Partial Wave Analysis, Bessel Functions, and
Scattering in Quantum Mechanics
Noah Nall
Dr. O’Neil
December 4, 2015
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Abstract
The Schrodinger equation provides a general method of calculating energy eigen
functions and their corresponding energy levels. Within the Schrodinger equation there is a wave
function that can often difficult to understand before it is expanded as a sum over angular
momentum or other quantum numbers. The wave function can be investigated using a method of
partial wave analysis. Partial wave analysis is used throughout quantum mechanics to decompose
waves into smaller portions that combine to give one the full description of the physical system.
Within this project, this method of partial wave analysis is used to verify that for short range
potentials, only low values of l get close enough to the origin to be affected in scattering
problems.
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Table of Contents
I. Introduction …………………………………………………………………………………….4
II. Theory …………………………………………………………………………………………7
III. Results and Discussion ……………………………………………………………………...12
IV. Conclusions ………………………………………………………………………………….13
V. References ……………………………………………………………………………………14
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I. Introduction
Investigation into the wave nature of light began in the 17th century continuing to grow
and expand into what is now known as quantum mechanics or quantum theory. Quantum
mechanics is a fundamental field of physics concerned with processes involving atoms and
photons. These processes have been observed to be quantized, meaning the action of measuring
has been observed to only be in integer multiples of Planck’s constant. Thus, the energy levels
for a bound system are found to be quantized. Planck’s constant is denoted with a lower case h
and equals 6.626x10-34 Js. One common problem which is investigated within the study of
quantum mechanics is an infinite square well. This problem is one which has some potential
within given boundaries but at those boundaries the potential is infinite outside the boundaries.
In order to solve this problem one would use the Schrodinger equation.
Schrodinger developed a differential equation for the time development of a wave
function which is an operator statement that the kinetic energy plus the potential energy equals
the total energy of the system.[1] This means that he developed a way to find the energy of given
physical situations. Using the Schrodinger equation one can solve a variety of different problems
in quantum mechanics. It is especially versatile because it can be applied to both one
dimensional situations such as an infinite square well or it can be transformed using spherical
coordinates, allowing one to solve three dimensional situations. One such situation is an infinite
spherical well or situations involving scattering. For example, when scientist are studying the air
quality of a given area they will often use a high energy laser and fire it directly up into the air.
They are then able to collect data on the number of photons that are scattered directly back
toward the initial position. This allows them to measure the levels of different gases in the air
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above them and draw conclusions about the quality of the air, among many other aspects. This
process is known as back scattering.
In a sense the beginnings of scattering are considered to be more classical mechanics than
quantum mechanics but they are relevant as it is a good starting point for discussion and the
concepts build on one another. Classically, one would have a contained region of space that was
filled with targets (in our case, particles), a number density would be considered to be the ratio of
targets to given space. The number density along with the distance the particle has traveled are
proportional to the probability that the particle will come in contact with the targets or other
particles within the space.[7] However, there is a little more to it than just multiplying the number
density and the distance the particle has traveled into the space. A probability is a dimensionless
number resulting in a constant of proportionality needing to be introduced. This quantity is
referred to as the cross section for scattering and is labelled with σ.[7] For a classical situation this
cross section is the area of each particle in the path of the incident particle. Incident throughout
the remainder of the paper will be used when referencing the entering particle or wave, the
incoming wave. However, the cross section is not so simple when we are dealing with particles
which may not be directly coming in contact with the target but instead interacting with it
because of a long range force as is the case when dealing with a particle scattering.
Another quantity, 𝑑𝜎⁄𝑑Ω is called the differential cross section. This is a function of
both θ and 𝜙 and it allows the determination of the probability that the incident particle will after
interaction be scattered in the θ and 𝜙 direction as those two quantities refer to angles.[7] Thus, in
quantum mechanics scattering, it often becomes a calculation of the differential cross section.
The purpose of this investigation is to use partial wave analysis, as a way of
understanding low energy scattering and short range potentials as this is when the process of
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partial wave analysis is especially useful. In order to understand why it is that only low l values
get close enough to the origin to be affected by the potential one must first understand the system
itself and how partial wave analysis breaks down the incoming wave.
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II. Theory
Schrodinger’s equation has become the fundamental equation for describing quantum
mechanical behavior because it allows one to describe how the wave function of a physical
system evolves over time. For one dimensional problems, such as the infinite square well,
Schrodinger’s equation is written as shown below in eq. (1),
ℏ2 𝜕 2 𝜓
𝜕𝜓
𝑖ℏ 𝜕𝑡 = − 2𝑚 𝜕𝑥 2 + 𝑉𝜓,
(1)
where 𝜓 is the wave function consisting of both x (distance) and t (time) components shown in
eq. (2), ħ is equal to h/2𝜋.
𝜓(𝑥, 𝑡) = 𝑋(𝑥)𝑇(𝑡)
(2)
Schrodinger’s equation is solved using a process of separation of variables in order to reduce the
equation into separate time and x coordinate parts. Expanding the Schrodinger equation into
three dimensions, specifically converting to spherical coordinates allows one to investigate the
behavior of more complex physical situations such as a spherical potential well or scattering
problems. The separation of variables for three dimensional spherical coordinates is,
𝜓(𝑟, 𝜃, 𝜙) = 𝑅(𝑟)𝑌(𝜃, 𝜙),
(3)
where we have a radial part R and a angular dependent part Y. The resulting Schrodinger
equation in spherical coordinates is,
ℏ2
𝑌 𝜕
𝜕𝑅
𝑅
𝜕
𝜕𝑌
𝑅
𝜕2 𝑌
− 2𝑚 [𝑟 2 𝜕𝑟 (𝑟 2 𝜕𝑟 ) + 𝑟 2 sin 𝜃 𝜕𝜃 (sin θ 𝜕𝜃) + 𝑟 2 𝑠𝑖𝑛2 θ (𝜕𝜙2 )] + 𝑉𝑅𝑌 = 𝐸𝑅𝑌,
(4)
where R and Y are defined as above and V is a given potential. In order to reach this one would
use separation of variables as can be done for a one dimensional problem.
In both investigations involving Schrodinger’s equation and partial wave analysis one can
only solve to a certain point before initial boundary conditions have to be applied. Also while
investigating these problems one comes across spherical Bessel functions. Bessel functions are
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defined as solutions to the differential equation shown below in eq. (5). Eq. (5) shows the one
dimensional differential equation for which Bessel functions are the solution to. In this project
however, one needs Spherical Bessel functions which are given in eq. (7), which shows what the
solutions to the three dimensional version of eq. (5)
𝑑2 𝑦
𝑑𝑦
𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 + (𝑥 2 − 𝑛2 )𝑦 = 0
(5)
Bessel functions of the first kind are denoted Jn(x) which will be the kind that shows up
in the theory below. The general solution can be seen in equation four. There are also Bessel
functions of the second kind Yn(x), and a third kind, commonly referred to as Hankel functions
which are a special combination of the first and second kind. Partial wave analysis is the
technique for investigating scattering problems by decomposing each wave into its constituent
angular momentum components and then solving by applying boundary conditions.
The basis of this investigation into partial wave analysis stems from a statement found in
commonly in quantum mechanics texts for short range scattering processes which states that
“Only the low l partial waves get close enough to the origin to be affected by the potential”
[1][4][6][7]
. The investigation began by looking at Bessel functions and spherical Bessel functions
as they were going to appear in the research of scattering. In the research we focused solely on
Bessel functions of the first kind. After separation of variables for the three dimensional in
spherical coordinates the radial part of the wave function depends only on l which is our value
for angular momentum. This allows one to find solutions to the radial part of the wave function
in terms of the angular momentum solely. The solutions we came across in our research lead us
to a common term of kr which will be denoted as 𝜌 throughout the remainder of the paper. This
quantity represents the product of the wave number (k), the spatial frequency of a wave, and the
radius (r). Having the wave number present is important because it provides us with a clear link
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to the energy as there is a connection between the wave number and energy as shown below in
eq. (6). The 𝜌 value is particularly important as well because it is the parameter which we can
manipulate in order to verify the following statement about low l waves:
𝐸=
ℏ2 𝑘 2
2𝑚
.
(6)
Using spherical Bessel functions we can write solutions to the radial part of the wave
equation for a constant potential and given value of angular momentum 𝑙.[1] The full solution for
such conditions is
𝑙 sin 𝜌
1 𝑑
𝑗𝑙 (𝜌) = (−𝜌)−1 (𝜌 𝑑𝜌)
𝜌
.
(7)
The spherical Bessel function has interesting behavior at both small r and limits for large r.
Recall that r is found within the rho (𝜌) value as described above. For small values of r, and a
constant potential for a given 𝑙 the spherical Bessel function exhibits the following behavior,
𝑗𝑙 (𝜌) →
𝜌𝑙
.
(2𝑙+1)!
(8)
Notice that when working with not just the Schrodinger equation but these Bessel functions, the
importance of initial conditions. For the case of (𝜌 ≫ 𝑙), meaning a large r value compared to the
state of angular momentum the Bessel functions shows the following limit,[1]
𝑗𝑙 (𝜌) →
𝑙𝜋
2
sin(𝜌− )
𝜌
.
(9)
Further, it is seen in texts that one can decompose the sine wave contained in Bessel function
shown in eq. (10).[1][8] The result of doing so is
1
𝑙𝜋
𝑙𝜋
𝑗𝑙 (𝜌) → − 2𝑖𝑘𝑟 [𝑒 −𝑖(𝑘𝑟− 2 ) − 𝑒 𝑖(𝑘𝑟− 2 ) ],
(10)
where we have an incoming and outgoing spherical wave of the same magnitude. Flux is defined
as the amount of something passing through a given surface. In terms of scattering, the flux
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would be given as the amount of particles in the incident wave coming in or those which are
being scattered away from the potential. The fact that the incoming and outgoing spherical waves
have the same magnitude means that they have equal fluxes. If the waves did not have equal
fluxes then not all of the particles would be scattered after coming in contact with the potential.
This would result in some of the incoming particles remaining at the origin and creating a
probability build up around the origin. Eq. (10) above allows one to expand the incident wave as
we look at scattering from a three dimensional potential.
The incoming plane wave along the z direction is given in equation eight below. Recall
the importance of initial conditions. For this scattering situation, the potential, V, is equal to zero
far from the origin. Picture a sphere with a concentrated potential at the origin of said sphere.
Travelling radially outward from the origin the effects of the potential at the center will be felt
less. By making this assumption we are allowing the incoming and outgoing waves to be
expressed in terms of constant potentials as was the case when we decomposed eq. (11) into eq.
(12).[1]
𝑙
𝑒 𝑖𝑘𝑧 = 𝑒 𝑖𝑘𝑟 cos 𝜃 = ∑∞
𝑙=𝑜 √4𝜋(2𝑙 + 1) 𝑖 𝑗𝑙 (𝑘𝑟)𝑌𝑙0
(11)
In equation eight we see the Bessel function written in terms of kr, which is what we labelled
with our Rho value. Inserting eq. (10) into eq. (11) makes the plane wave look like eq. (12),
below. The scattering for each partial wave is able to be computed separately and
independently.[1][7]
1
𝑙𝜋
𝑙𝜋
−𝑖(𝑘𝑟− )
𝑖(𝑘𝑟− )
𝑙
2 −𝑒
2 )𝑌
𝑒 𝑖𝑘𝑧 → − ∑∞
𝑙0
𝑙=0 √4𝜋(2𝑙 + 1)𝑖 2𝑖𝑘𝑟 (𝑒
(12)
The Yl0 at the end of eq. (12) refers to the angular portion of the plane wave and l is once more
the angular momentum. Each angular momentum is called a partial wave and that is why when
you sum over all possible angular momentums we obtain the entire wave. The potential interacts
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with the plane wave shown in equation nine. The result of this interaction is seen in the outgoing
portion of the wave, hence the effect of scattering. Recall, the flux in must equal the flux out in
order to avoid build up about the origin. Thus, the effects of scattering can be nothing more than
a change in relative phase shift from the incoming wave to the outgoing wave.[1] This effect was
given as
𝑙𝜋
1
𝑙𝜋
𝑅𝑙 (𝑟) → − 2𝑖𝑘𝑟 (𝑒 −𝑖(𝑘𝑟− 2 ) − 𝑒 2𝑖𝛿𝑙 (𝑘) 𝑒 𝑖(𝑘𝑟− 2 ) ),
=
𝑙𝜋
2
sin(𝑘𝑟− +𝛿𝑙 (𝑘))
𝑘𝑟
(13)
𝑒 𝑖𝛿𝑙(𝑘) ,
(14)
where 𝛿𝑙 (𝑘) is called the phase shift of the partial wave for a given value of angular momentum.
Knowing this phase shift allows one to calculate the differential cross sectional area as was
discussed in the introduction. A general form of the differential cross section is as follows
𝑑𝜎
≡
𝑑Ω
𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 𝑓𝑙𝑢𝑥 𝑖𝑛𝑡𝑜 𝑑Ω
𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑓𝑙𝑢𝑥
.
(15)
Eq. (15) in terms of phase shifts is given by reference 1 in eq. (16). Equation thirteen gives the
furthest step able to be found without having a specific potential energy.
𝑑𝜎
𝑑Ω
=
1
𝑘2
𝑖𝛿𝑙
|∑∞
sin(𝛿𝑙 ) 𝑃𝑙 (cos 𝜃)|
𝑙=0(2𝑙 + 1)𝑒
2
(16)
It is at this point also that a number of different references made statements about only needing
the 𝑙 = 0 term for instances where you are working with low energy scattering and short range
potentials.[1][4][6][7]
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III. Results and Discussion
This project was focused on research and personal understanding and did not have an
actual project or simulation that was being created. It could be expanded with more
understanding of concepts and programming to have simulations created that could model the
effects of scattering based on a set of initial conditions and a physical situation that had a specific
potential. If a specific potential was supplied, using the assumptions discussed in the previous
two sections, then the phase shifts that the incoming particles experienced could be calculated.
Once those are computed the differential cross section would also be able to be computed.
Comparison of the results of different potential values would be able to be conducted and further
investigation into the cause for only low values of angular momentum would be able to be
drawn. The differential cross section provide the number of interactions per target particle that
lead to scattering with a given scattering angle. It allows one to look at the effects of scattering.
As far as the investigation into the causes for only the first term of 𝑙 = 0 term being
needed is concerned, the initial results of calculations for hard scattering confirmed this to be
true as the effect the other terms contributed was minimal compared to that of the 𝑙 = 0 term.
Further calculation would need to be conducted to truly confirm this statement but initially it
holds. Note, these initial calculations were conducted for hard scattering because it provides a
starting point for understanding of the mechanics of scattering.
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IV. Conclusions
I neither confirmed nor denied my goal of coming up with reasoning for only the l = 0
term being needed to solve problems with low energy and short range scattering. Initial
understanding leads me to believe that it is a result of the phase shift being contained in the
exponential and thus as the angular momentum value increases the contribution of each partial
wave decreases exponentially thus creating a situation where the result is dominated by the lower
values of angular momentum especially that of the l = 0 term.
Through this project I was exposed further to Bessel functions and introduced into the
methods of partial wave analysis.
Due to time constraints and needing to spend more time than expected learning the basics
of the concepts I was unable to confirm the driving statement of the project or simulate it using
Mathematica. Further research into the concepts outlined in this paper would be needed.
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V. References
[1] Branson, James. Online notes for Quantum Mechanics 130. Retrieved from:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node1.html
[2] Brandt, Siegmund & Dahmen, Has Dieter. (1985). The Picture Book of Quantum Mechanics.
New York: John Wiley & Sons.
[3] Fuller, Robert & Byron, Frederick. (1969). Mathematics of Classical and Quantum Physics.
New York: Dover Publications.
[4] Griffiths, David. (1995). Introduction to Quantum Mechanics. New Jersey: Prentice Hall.
[5] Messiah, Albert. (1965) Quantum Mechanics. Amsterdam: North-Holland Publishing
Company.
[6] Shankar, R. (1994). Principles of Quantum Mechanics. New York: Kluwar Academic/
Plenum Publishers.
[7] Scherrer, Robert. (2006). Quantum Mechanics An Accessible Introduction. San Francisco:
Pearson.