International Journal of Pedagogics
46
https://theusajournals.com/index.php/ijp
VOLUME
Vol.05 Issue07 2025
PAGE NO.
46-52
10.37547/ijp/Volume05Issue07-11
Methodological Aspects of Teaching Quantum Mechanics Using
the Example of The Hydrogen Atom
Mukhtarov E.K.
Andijan State University, Uzbekistan
Received:
17 May 2025;
Accepted:
13 June 2025;
Published:
15 July 2025
Abstract:
This article explores methodological aspects of teaching quantum mechanics in higher education within
the framework of physics and mathematics programs. Particular attenti
on is given to the challenge’s students face
in mastering fundamental concepts due to the high level of abstraction inherent in the subject. The hydrogen
atom model is considered as a focal example
–
being both conceptually significant and visually illustrative in
demonstrating the principles of quantum theory.
The core emphasis is placed on the interpretation of the wave function and its physical meaning through the
concept of probability density. Methods for visualizing electron wave states are discussed, alongside graphical
representations of probability densities corresponding to various quantum numbers. The relationship between
the analytical solutions of the Schrodinger equation and the physical interpretation of these results is
substantiated.
Special
attention is also devoted to developing students’ understanding of the probabilistic nature of quantum
systems, as well as their ability to apply mathematical tools for modeling and interpreting quantum states.
Didactic approaches are proposed to enhance
students’ abstract thinking and intuitive grasp of quantum
principles.
Keywords:
Information technology, quantum mechanics, teaching methods, educational process, integration,
hydrogen atom, wave functions, Schrodinger equation.
Introduction:
The hydrogen atom represents one of
the simplest systems in quantum mechanics, making it
an ideal object for studying the fundamentals of
quantum theory. However, despite the apparent
simplicity of the problem, it poses significant challenges
for students, as quantum mechanics is fundamentally
opposed to the intuitive perception of the world
shaped by classical physics. Teaching the hydrogen
atom requires not only an understanding of the
mathematical aspects of the theory but also the ability
to present abstract ideas of quantum mechanics to
students.
This article explores the main pedagogical challenges
faced by instructors when introducing students to the
topic “The Hydrogen Atom in Quantum Mechanics”
and proposes methods for effectively overcoming
these difficulties.
Literature Analysis
Classical mechanics is generally intuitive for students,
as it describes phenomena they encounter in everyday
life. However, when transitioning to quantum
mechanics, this understanding breaks down, and
students are faced with abstract concepts such as wave
functions, probabilistic interpretations, and energy
quantization.
Quantum mechanics requires students to work with
probability distributions instead of precise particle
trajectories. In quantum mechanics, the electron does
not move along a defined orbit; rather, its state is
described by a wave function that gives the probability
of finding the electron at a particular point in space [1].
To overcome this difficulty, instructors should use
examples that help visualize quantum phenomena,
such as analogies with waves, interference, or the
superposition of states. At the initial stages, it is
important to emphasize the differences between
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International Journal of Pedagogics (ISSN: 2771-2281)
classical and quantum mechanics and to explain that
energy quantization and the uncertainty principle are
not just mathematical abstractions but real
phenomena observed in nature [2].
Solving the Schrodinger equation for the hydrogen
atom requires skills in working with differential
equations and knowledge of function theory. Students
may find it difficult to grasp the physical meaning of
mathematical solutions such as wave functions and
their interpretation.
Educators must clearly separate the mathematical
aspect from the physical content of the solutions. It is
important to highlight each step in solving the
Schrodinger equation and explain the physical meaning
of each term. A detailed explanation should be given of
how wave functions represent the probability of
locating an electron at various points in space, and how
the quantum numbers n, l, and m determine energy
levels and orbital symmetry [3,4].
One of the most challenging aspects of the topic is the
concept of energy quantization, which contradicts the
classical idea of continuous energy. Students may
struggle to understand that the energy of an electron
in a hydrogen atom can only take on discrete values,
and that transitions between these levels lead to the
emission or absorption of photons with specific
energies.
To explain energy quantization, it is helpful to use
analogies such as vibrating strings or other systems
where energy discretization is also observed. It is
important to stress that quantization is not due to a
lack of knowledge, but is a fundamental property of
nature that explains many observed phenomena, such
as atomic spectra [5].
One of the most effective teaching methods is the use
of visual materials such as graphs, animations, and
simulations. For the hydrogen atom, this might include
images of orbitals, transition spectra, and visualizations
of wave functions. Modern simulators and computer
programs allow students to observe processes such as
quantum transitions and to construct wave functions
themselves.
METHODOLOGY
The methodology is based on a comprehensive
approach that integrates theoretical analysis,
mathematical modeling, and digital visualization to
enhance the effectiveness of teaching the “Hydrogen
Atom” topic in quantum mechanics.
The study includes:
–
theoretical analysis of pedagogical challenges related
to understanding wave functions and probability
density;
–
analytical and numerical solution of the Schrodinger
equation for the hydrogen atom;
–
development of a digital educational module for
visualizing
quantum
states
and
probability
distributions;
–
implementation of computational tasks with
graphical analysis and interpretation of physical
meaning;
–
inclusion of control questions and tasks aimed at
developing students' skills in independent analysis.
This approach enhances students’ conceptual
understanding of quantum mechanics and promotes
the formation of strong foundational knowledge.
RESULT AND DISCUSSION
To improve the quality of education, to foster deep
theoretical knowledge and practical skills among
students, as well as to model the quantum state of an
electron in a hydrogen atom, a custom-designed
electronic educational program based on digital
technologies was developed [6,7]. This program
enables the visualization of the most probable states of
the electron’s wave function, the analysis of probability
density distributions, and the interpretation of
quantum states based on the latest scientific advances.
Additionally, the program is aimed at creating a digital
learning environment that promotes the development
of students' skills in independent analysis, calculation,
and drawing conclusions within the educational
process.
The
functional
and
methodological
advantages of this electronic educational resource are
clearly demonstrated in the following examples [8].
1. In the ground state of the hydrogen atom, the
electron’s wave function is given by:
0
( )
exp
r
r
A
a
=
−
where
𝐴
is a constant,
𝑎
0
is the first Bohr radius.
Find:
a) The most probable distance
r
ₘₐₓ
between the electron and the nucleus;
b) The probability of finding the electron in the region
r < r
ₘₐₓ
.
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International Journal of Pedagogics (ISSN: 2771-2281)
Solution:
The physical meaning of the wave function is that the square of its modulus gives the probability density of the
particle’s location. Thus,
the probability of finding the particle in a spherical shell of radius
r
and thickness
dr
is
given by [9]:
2
2
2
2
0
2
4
4
exp
r
d
r dr
A r
dr
a
=
=
−
(1)
From expression (1), it follows that the probability of finding the electron in a spherical shell of unit thickness is:
2 2
0
2
( )
4
exp
r
r
A r
a
=
−
(2)
Let us find the value
𝑟 = 𝑟
𝑚𝑎𝑥
at which function (2) reaches its maximum [10]:
2
2
2
0
0
0
0
0
2
2
2
2
4
2 exp
exp
8
exp
1
r
r
r
r
A
r
r
A r
r
a
a
a
a
a
=
−
+
−
−
=
−
−
max
max
max
0
0
0
1
0
r r
r
r
a
r
a
=
=
−
=
=
(3)
Thus, the most probable distance of the electron from the nucleus in the ground state corresponds to the first
Bohr radius. The graphical representation of the function
𝜌(𝑟)
is shown in Fig. 1.
Fig.1.Probability density of an electron in a hydrogen atom (n=1, l=0).
The probability of finding the electron in a spherical shell of unit thickness is equal to [11]:
2
2
2
3
0
0
0
2
4
2
( )
4
exp
exp
r
r
r
A r
r
a
a
a
=
−
=
−
(6)
The maximum value of the probability density for the electron in the ground state of the hydrogen atom is
determined as follows:
2
2
10
0
0
0
3
0
0
0
2
4
4
(
)
exp
1, 03 10 .
a
a
a
e
a
a
a
−
=
−
=
Based on the graph (Fig.1), it is evident that the function
𝜌(𝑟)
reaches its maximum value at
𝑟 = 𝑎
0
, which
indicates the correct operation of the software module and the adequacy of the numerical modeling.
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Let us find the probability of locating the particle in the region
𝑟 < 𝑎
0
. To do this, it is necessary to
evaluate the following integral:
0
2
1
3
2
0
0
0
4
2
5
exp
1
0,323
a
r
r
dr
a
a
e
=
−
= −
(7)
Thus, the probability of finding the electron at a distance less than one Bohr radius from the nucleus is
approximately 32.3%. This result has important methodological significance, as it demonstrates the statistical
nature of describing microparticles in quantum mechanics and allows for the visualization of the probability
distribution in familiar coordinates. A visual representation of the integrated region is shown in Fig. 2.
Fig.2. The probability of finding the electron in the region
𝐫 < 𝐚
𝟎
(highlighted as the
shaded area).
To determine the probability of finding the particle in the region
𝑎
0
< 𝑟 < ∞
, it is necessary to calculate the
following integral, which represents the area under the probability density function within the specified interval
(Fig.3):
0
0
2
2
3
2
0
0
0
2
3
2
0
0
0
4
2
2
5
exp
exp
0, 68.
2
4
4
a
a
a r
a r
a
r
r
r
dr
a
a
a
e
=
−
=
−
+
+
=
=
Fig.3. Probability density of an electron in a hydrogen atom (n=1, l=0).
The probability of detecting a particle in the range from
𝑎
0
to ∞ is shown
in red on the graph.
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International Journal of Pedagogics (ISSN: 2771-2281)
Consequently, the relationship
ω
1
+
ω
2
=1, is satisfied, which indicates compliance with the boundary condition of
the wave function. This result is also confirmed by graphical analysis: the integral area under the probability
density curve covers the entire domain (Fig. 4), which demonstrates the fulfillment of boundary conditions and
the accuracy of the numerical modeling.
Fig.4. Probability density of an electron in a hydrogen atom (n=1, l=0).
The probability of detecting a particle in the range from
0
to ∞ is shown
in red on the graph.
The wave function
𝜓(𝑟)
has a maximum at
𝑟 = 0
and decreases exponentially with increasing distance
𝑟
from
the nucleus. This corresponds to the analytical form of the function [12,13]:
0
( )
exp
r
r
A
a
=
−
The graph (Fig. 5) shows that the probability of finding the electron is highest near the nucleus (at point
𝑟 = 0
) and
then rapidly decreases as the distance from the nucleus increases.
Fig.5. Wave function of the 1s electron in a hydrogen atom
Although
𝜓(𝑟)
is maximum at
𝑟 = 0
, the maximum probability of finding an electron is not at the center, but at a
distance
𝑟 = 𝑎
0
, as can be seen from the probability density graph [14]:
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2
2
( )
( )
r
r
r
=
which has a maximum at
𝑟 = 𝑎
0
. Therefore, for a complete picture, it is necessary to analyze not only
𝜓(𝑟)
, but
also
𝜌(𝑟)
.
The graph (Fig.5) confirms that the 1s state is the most stable and symmetric state of the electron, in which
the electron is, on average, located at a distance approximately equal to one Bohr radius from the nucleus. This
corresponds to the fundamental principles of quantum mechanics and the hydrogen atom model.
The given example demonstrates that the wave function not only formally describes the state of the
electron but also provides quantitative information about the probabilistic distribution of its position. Integrating
graphical materials and analytical calculations into the educational process contributes to a deeper understanding
by students of such fundamental concepts as probability density and quantization [15].
The above problems allow students to consolidate key ideas and methods used in the quantum mechanical
description of the hydrogen atom. They help develop an understanding of the probabilistic nature of electron
localization in various regions of the atom, as well as the role of the wave function in the distribution of probability
density. Solving these problems covers both the basic theoretical principles and the development of practical skills
in applying mathematical tools of quantum mechanics.
To deepen and reinforce the theoretical concepts covered in the given topic, the following problems are
provided:
1. Calculate the probability of finding the electron in a hydrogen atom in the ground state (
𝑛 = 1
) within the
interval
𝑎
0
< 𝑟 < 3𝑎
0
. Compare the obtained value with the graphical representation shown in Fig.6.
2. Determine the values of the dimensionless variable
𝑥 = 𝑟/𝑎
0
at which the probability density function of the
electron in the ground state of the hydrogen atom reaches the value
10
( )
0,6 10
x
=
.
3. Calculate the probability of finding the electron in the interval
𝑎
0
< 𝑟 < 2𝑎
0
(
ω
1
) and in the interval
2𝑎
0
< 𝑟 <
3𝑎
0
(
ω
2
) for the quantum state with
𝑛 = 1
(Fig. 6). Find the sum of probabilities
ω
1
+
ω
2
and compare the result
with the answer obtained in Problem No1.
Fig.6.
Below are control questions that can be used to assess
students' knowledge on this topic:
1. What is a wave function and what information does
it provide about the system?
2. Write the Schrodinger equation for the hydrogen
atom and explain the meaning of its terms.
3. What physical quantities can be obtained from the
solution of the Schrodinger equation for the hydrogen
atom?
4. How can the probability of finding an electron within
a radius r from the nucleus of a hydrogen atom be
interpreted?
5. What is the probability density function for an
electron in a hydrogen atom, and how is it derived from
the wave function?
6. What is the wave function of an electron in a
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International Journal of Pedagogics (ISSN: 2771-2281)
hydrogen atom and how is it related to the probability
of finding the electron at a specific point in space?
7. What physical information is contained in the square
of the modulus of the wave function?
8. What are the main differences between the classical
model of the atom and the quantum mechanical model
of the hydrogen atom?
CONCLUSION
The hydrogen atom in quantum mechanics serves as an
ideal model for studying the fundamentals of quantum
systems. It acts as a starting point for understanding
key
concepts
such
as
energy
quantization,
superposition of states, the uncertainty principle, and
the wave nature of particles. However, teaching this
topic may present pedagogical challenges. The main
difficulties include the transition from classical
representations
to
quantum
concepts,
the
mathematical complexity of the Schrodinger equation,
energy quantization, and the abstract nature of wave
functions and the uncertainty principle.
To overcome these challenges and help students
master the fundamental principles of quantum
mechanics, instructors should use a variety of methods
and approaches, such as visualization, analogies, and
the gradual introduction of new concepts.
Solving problems in quantum mechanics requires
students to have a solid foundation in mathematics,
including topics such as trigonometric functions,
differentiation, integration, and graphing complex
functions.
Pedagogical research aimed at improving the quality of
teaching the topic “Hydrogen Atom” in the context of
quantum mechanics will continue.
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https://doi.org/10.47390/1342V3I3Y2023N
