computer simulations of quantum theory of hydrogen atom...

Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012

1

COMPUTER SIMULATIONS OF QUANTUM THEORY OF HYDROGEN ATOM FOR

NATURAL SCIENCE EDUCATION STUDENTS IN A VIRTUAL LAB

Gurmukh Singh Ph.D.

Department of Computer and Information Sciences

State Unveristy of New York at Fredonia, Fredonia, NY 14063

[email protected]

Abstract The present scholarly article is targeted for the advanced college/university undergraduate students of

chemistry/physics education, computational physics/chemistry, and computer science. The most recent

software system such as MS Visual Studio .NET version 2010 is employed to perform computer

simulations for modeling Bohr’s quantum theory of hydrogen (H) atom in classroom-setting of a virtual

laboratory. The necessary computer algorithm is developed to compute discrete values of the orbit radius,

and stationary energy levels of Bohr’s H-atom. More than 2000 computer simulations are performed to

investigate the quantum model behavior starting from the ground state of H-atom until we reached the

energy continuum. One of the natural consequences of Bohr’s model is that it could provide a perfect

corroboration of the experimentally observed spectrum of H-atom with that empirically obtained from

formulas derived by famous scientists of 19th and 20

th centuries. Using old theory of classical

electrodynamics, it was not possible to explain the observed line spectrum of H-atom. Bohr’s quantum

model of H-atom set the stage for development of a modern branch of science in microscopic world, the

so-called quantum mechanics, and very recently of a new computing technique known as quantum

computing.

1. HISTORICAL BACKGROUND AND NECESSITY OF QUANTUM MECHANICS IN

MICROSCOPIC WORLD

Considerable amount of modern scientific progress that took place during 20th century can be

summarized in a short list as follows: (i) general theory of relativity [1], (ii) quantum mechanics

[2-5], (iii) big bang model of cosmology [6], (iv) the unraveling of the genetic code [7], (v)

evolutionary biology [8], (vi) nanotechnology [9] and (vii) may be a few other topics depending

on reader's personal taste and choice. In this short list, quantum mechanics has a unique status

due to its profound quality work. Quantum mechanics forced physicists and chemists to change

their view points of deterministic reality, and made them to rethink the nature of things at the

microscopic level in terms of probabilistic events rather than their deterministic attributes.

Therefore, scientists had to revise their classical concepts of position, velocity, momentum

vector as well as their notions of cause and effect in order to understand quantum mechanics

implications [10].

Although the basic aim of the formalism of quantum mechanics was to fully describe an abstract

microscopic, atomic world far away from realm of daily-life experience, however, its immense

impact on human society had been really very pronounced. The spectacular advances in modern

physics, chemistry, biology, and medicine, and in essentially every other scientific field, could

not have taken place without the tools that quantum mechanics has provided in modern age.

Without quantum mechanics it is impossible to talk about global economy, since the electronics

revolution that brought us in the extremely fast moving computer age is fundamentally an

outcome of quantum mechanics. In the same way is the photonics revolution that brought us in

the Information Technology Age: especially fiber optics, e.g., optical fibers being used for


2

information and data transmission over worldwide area network (WAN) and local area network

(LAN) communications. The development of quantum physics has transformed our world,

bringing with it all the benefits of scientific revolutions and advancements. To get an

understanding on the successes of quantum mechanics by an educated layman, a reference may

be made to a very recently published, best selling and very inexpensive textbook in which careful

explanations of the concepts and details of quantum mechanical phenomena are presented in a

nonmathematical form [11].

When scientists analyzed the light emitted by the simplest element of periodic table, e.g., H-

atom, surprisingly they were unable to observe an entire spectrum of continuous colors as seen in

a rainbow. Instead they just observed a few discontinuous bright lines of certain colors with

discrete wavelength or frequency. That would mean that the H-atom was only emitting

electromagnetic waves of discrete frequencies in the ultra-violet (UV), visible and infra-red (IR)

region. Each atom in periodic table emitted a unique set of electromagnetic waves of different

frequencies, and consequently of different wavelengths, which are now known as the spectral

lines. These spectral lines are a quantitative measure of the discrete energy states of an atom, and

consequently a kind of "signature" of its intrinsic nature. The present article is devoted to

computer simulation of discrete energy states of H-atom using the most recent version of MS

Visual Studio .NET 2010 software system in a virtual lab and its educational technology benefit

to the junior and senior college/university undergraduates.

Organization of current article is done as follows: Section 2 discusses how to adapt the MS

Visual Studio .NET 2010 as an effective teaching/research tool in a virtual. Section 3 deals with

the target audience that could use it and its degree of success. Section 4 focuses on fundamental

limitations of Rutherford’s model of atom [12, 13], and a very brief history of Bohr’ quantum

theory of H-atom [14]. Section 5 presents the actual computer simulations of H-atom and the

predictions of Bohr’s model. In Section 6, we will also have a comprehensive discussion on how

this simple quantum model could explain the observed spectral lines such as Lyman [15], Balmer

[16], Paschen [17], Brackett [18], Pfund [19] and Humphrey [20] series emitted by H-atom in

highly excited state. Finally, conclusions and implications of the current investigation will be

presented in Section 7 in light of its usefulness for natural science education students and

instructors.

2. ADAPTION AND TEACHING WITH MS VISUAL STUDIO .NET 2010

Although, there are several available software systems like MS Office [21], OpenOffice [22],

Lotus 123 [23], QuattroPro. [24], Mathematica [25], Maple [26], Linux and Unix OS [27] based

computing machines etc., we preferred to employ the most recent version of Microsoft Visual

Studio .NET 2010 for the current investigation. Further details about the MS Visual Studio can

be found in Ref. [21]. This is due to two main reasons: (i) MS Visual Studio .NET 2010 is a

wholesale package of numerous built-in computing languages and information technology tools

in its Default Collection of Settings: (i) General Development Settings, (ii) Project Management

Setting, (iii) Visual Basic Development Settings,(iv) Visual C# Development Settings, (v) Visual

C++ Development Settings, (vi) Visual F# Development Settings, (vii) Web Development and

(viii) Web Development (Code Only). Therefore, the pertinent user such as an educator or a

learner can pick-up any computational tool for the teaching/learning purpose. In addition, SUNY

Fredonia including many New York State colleges/universities have MS Visual Studio .NET


3

2010 license, and thus instructors/students have an easy access to this software system in their

respective educational institutes. (ii) MS Visual Studio has a user friendly Graphical User

Interface (GUI), so that its pertinent user can easily master its usage within a few hours [21]. For

the present investigation we used setting (iii) as cited above.

From past more than ten years experience of employing MS Visual Studio .NET software system

as a very useful teaching/learning and research tool in the Department of Computer and

Information Sciences, it is our belief that Visual Studio .NET could be adapted as a powerful

teaching tool both by instructors and students to create simulations of several natural science,

engineering, medical, biological, web based etc. application principles. To make an effective use

of this software system for the development of applications for advanced science, engineering

and medical instructors/learners, we advise that the interested users must be proficient in the

basic knowledge of university calculus. The current article is targeted for such like audience who

had prior, sound knowledge of integral and differential calculus, and it provides an in-depth

exploration of one virtual classroom teaching/research application for its educational technology

benefit. In other words, both instructor and learners have an ample opportunity to explore

methodologies and strategies for applying MS VB Studio .NET to their mutual teaching/learning

benefit, in order to create technology-enhanced learning experiences, to assess learning, and to

facilitate collaboration and cooperative learning experiences. Through hands-on experience with

such an important application principle such as quantum theory of H-atom discussed in this

article, instructors/learners would learn how to employ MS VB Studio .NET in order to develop

instructional tools and application principles that could be effectively used in the virtual

classroom setting. We honestly believe that current investigation is an attempt to increase

educator’s and learner’s skill who are very much inclined and interested to seek new ways to

expand their teaching/learning expertise by applying enhanced technology skills as a

constructive means to enhance and improve teaching/learning process.

3. TARGET AUDIENCE AND ITS DEGREE OF SUCCESS

Before assigning Bohr’s quantum theory of H-atom as an independent study or group research

project to the advanced junior and senior college and university undergraduates, they are

required to take at least four semesters of programming courses such as C++, Java, Visual Basic

I & II, data structures etc. so that the students have sufficient programming knowledge and

experience to work independently. Usually, the tradition in SUNY Fredonia and other

educational institutes where I worked is to form a group of two students if the project is to be

completed in time duration of one semester. The participants in a group project are required to

search the literature on internet independently using google.com and they could explore the

university library resources. Instructor may provide the students with relevant textbooks or

research papers concerning the Bohr’s theory of H-atom and the literature dealing with existing

data on the experimental verification of model. In case of difficulty in any phase of group

project, students are allowed to take instructor’s help and advice to get the project accomplished

before the deadline. So far, this project has been assigned to more than a dozen

college/university undergraduates in different institutes in which the author has taught and the

degree of its success has been more than 98%. We believe this scholarly project could be tried in

other educational institutes with the same or greater degree of success.

4. BOHR’S QUANTUM THEORY OF H-ATOM


4

Although the planetary model of the atom presented by Rutherford was extremely simple to

visualize, still it was very successful to explain the scattering of alpha particles from the nucleus

[12, 13]. It was not, however, understood how electrons could continuously orbit the nucleus

without radiating energy, as required by classical electrodynamics [28]. Ten years earlier to

presenting of Bohr’s quantum model [14], Max Planck had proposed that radiation emitted or

absorbed by a perfect black body should always be in discrete quanta of electromagnetic energy

[29]. Bohr postulated that an atom may occupy only a certain number of stable energy states,

each with a certain amount of discrete energy, in which electrons would orbit the nucleus without

emitting or absorbing radiation. When an atomic transition occurs, an electron jumping from a

lower energy state to a higher energy state should absorb energy in discrete amount. Similarly,

an electron jumping from a higher energy state to a lower energy state must emit energy in

discrete amount. During such atomic transitions, an electron will emit or absorb energies

corresponding to a particular set of quantum numbers, n and these quanta of electromagnetic

energy are emitted or absorbed at a particular set of frequencies, ν or wavelengths, λ. According

to Einstein [30], energy of a radiated photon must be equal to the energy difference, ∆E, between

the two stable quantum states, hE , where h is called Plank’s constant. According to dual

nature of light, the frequency of emitted or absorbed photon is related to its wavelength, λ, by

c , where c is the speed of light in free space. The mathematical formula for energy

difference E and that for Bohr’s radius can be found in Ref. [31].

5. COMPUTER SIMULATION USING MICROSOFT VISUAL STUDIO .NET 2010

In MS Visual Studio, the programming technique is implemented from a task-driven point of

view rather than command-driven approach. There are two main aspects used in the designing of

Visual Basic applications, which is a two-step building process:

5.1 Designing of the Graphical User Interface (GUI): The first part of application design is to

create its GUI, which is done in the “Design Window”. This window is shown in Fig. 1 in the

middle of diagram. To design the GUI of an application is a fun part and students/users always

enjoy creating it. It can be done very easily, just by clicking on an object from several choices in

the “ToolBox” window (e.g., see left hand side window in Fig. 1), dragging it and then releasing

it onto something called “Form” object sitting in the center of user interface.

5.2 Writing of the Code: The second part is to write the code of an application, which is done in

the “Code Window”. This window can be accessed by clicking on “View” and then on “Code” in

the user interface shown in Fig. 1. The writing of code needs practice, involves critical thinking

and prior knowledge of college/university mathematics and object-orient programming. If a

student can master the code writing part, there may be a possibility that after graduating he/she

could land job in a company where the knowledge of coding and designing of Visual Basic (VB)

applications could be utilized.

In writing VB code, one has to employ common mathematical and logical operators, and

numerous built-in functions in MS Visual Studio. To simulate Bohr’s radius and its

corresponding discrete energy state value [31] with the help of MS Visual Studio .NET version

2010 [21], the following two basic equations in the quantum theory H-atom should be used to

develop the required source code in the “Code Window”:


5

rn = (є0h2n2)/(πµeZe2) …………………………………………..(1)

En = (µeZ2e4)/(8є02h2n2) ………………………………………(2)

Fig. 1: Actual screen-shot of graphical user interface (GUI) in Microsoft VB .NET, version 2010.

Here, µe = reduced electron mass, e = electron charge, Z = mass number of H-atom, h is known

as the Plank’s constant, and ε0 is the permeability of free space. One may perceive that the

running index in above Eq. (1) and Eq. (2) is the principal quantum number n. For H-atom with

atomic charge, Z = 1, only two equations are sufficient to simulate Bohr’s radius and its

corresponding stationary energy state by varying the principal quantum number n = 1 through n

= 2000 in steps of unity. An actual screen-shot of GUI for simulation run performed with

Microsoft Visual Studio .NET 2010 is shown in Fig. 2 in which only first nineteen simulated

values of the Bohr radius and its corresponding discrete energy state value are visible in the list

box, although 2500 simulations have been done. For n = 1, one can obtain the radius of H-atom

in its ground state. From the current simulation work, the computed radius of H-atom in its

ground state is: r1 = 5.292786 x 10-11

m ≈ 10-10

m, which agrees quite well with the recently


6

determined experimental value for the size of H-atom [32], indicating that the accuracy in

computed value of atomic radius of H-atom is exceedingly good.

Fig. 2: Actual screen-shot of simulation with atomic charge Z = 1 of H-atom and principal quantum

number n = 1 – 2000.

6. DISCUSSION OF RESULTS

We draw an energy level diagram for the discrete energy states of H-atom in Fig. 3. For the sake

of clarity, we depict only transition lines corresponding to three spectral series, namely: the

Lyman [15], Balmer [16] and Paschen [17] series. Lyman series [15] lies in the UV region of the

electromagnetic spectrum. However, the Paschen [17], Brackett [18], Pfund [19] and Humphrey

[20] series are confined to the IR part of the electromagnetic spectrum. Only a few spectral lines

of the Balmer series could be seen in the visible part of the electromagnetic spectrum, and those

five spectral lines are experimentally photographed, which is shown Fig. 4 for transition between

quantum states with n = 2 and n = 3, 4, 5, …. Wavelength of each of these lines is computed

using Equations of Ref. [31] and is listed in Table 1. The computed values of spectral lines agree

very well with those obtained from the empirical formulas of the Lyman, Balmer, Paschen,

Brackett, Pfund and Humphrey (not shown here). Wavelengths of first five spectral lines of the


7

Balmer series are generally represented by λα, λβ, λγ, λδ, and λε, respectively, which lie in the

visible part of the electromagnets spectrum, and the last spectral line of wavelength λ∞ lies in the

UV region.

Fig. 3: Computed spectral lines of H-atom for only three series, i.e., the Lyman, Balmer and Paschen are

depicted. The Pfund and Humphrey series are not plotted in this figure for the sake of clarity.

Experimentally determined wavelengths of these five spectral lines of H-atom are displayed in

Table 1. In literature these lines are represented by a special notation: H-α, H-β, H-γ, H-δ, H-ε,

H-ζ, H-η, and H-∞, since these spectral lines are for the H-atom. Experimental value of each

wavelength [32] given in Table 1 is very close to the computed value of corresponding

wavelength, which indicates Bohr’s quantum model works exceptionally good for explaining the

experimental H-atom spectrum. To strengthen our argument, in Fig. 4, we present an actual

photograph of the experimental spectrum of H-atom for the first five spectral lines in visible

region of the electromagnetic spectrum. First spectral line is of red color, and consequently of

longest wavelength. Fifth line is of violet color, and therefore of shortest wavelength among

these five theoretically computed and experimentally observed spectral lines in the Balmer


8

series. Last three spectral lines in Table 1 lie in the UV region and their wavelength has been

determined with a special instrument called UV Spectrophotometer.

Table 1: Computed values of wavelength, λα, λβ, λγ, λδ, λε, and λ∞, for the Lyman, Balmer, Paschen,

Pfund and Humphrey series for H-atom.

Model

Lyman

Series (nm)

Model

Balmer

Series (nm)

Model

Paschen

Series (nm)

Model

Brackett

Series (nm)

Model

Pfund

Series (nm)

Experimental

Balmer Series

(nm) [32]

λα 121.611 656.700 1876.287 4053.707 7462.505 656.285 (Red)

λβ 102.609 486.445 1282.618 2626.802 4655.429

486.133

(Blue-green)

λγ 97.289 434.326 1094.501 2166.890 3741.883 434.047 (Violet)

λδ 95.009 410.438 1005.573 1945.779 3298.161 410.174 (Violet)

λε 93.814 397.263 955.201 1818.555 3040.280 397.007(Violet)

λ∞ 91.208 364.834 820.876 1459.334 2280.210 -

nf = 2, 3, 4,.

→ ni = 1

nf = 3, 4, 5,.

→ ni = 2

nf = 4, 5, 6,.

→ ni = 3

nf = 5, 6, 7,.

→ ni = 4

nf = 6, 7, 8,.

→ ni = 5

nf = 3, 4, 5,. →

ni = 2

Fig. 4: Actual experimentally observed five wavelengths, H-α (red), H-β (blue-green), H-γ (violet), H-δ

(violet), and H-ε (violet) of the Balmer series for H-atom in visible part of the electromagnetic spectrum

[32].

From the equation of total energy of H-atom [31], it is also possible to determine the theoretical

value of the Rydberg constant by plugging into the values of fundamental physical constants

such as: electric charge, e = 1.602 × 10-19

C, Plank’s constant, h = 6.626 × 10-34

J.s and speed of

light in free space, c = 2.99792458 × 108

m/s. Theoretical value of Rydberg’s constant

determined in the present investigation is: RH = 1.096987 × 107

m-1

, which is very near to its

corresponding experimentally observed value [32]. Therefore, it is possible that one could

determine an independent, experimental value of Rydberg’s constant from the experimental line

spectrum of H-atom in laboratory and could compare it with the corresponding theoretically

computed value, which can be easily done in undergraduate chemistry and physics labs in our

country.

Ionization energy or potential of an atom is defined as the amount of energy required to dislodge

an electron from the outer most orbit of an atom, which can be computed theoretically from the

total energy En by inserting ni = 1, nf = ∞ and other pertinent physical parameters of H-atom,

along with some fundamental physical constants as discussed in the computation of wavelengths

of spectral lines for the Blamer series. The theoretically computed value of the ionization energy

for H-atom is 13.6193 eV, which agrees very nicely with that of its corresponding experimental

value [33, 34].


9

7. CONCLUSIONS AND IMPLICATIONS OF THE PRESENT WORK

We may conclude this article with the following noteworthy remarks: We presented a brief

historical background of Bohr’s quantum theory of H-atom and its necessity for the natural

science education students and instructors. We believe that this article would be very beneficial

for physics, chemistry education, and computational physics and chemistry and computer science

college and university majors. This would also present them with interesting aspects and

outcomes of Bohr’s quantum theory in a virtual lab. We employed Microsoft Visual Studio .NET

version 2010 software system to perform the simulations of Bohr’s quantum theory of an atom.

More than 2000 simulations have been performed to compute the values of nuclear radius and its

corresponding stationary energy value. Bohr's quantum model of the atom successfully described

the electron motion in discrete, precisely defined circular orbits around the nucleus. It also offers

one possible explanation for the emission spectrum observed from H-atom. This simple Bohr’s

quantum theory lead to a new branch of natural science called quantum mechanics, and very

recently to a modern computing technique known as quantum computing [35].

With the help of Bohr’s quantum model, one could predict and compute for the H-atom,

wavelengths of many spectral lines such as the Lyman, Balmer, Paschen, Brackett, Pfund,

Humphrey series. Since Lyman series lies in the UV region, and whereas Paschen, Pfund and

Humphrey series are confined in the IR part of the electromagnetic spectrum, and therefore in

laboratory, it will not be feasible to observe the above mentioned line spectra. However, physics

and chemistry students and including their instructors would definitely be able to measure

experimentally the spectral lines of Balmer series in the visible part of the electromagnetic

spectrum for H-atom, and thus could verify the predictions of Bohr’ quantum model. We

compared experimentally determined [32] and theoretically computed values of the wavelengths

of fives lines of the Balmer series and found an excellent agreement between Bohr’s theory and

experimental line spectrum of H-atom, e.g., see Table 1 Section 6.

Another advantage of Bohr’s model of an atom is to predict theoretically the ionization energy or

potential of H-atom and its computed value is 13.6 eV, which is very close to its experimentally

determined value [34, 35]. This simple quantum model of H-atom has been assigned to more

than one dozen of senior undergraduates of different colleges and universities as independent

study or group project and its degree of success had been found to be more than 98%. Initially

when this project was assigned many students were worried if they could finish it before the

deadline date, but once they completed it with my help, they felt pretty good about their

accomplishment.

Finally, we believe that this scholarly article would be appreciated by both university and college

instructors as well by graduate and undergraduate students because of the concept of using an

available software system like Microsoft Visual Studio .NET, which most users these days could

install on their Windows, Mac computers and laptops for scientific and engineering

computations. Consequently, we hope this will help make science more accessible to a wider

range of college and university students, instructors and members of the general audience. Our

user friendly explanations of how to use various built-in Microsoft Visual Studio .NET 2010

functions to carry out numerical calculations that are ordinarily done using computing languages

such as Fortran-2003, C, C++ or Java, will be very helpful to students and instructors alike. The

hydrogen spectrum and the Bohr quantum model which explains it with wonderful precision are


10

excellent choices for introducing non-physicists to the basic ideas and methods of quantum

physics, and the interplay between experiment, theory, and computation. It is worthwhile to

mention here that it is also possible to perform Bohr’s quantum model simulations for H-atom

with an exactly same accuracy and precision with a lower of version of Microsoft Visual Studio

.NET 2008, 2005, Microsoft Excel software systems, and Linux or Unix based computing

machines [36, 37].

ACKNOWLEDGMENTS

I am really grateful to Dr. Reneta Barneva, Professor and Chair, Department of Computer and

Information Sciences, SUNY at Fredonia, Fredonia, NY, for providing me with the necessary

computational facilities. Special thanks are due to my former colleague and collaborator, Dr.

Richard J. Gonsalves, Department of Physics, State University of New York at Buffalo, Buffalo,

NY, for his useful comments on the manuscript.

REFERENCES

1. A. Einstein, Relativity: The Special and General Theory, Springer Publisher (1916).

2. E. Schrödinger, Encyclopedia Britannica, (2009). Encyclopedia Britannica, online, 18 Nov.

2009.

3. W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer

Beziehungen, Zeitschrift für Physik, 33, 879-893 (1925). The paper was received on 29 July

1925. [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics

(Dover Publications, 1968) ISBN 0-486-61881-1 (English title: Quantum-Theoretical Re-

interpretation of Kinematic and Mechanical Relations).] This is the first paper in the famous

trilogy which launched the matrix mechanics formulation of quantum mechanics.

4. M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, 34, 858-888 (1925).

The paper was received on 27 September 1925. [English translation in: B. L. van der

Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-

61881-1 (English title: On Quantum Mechanics).] This is the second paper in the famous

trilogy which launched the matrix mechanics formulation of quantum mechanics.

5. M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35,

557-615 (1925). The paper was received on 16 November 1925. [English translation in: B. L.

van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN

0-486-61881-1] This is the third paper in the famous trilogy which launched the matrix

mechanics formulation of quantum mechanics.

6. B. Feuerbacher and R. Scranton, Evidence for the Big Bang (25 January 2006): E. L. Wright,

What is the evidence for the Big Bang? (9 May 2009).

7. http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/C/Codons.html

8. J. Douglas, Evolutionary Biology. Sunderland, Mass.: Sinauer Associates (1997); M. Ridley,

Evolution, Boston: Blackwell Scientific (2003); B. William, The Origins of Theoretical

Population Genetics, University of Chicago Press, Chicago (2001).

9. http://www.nanotech-now.com/

10. D. Kleppner and R. Jackiw, One Hundred Years of Quantum Physics, American Association

for the Advancement of Science (2000).


11

11. T. Padmanabhan, Quantum Themes: The Charms of the Microworld, World Scientific,

Imperial College Press, London (2009).

12. E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom,

Philosophical Magazine, 6, 21 (1909)

13. H. Geiger and E. Marsden, On a Diffuse Reflection of the α-Particles, Proceedings of the

Royal Society, A 82, 495–500 (1909).

14. N. Bohr, On the Constitution of Atoms and Molecules, Part I, Philosophical Magazine 26: 1–

24 (1913).; N. Bohr, On the Constitution of Atoms and Molecules, Part II, Systems

Containing Only a Single Nucleus, Philosophical Magazine 26, 476–502. (1913); N. Bohr,

On the Constitution of Atoms and Molecules, Part III Systems containing several nuclei,

Philosophical Magazine 26, 857–875 (1913).

15. T. Lyman, The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length,

Memoirs of the American Academy of Arts and Sciences, New Series 13 (3), 125–146 (1906);

T. Lyman, An Extension of the Spectrum in the Extreme Ultra-Violet, Nature 93, 241, (1914).

16. J. J. Balmer, Notiz uber die Spectrallinien des Wasserstoffs, Annalen der Physik 261 (5), 80–

87, (1885).

17. F. Paschen, Zur Kenntnis ultraroter Linienspektra. I. (Normalwellenlängen bis 27000 Å.-E.),

Annalen der Physik 332 (13), 537–570 (1908)

18. F. S. Brackett, Visible and infra-red radiation of hydrogen, Astrophysical Journal 56, 154,

(1922),

19. A. H. Pfund, The emission of nitrogen and hydrogen in infrared, J. Opt. Soc. Am. 9 (3), 193–

196, (1924).

20. C. J. Humphreys, Humphreys Series, J. Research Natl. Bur. Standard,s 50 (1953).

21. http://www.microsoft.com/en/us/default.aspx

22. http://www.openoffice.org/

23. http://www-01.ibm.com/software/lotus/products/123/

24. http://office.microsoft.com/en-us/excel-help/opening-quattro-pro-files-in-excel-

HA001044873.aspx

25. http://www.wolfram.com/solutions/education/students/

26. http://www.maplesoft.com/products/maple/academic/

27. http://www.ubuntu.com/

28. J. D. Jackson, Classical Electrodynamics, 3rd

ed., John Wiley & Sons (1999).

29. M. Planck, "Ueber das Gesetz der Energieverteilung im Normalspectrum", Ann. Phys. 309

(3): 553–63, (1901).

30. A. Einstein, The Development of Our Views on the Composition and Essence of Radiation,

Physikalische Zeitschrift, 10 (22), 817–825 (1909).

31. D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, Extended 8th

ed., Wiley

2008.

32. C. R. Nave, HyperPhysics: Hydrogen Spectrum, Georgia State University (2008); Eisberg

and Resnick, Quantum Physics. John Wiley & Sons, pp. 97 (1985). Also see, e.g., Spectrum

of the Hydrogen Atom, Department of Chemistry, University of Massachusetts, Boston, Fall

2009.

33. L. Julien et al., Experimental determination of the Rydberg constant using two-photon

spectroscopy of atomic hydrogen, J. Quantum Electron 18, 763-765 (1988).

34. N. Bohr. Atomic Theory and the Description of Nature. New York, Macmillan (1934).

http://www.microsoft.com/en/us/default.aspx

http://www.openoffice.org/

http://www-01.ibm.com/software/lotus/products/123/

http://office.microsoft.com/en-us/excel-help/opening-quattro-pro-files-in-excel-HA001044873.aspx

http://office.microsoft.com/en-us/excel-help/opening-quattro-pro-files-in-excel-HA001044873.aspx

http://www.wolfram.com/solutions/education/students/

http://www.maplesoft.com/products/maple/academic/

http://www.ubuntu.com/

http://www.wiley.com/WileyCDA/Section/id-302475.html?query=David+Halliday

http://www.wiley.com/WileyCDA/Section/id-302475.html?query=Robert+Resnick

http://www.wiley.com/WileyCDA/Section/id-302475.html?query=Jearl+Walker

http://hyperphysics.phy-astr.gsu.edu/Hbase/hyde.html#c4


12

35. Moving Toward Quantum Computing - Science in 2011 – New York Times and its website:

http://www.nytimes.com/2010/11/09/science/09compute.html;

http://www.physorg.com/news/2011-10-quantum-qubit.html

36. S. R. Goode and L. A. Metz, J. Chem. Ed., 80 (12), 1455-1459 (2003); L. R. Khundkar, J.

Chem. Ed., 73 (11), 1055-1054 (1996); B. R. Ramachandran and A. M. Halpern, J. Chem.

Ed. 76 (9), 1266-1268 (1999).

37. G. Singh, and A. Mukhopadhyay, Modeling Bohr’s Theory of Hydrogen Atom for Physics,

Chemistry and Computer Science Graduates, Proceedings of International Conference:

Applications of Computer and Information Sciences to Nature Research (ACISNR-2010),

ISBN 978-1-60558-918-3, p. 59-63, SUNY at Fredonia, Fredonia, NY, May 5-7, 2010.

www.nytimes.com/2010/11/09/science/09compute.html

http://www.physorg.com/news/2011-10-quantum-qubit.html

computer simulations of quantum theory of hydrogen atom...

Documents