Java程序辅导

Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
singh@fredonia.edu 
 
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 19
th
 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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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”: 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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 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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 × 10
8 m/s. Theoretical value of Rydberg’s constant 
determined in the present investigation is: RH = 1.096987 × 10
7 
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].  
 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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 
Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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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. 
 
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Selected for publication: Journal Educational Technology Systems, Vol. 40(3), 2011-2012 
 
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35. Moving Toward Quantum Computing - Science in 2011 – New York Times and its website: 
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