CO22 Solutions of Schrödinger’s equation by numerical integration

Abstract

The aim of this program is to solve Schrödinger’s equation via a numerical method and to compare our results with the analytical result. The harmonic oscillator potential was taken as the potential in the equation. This was used because the solutions to this one dimensional system are well known, and an analytic solution is relatively easy to program. The idea is that once we have verified that our numerical approach works for this system it can be extended to work for any one dimensional potential that we want, even if the analytic solutions are not known.

Introduction

The time independent Schrödinger’s equation in one dimension is:

\* MERGEFORMAT

We are looking at the simple harmonic oscillator for which , where k is the spring constant. To make the equation easier to solve, we use non dimensional variable:

and

Therefore

Equation (1) reduces to:

\* MERGEFORMAT

Analytic solutions exist for equation (2), that take the form:

Where are Hermite polynomials for n positive integers and zero given by the relation:

With corresponding eigenvalues:

With this I could then write a program to show the analytic solution of this problem. For the numerical solution I used Numerov’s method. This says that a differential of the form:

Has a solution, for a uniformly spaced set of points of difference δ the value of ψ at point v+1 is approximately related to the points v and v-1 by the equation:

Where and are the values of and at . Due to the symmetry of the potential there will be both even and odd solutions. The boundary conditions will be:

............................................................................................. 41 Bijlage 3: Trapeziumregel .................................................................................................................. 42 Bijlage 4: Simpson regel .................................................................................................................... 43 Bijlage 5: Eigen regel ......................................................................................................................... 44 Bijlage 6: hoofdprogramma............................................................................................................... 45
3|P a gin a Numerieke Methoden - Nulpuntsbepaling
Literatuurlijst
Numerical Analysis, Burden & Faires, 2005, Thomson Brooks Corporation, Belmont The Student Edition of Matlab, The Math Works, 1995, Prentice-Hall, Englewood Cliffs ‘Numerieke integratie’, Vrije Universiteit Brussel, onbekende auteur, 2003
4|P a gin a Numerieke Methoden - Nulpuntsbepaling
Lijst van figuren en tabellen
Figuur 1 - Weergave van de enkelvoudige trapeziumregel ..................................................................7 Figuur 2 - Afbeelding van de trapeziumregel .......................................................................................8 Figuur 3 - Afbeelding van de regel van Simpson ................................................................................ 11 Figuur 4 - Verdeling van...

...
Psychology and Christianity: Initial Integration Paper
Azusa Pacific University
Psychology and Christianity: Initial Integration Paper
Psychology and Christianity: two subjects that have seem to become almost impossible to talk about hand in hand. Psychology has become one completely different subject than Christianity (theology) and both have lost all contact with the other. Psychology is strictly a science and Christianity is solely based on faith and religion and the two can not be integrated together. The real question is why? Why can’t psychology and Christianity be integrated and used together as one? This paper will give some dictionary definitions of both subjects, will provide definitions based on the authors opinion, and will discuss the authors viewpoint on the integration of the two.
The American Heritage dictionary defines psychology as “the science that deals with mental processes and behavior” (American Heritage Dictionary, 2000) and defines Christianity as “the Christian religion, founded on the life and teachings of Jesus” (American Heritage Dictionary, 2000). As you could see, both definitions used the words science and religion placing both subjects in their own categories. In modern days, religion and science can not be compared but rather contrasted. The integration of the two does not make sense for those who believe there should be...

...of last resort for CARICOM Member States. The
ﬁrst proposal for such a court was made at the Sixth Meeting of Heads
of Government of the Caribbean Community in 1970 and a call for its
establishment was renewed by the West Indian Commission’s 1992 report
“Time For Action”. The CCJ was eventually established in 2001 and inaugurated
in April 2005 as a court of both ﬁnal appeal from member states and original
jurisdiction for disputes arising out of the Revised Treaty of Chaguaramas.
The ﬁfty-eight year road to the Court’s launch was lined with challenges.
Nonetheless, by 2001 the need to complete the independence of the
Commonwealth Caribbean States and a greater appreciation of the advantages
to be derived from closer regional integration laid a ﬁrm basis for the
establishment of the CARICOM Single Market and Economy with the creation of
the CCJ as its centre-piece. The idea of building a “Caribbean jurisprudence”
based on international standards had become a critical objective of Caribbean
states. In his lecture to the UWI Law Society in November, 2011, the Right
Honourable Sir Dennis Byron, President of the CCJ noted that when people,
…speak of the development of a Caribbean jurisprudence in the same breath as
the establishment of the CCJ, what they mean is that, at the level of the top
judicial tier, the CCJ aﬀords the opportunity, by its judgments, more closely to
align the trajectory of Caribbean jurisprudence with the mores, values,...

...have never entered into force. Germany formed an army anyway and 6 countries + UK observed it.
ECSC was operating well, but it’s political and economic impact wasn’t that high. Back then intergovernmental cooperation spills over into other areas. ACSC is not enough. President of ACSC (French) wanted deeper integration in Europe. Coal was rapidly loosing it’s position. His idea - Euroatom. 6 member states agreed on the idea. At the same time decided to create EEC. These 2 created by the Treaties of Rome in 1957-58. About EEC and Euroatom. The process is time consuming, because the process of ratification varies from country to country. We had 3 organizations: ECSC 1952, Euroatom 1957, EEC 1957.
Treaties of Rome are basic as it comes to European integration process. Afterwards we had few new treaties. Most important is that when we negotiate every new treaty after the Treaties of Rome - every other treaty amended a previous ones.
Main reason of integration in Europe - economic cooperation. Nowadays we have more cooperation than that, still everything started from economic cooperation.
Phases of development:
1. 1950s-1970s: ECSC is a success - starting point for integration. EDC - failure. Succeeded in economic cooperation, but failed in common defense establishing. States were not ready to cooperate strongly on political level. Ideology that gathered all European states against USSR. Economic cooperation was #1,...

...Chapter 7
Advanced Integration Techniques
Before introducing the more advanced techniques, we will look at a shortcut for the easier of the substitution-type integrals. Advanced integration techniques then follow: integration by parts, trigonometric integrals, trigonometric substitution, and partial fraction decompositions.
7.1
Substitution-Type Integration by Inspection
In this section we will consider integrals which we would have done earlier by substitution, but which are simple enough that we can guess the approximate form of the antiderivatives, and then insert any factors needed to correct for discrepancies detected by (mentally) computing the derivative of the approximate form and comparing it to the original integrand. Some general forms will be mentioned as formulas, but the idea is to be able to compute many such integrals without resorting to writing the usual u-substitution steps. Example 7.1.1 Compute cos 5x dx.
Solution: We can anticipate that the approximate form1 of the answer is sin 5x, but then d d sin 5x = cos 5x · (5x) = cos 5x · 5 = 5 cos 5x. dx dx Since we are looking for a function whose derivative is cos 5x, and we found one whose derivative is 5 cos 5x, we see that our candidate antiderivative sin 5x gives a derivative with an extra factor of 5, compared with the desired outcome. Our candidate antiderivative’s derivative is 5 times too large, so this candidate antiderivative,...

... Compare and contrast the ways that many Americans expressed their opposition to immigrants in the 1840s-1850s with the ways that many Americans expressed their opposition to immigrants in the 1900s-1920s.
Through history immigration has always caused a problem within the U.S. Americans believed that immigrates were taking all the available jobs, and were overpopulating many of the cities they settled in. In the 1840s-1850s and the 1910s-1920s, Americans always believed that if you weren’t born in America then you weren’t allowed to have the same privileges as them. In the 1840s-1850s, Americans opposed those that migrated from China, Mexico, Ireland, and of different religions but in the 1910s-1920s many opposed those that weren’t American. Immigration has been taken in our country since 1840, and although immigration has changed in whom Americans oppose, there’s one thing that’ll never change; Americans don’t see immigrates as equals.
During the periods of 1840-1850 and 1910-1920, the views of immigration were similar. There was a strong rise of nativism during both times, but for different reasons. From 1840-1850, native-born Americans were afraid of the immigrants coming in and "stealing" their already low-wage jobs. Since factory jobs were becoming very popular due to the industrial revolution, the skill level for those jobs went down which dramatically changed the chances of getting laid off. Also, because it was so easy to lose a job, immigrants were able to...

...
ECON 4020
Spring 2008
Midterm exam 1
Name __________________________________________
This is a closed book exam.
Exam is out of 100 points.
Answer all the questions.
Show all your calculations.
Explain your answers in the essay type questions.
You have 50 minutes to finish this exam.
Academic Integrity: In cases of a violation of academic integrity it is the policy of the Department of Economics to impose the most severe penalties that are most consistent with University guidelines.
GOOD LUCK!
1. (4 points) Average labor productivity is the
(a) amount of workers per machine.
(b) amount of machines per worker.
(c) ratio of employed to unemployed workers.
(d) amount of output per worker.
Answer: D
2. (4 points) A country has a trade deficit when
(a) imports exceed exports.
(b) imports equal exports.
(c) exports exceed imports.
(d) export is zero.
Answer: A
3. (4 points) Classical economists argue that
(a) the government should have an active role in the economy.
(b) government policies will be ineffective and counterproductive.
(c) the government should actively intervene in the economy to eliminate business cycles.
(d) wages and prices don’t adjust quickly, so the economy is slow to return to equilibrium.
Answer: B
4. (4 points) The country of Old Jersey produces milk and butter, and it has published the following macroeconomic data, where quantities are in gallons and prices are...

...http://sahatmozac.blogspot.com
ADDITIONAL MATHEMATICS FORM 5
MODULE 4
INTEGRATION
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CHAPTER 3 : INTEGRATION
Content
Concept Map
page
2 3–4 5 6 7 8–9 10 – 11 12
4.1 Integration of Algebraic Functions Exercise A 4.2 The Equation of a Curve from Functions of Gradients. Exercise B SPM Question Assessment Answer
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Indefinite Integral
a)
ò
ò
a x
n
a dx = ax + c.
xn+ 1 + c. n+ 1
b)
x n dx =
c
)
ò
d
x
=
a
ò
x
n
d
x
=
a n
x +
n
+
1
1
+
c
.
Integration of Algebraic Functions
e) ) The [f (x) ± g(x) ]dx = ò f (x) dx ± d ò Equation of a Curve from Functions of Gradients
ò g(x)dx
y = y =
ò
f '( x ) d x c,
f (x) +
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INTEGRATION
1. Integration is the reverse process of differentiation. dy 2. If y is a function of x and = f '( x) then ò f '( x)dx = y + c, c = constant. dx If
dy = f ( x ), then dx
ò f ( x)dx =
y
4.1.
Integration of Algebraic Functions
Indefinite Integral
a)
b)
ò
ò
a dx = ax + c.
n
a and c are constants
xn+ 1 x dx = + c. n+ 1
n
c is constant, n is an integer and n ≠ -
c)
ò ax
dx = a ò
ax n + 1 x...

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