBroglie.asp

The particle in a box is free (there are no forces acting upon it) but
is limited spatially. We can use this model to examine and define the wave that
an electron makes in orbit around a nucleus. Once we know the wave function of
a particle we can then find the energy and momentum of the particle. Models,
such as this one, can aid us in interpreting data gathered from actual
experiments.

Part B: How does a particle in a box behave?

There are certain requirements of behavior for a particle in a box. We
can visualize these requirements by again looking at the behavior of a standing
wave on a string.

Figure 2 A standing
wave with points of minimum amplitude (nodes) and maximum amplitude (antinodes)

Source of
figure: http://www.cord.edu/dept/physics/p128/lecture99_35.html

A standing wave must meet two conditions. 1) At both ends of the string,
a standing wave exhibits a node (point of zero amplitude). 2] The length of the
standing wave is broken up into an integral number of half-wavelengths. The
matter wave of a particle in a 3-D box has these same characteristics.

A particle starts from one side of the box at zero amplitude, hits the
opposite side of the box (also at zero amplitude) and must return to its
starting point, continuing the pattern. To describe the wave of the particle we
must find a wave function that properly describes the motion of the particle.
What kind of wave starts at zero amplitude and ends at zero amplitude? A sine wave! The matter wave of a particle
inside a box is a sine wave just as the standing wave on a string is a sine
wave. We have found a wave function that meets the first condition (from above)
which is sin x. The wave function of form sin x should describe the wave at any
point x in 1-D. If our box is 3-D, our wave function would be of the form
sin(x)sin(y)sin(z) and would describe the wave at any point (x,y,z) in 3-D.

In order for the sine wave to be at a point of zero amplitude at each
side of the box, the length of the sine wave in the box must be limited to ½ a
wavelength, 1 wavelength, 1 ½ wavelengths, etc. which can be more succinctly
written as n/2 wavelengths where n is 1, 2, 3, etc. We need to include this
characteristic in our wave function. Defining the wave number, k (the number of
radians of the wave cycle per unit length), to be (where is 180 degrees of the wave or one-half of a
full wave and L is the length of one side of the box) we restrict
the number of wavelengths in the box to an integer number of half wavelengths.
Our wave function is now of the form or sin(kx). This wave function meets both
conditions one and two.

**Figure 3** Infinite potential well with wavelengths

Source of figure: http://hyperphysics.phyastr.gsu.edu/hbase/quantum/pbox.html

The final step in properly defining the wave function of a particle in a
box is to normalize the wave function. The probability of finding a particular
particle in all space is 1 (the
particle exists). [Aside: *Probability is a range from 0 to 1 where 0 means
that a particular event will not occur and 1 indicates that a particular event
is certain to occur.*] A quantum mechanical property of a wave function is
that the probability of finding a particle at some point in space is the
absolute value of the wave function squared at the point of interest (x, y, z).
The sum of the probabilities over all points in space should equal one. Thus,

where is our wave function in 3-D

In order for our wave function to meet this requirement we must tack on a
coefficient (normalization coefficient). Our normalized wave function for a 1-D
box is Asin(kx) where A is
our normalizing coefficient and k = n/L where n=1,2,3,
For a 3-D box, our
normalized wave function would be Asin(kx) Bsin(ky) Csin(kz) with A,B,C acting as
normalizing coefficients.

Notice that
the requirements needed to produce a wave function suitable to a box has
produced quantization. The wave number (k) increases by , but no values in between. Whereas an unbounded free particle
has no such restrictions on k. Likewise,
the energy of a particle, which depends on k, is also quantized for a particle
in a box and continuous for an unbound free particle.

The particle in a box problem will be solved with more detail in the
next section.

** **

This example will illustrate a method of solving the 3-D Schrodinger equation to find the eigenfunctions for a infinite potential well, which is also referred to as a box.

A particle of
mass m is captured in a box. This box can also be thought of as an area of zero
potential surrounded by walls of infinitely high potential. The particle cannot
penetrate infinitely high potential barriers. The box is of length a along the
x axis, length b along the y axis and length c along the z axis.

This potential
is described as follows:

V(x,y,z)=0 if 0<x<a ,
0<y<b, 0<z<c Region
I

V(x,y,z)= elsewhere Region II

**Figure 4** Slice of 3-D infinite potential well

Source of figure: http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/models/piab1/piab1prb.htm

The energy
operator, , (the quantum mechanical operator we will use to find the
energy of the particle) for a single particle of mass m, in a potential field
V(x,y,z) is:

[Eq-1]

where is the momentum operator and is equal to

ASIDE**:**

*Total
Energy = Kinetic Energy + Potential Energy*

*In
symbolic form: E = T + V . In operator
form: **.*

*Kinetic
Energy (T) =** ** *

* *

* *

The
time-independent Schrodinger equation, which is used to find the possible
energies, E, that the particle may have, is

[Eq-2]

is called the eigenfunction. It is the wave function that satisfies Eq-2.

Eq-2
is often called an eigenvalue equation. The eigenfunction for Eq-2 is to be
found. This eigenfunction is used to determine the energies possible for the situation
(generically referred to as the eigenvalues).

In
Region II, V is infinite and the Hamiltonian, , is
infinite.

The wave function
and the energy are finite. Thus is zero in this

region. Since , there is zero probability that the particle will be found
in this region.

__ __

In Region I, V
is zero and the Hamiltonian is purely kinetic.

Plugging this
Hamiltonian into the eigenvalue equation [Eq-2], we find

[Eq-3]

The subscript n is used in anticipation of discrete values that depend upon some integer n.

Now we must
find a wave function that solves the eigenvalue equation, where is a
number.

The wave function
must be continuous across the regions. Therefore, at the walls, the wave
function inside the box must equal the wave function outside the box. Since we
have already determined that the wave function outside the box must be zero,
the wave function inside the box must go to zero at the walls.

**Figure 5** Slice of 3-D infinite potential well

Source of figure: http://scienceworld.wolfram.com/physics/InfiniteSquarePotentialWell.html

The walls of
the box are located at x=0 and x=a; y=0 and y=b; z=0 and z=c. Thus,

[Eq-4]

** **

Since the wave
function does not exist beyond the walls of the box (created by the infinite
potential barrier) we are not concerned with the behavior of the wave function
across the barrier. However, in other types of problems where the wave function
does exist on the other side of the barrier, we need to make sure that the
first derivative of the wave function is continuous across the barrier. [See
worked example B for an example of this procedure.]

Back to our
time-independent Schrodinger equation [Eq-3],

+ 0

The general solution to this homogeneous differential equation in 1-D is

+ where [Eq-5]

Our boundary
conditions [Eq-4] tell us that at x=0, 0. Therefore, B must equal 0 and . The same applies for the 3-D case.

** **

[Eq-6]

A is the normalization coefficient and the superscripts (1), (2), (3) signify that each dimension is independent.

From our
boundary conditions [Eq-4] we must also make the wave function equal zero at
x=a, y=b and z=c. The sine function is zero at integral multiples of . Therefore,

an; b n; c n where n = 0,1,2,3

The n subscript on k has been dropped to improve clarity, but is technically still there.

Rearranging gives; ; [Eq-7]

We can now
plug k into Eq-5 to find the possible values of energy for a particle in this
box. Rearranging Eq-5 to solve for E gives

[Aside: ] Substituting in values for k

from Eq-7 one finds .

As a final step, we must normalize our wave function.

Our wave function [Eq-6] is .

To normalize, 1

This becomes 1

which yields

The eigenenergies, , and the normalized eigenfunctions, , for the 3-D box problem are:

where

This example will illustrate a method of solving the 1-D Schrodinger equation
to find the eigenfunctions for a finite potential well. The potential is
defined as follows:

** **V(x)= 0 if x<-a Region
I

V(x)=-Vo if Region II [Eq-8]

V(x)= 0 if x>a Region III

If E is
greater than 0, the wave function is unbounded as there are no boundary
conditions that would place any limitations on the wave function.

If E is less
than zero, boundary conditions are imposed upon the wave function. This is the
case that will be examined below.

**Figure 6** Finite potential well

Source of figure: http://scienceworld.wolfram.com/physics/FiniteSquarePotentialWell.html

Starting with
the time-independent Schrodinger equation:

where

Applying the conditions
of Eq-8, inside the well, the Schrodinger equation is:

for Region II for Regions I and III

Let and .

Inside the well,
the wave function has the general solution:

.

Outside the well, the general solution to the Schrodinger equation is: .

Now we must
apply boundary conditions to ensure that the composite wave function behaves
properly as it crosses boundaries.

There are
several boundary conditions must be imposed upon our wave function.

1. The
wave function for the particle outside the well must show that the likelihood
of finding the particle in these regions (I and III), decreases as the distance
into the barrier region increases.

2.
The wave functions must be continuous across the boundaries.

3.
The first derivatives of the wave function must be continuous across the
boundaries.

Applying the first
boundary condition, the wave function must decrease as the distance into the
barrier increases. For Region I (x is negative) . Likewise, in Region III (x is positive) .

Applying the
second and third boundary conditions to the boundary between Regions I and II,
where x = -a,

Aside:

sin(-A)=-sin(A)

cos(-A)=cos(A)

[Eq-9]

[Eq-10]

Applying the
second and third boundary conditions to the boundary between Regions II and
III, where x = a,

[Eq-11]

[Eq-12]

There now
exists a set of four equations ([Eq-9] - [Eq-12]) for our four unknown coefficients A, B, C, D. We can
solve for the coefficients through addition and subtraction of these equations
and normalization.

Adding [Eq-9]
and [Eq-11] produces

[Eq-13]

Adding [Eq-10]
and [Eq-12] results in

[Eq-14]

Subtracting
[Eq-11] from [Eq-9] gives

[Eq-15]

Subtracting
[Eq-12] from [Eq-10] yields

[Eq-16]

Dividing
[Eq-16] by [Eq-13] results in .

Inserting this
into [Eq-15] we obtain

In order for
this equality to be true, A=0 and C=D.

Substituting
these values into [Eq-13] and [Eq-15] (equations matching the wave functions at
the boundaries) we get

[Eq-13]

[Eq-15] 0

Thus we get an
even function inside the well.

Dividing
[Eq-14] by [Eq-15] results in .

Inserting this into [Eq-15] we obtain .

In order for this
equality to be true, B=0 and C=-D.

Substituting
these values into [Eq-13] and [Eq-15](equations matching the wave functions at
the boundaries) we get

[Eq-13]

[Eq-15] 0

Thus we get an
odd function inside the well.

The precise
values for the coefficients are found through normalization:

** **

** **

** **

**Helpful
References**

** **

- Particle
in Box: http://www.chem.uci.edu/education/undergrad_pgm/applets/dwell/dwell.htm
- Particle
in 2-D Box
__: http://www.falstad.com/qm2dbox/__ - Particle
in Finite Well:

__http://webphysics.davidson.edu/physletprob/ch10_modern/finitewell.html__

__ __

__ __

- Finite
Square Well, explanation:
__http://musr.physics.ubc.ca/~jess/p200/sq_well/sq_well.html__ - Finite
Square Well, even and odd solutions, parity: http://hyperphysics.phy_astr.gsu.edu/hbase/quantum/pfbox.html

__ __

__ __

- Elementary Level

Bernstein, J., P. Fishbane, S.
Gasiorowicz. *Modern Physics*.__ __(Upper Saddle River, NJ: Prentice
Hall, 2000).

Serway, Raymond. *Physics : For
Scientists and Engineers*, 4^{th} ed.(Philadelphia: Saunders College
Publishing,1996).

Transnational College of LEX;
translated by John Nambu. *What is Quantum Mechanics? A Physics Adventure*.
(Boston: Language Research Foundation,1996).

- Intermediate Level

Greiner, Walter. *Quantum Mechanics:
an Introduction*. (New York: Springer, 2001).

Hecht, K.T. *Quantum Mechanics*.
(New York: Springer, 2000).

Liboff, Richard. *Introductory
Quantum Mechanics*. 4^{th} ed. (New York: Addison-Wesley, 2003).

Singh, Jasprit. *Quantum Mechanics:
Fundamentals and Applications to Technology*. (New York: John Wiley and
Sons, 1997).

Thankappan, V.K. *Quantum Mechanics*,
2^{nd} ed. (New York: John Wiley and Sons, 1993).

Zettili, Nouredine. *Quantum
Mechanics: Concepts and Applications*. (New York: John Wiley and Sons,
2001).

__ __