Erwin Schrödinger, an Austrian physicist, used the wave-particle duality of the electron to develop and solve a complex mathematical equation that accurately described the behaviour of the electron in a hydrogen atom in 1926. The solution of Schrödinger’s equation resulted in the formation of the quantum mechanical model of the atom. Quantization of electron energies is a requirement in order to solve the equation. Through this article, one can learn all about the Quantum Mechanical Model of an atom. 

What is the Quantum Mechanical Model of Atom?

The Quantam Mechanical Model of an atom was proposed by Erwin Schrodinger in \(1926.\) He developed an atomic model taking into account both the wave and particle nature of the electron, which is known as the quantum mechanical model of the atom.

It describes the electron as a three-dimensional wave in the electronic field of the positively charged nucleus. The wave motion of the electron in this field is described with the help of a differential equation known as the Schrodinger wave equation.

Here,  \(x, y, z\) are the space coordinates.

\(m=\) mass of the electron; \(E=\) Total energy of the electrons

\(V = \) Total potential energy of the electrons

[math]\({\rm{\psi \;}}\left( {{\rm{Psi}}} \right) = \)[/math] Wave function of the electron.

It is also called Eigen-function.

For an electron with charge \(\left( { – e} \right)\) and nuclear charge \(\left( { + e} \right)\) (in case of \(H-\)atom), potential energy may be given as- \(V = – \frac{{{e^2}}}{r}\)

Therefore, the quantum mechanical model of atom Schrodinger wave equation may be written as:

Practice Exam Questions

Features of Quantum Mechanical Model

Schrodinger wave equation gives a better picture of the atom than Bohr’s model. Some of the salient features of the Quantum Mechanical Model of an atom are as follows-

• 1. The energy of the electrons in an atom is quantized, i.e., it can have only certain specific values.
• 2. The quantized energy levels in which electrons can be present are obtained from the solutions of the Schrodinger wave equation.
• 3. The exact position and the exact velocity (or momentum) of an electron cannot be determined simultaneously. Therefore, the path of the electron is only probable and not exact. This aspect has ultimately led to the concept of atomic orbitals.
• 4. The atomic orbital is represented by the wave function \({\rm{\psi }},\) which is also known as the orbital wave function. Since a number of such wave functions are possible for an electron, the corresponding atomic orbitals are also possible. The electron has definite energy in each orbital, and it cannot have more than two electrons.
• 5. The probability of finding an electron at a point in an atom is proportional to the square of the wave function, i.e., \({[{\rm{\psi }}]^2}.\) It is also called probability density and is positive. From the value of \({[{\rm{\psi }}]^2}\) at different points within the atom, one can predict the region around the nucleus where the electron will be most probably found or located

Dual Nature of Electrons (De Broglie’s Equation)

The properties of electrons indicate that they have a dual nature. An electron behaves both as a particle and as a wave. An electron has mass and possesses kinetic energy. Hence, it should be a particle, and at the same time, it can be diffracted in the same way as light waves, which are only possible when electrons have a wave nature.

Louis De-Broglie suggested that all material objects show a dual nature. Every object which possesses a mass and a velocity behaves both as a particle and as a wave.

According to de-Broglie, the wavelength \(\lambda \) of a particle of mass \(m,\) moving with a velocity \({\rm{\nu }}\) is given by-

\({\rm{\lambda }} = \frac{{\rm{h}}}{{{\rm{m\nu }}}}\)

Where \(h\) is Plank’s constant \( = 6.626 \times {10^{ – 34}}{\rm{\;Joule\;second}}.\) The quantity \({\rm{m\nu }}\) represents the momentum of the particle. Therefore, the above equation can also be expressed as-
\({\rm{\lambda }} = \frac{h}{p}\)

Heisenberg’s Uncertainty Principle

Werner Heisenberg, a German scientist, proposed a principle in \(1927\) known as Heisenberg’s Uncertainty Principle, which can be stated as-

It is impossible to simultaneously determine the position and momentum (or velocity) of a moving microscopic particle with absolute accuracy.

If \(\Delta x\) is the uncertainty in the determination of the position and \(\Delta p\) is the uncertainty in the determination of the momentum of a very small particle, then according to Heisenberg-

\(\Delta x{\rm{\;}}\Delta p \ge \frac{h}{{4{\rm{\pi }}}}\)

where h is Plank’s constant \(= 6.626 \times {10^{ – 34}}{\rm{\;J\;s}} = {\rm{\;}}6.626 \times {10^{ – 27}}{\rm{\;erg\;s}}\)

The above equation can also be represented as-

\(\Delta x{\rm{\;}}\Delta v \ge \frac{h}{{4{\rm{\pi m}}}}\)

Here \(\Delta v\) is the determination of the uncertainty in the velocity.

Electron is a tiny particle with a negligible mass of \(9.108 \times {10^{ – 28}}{\rm{g}}.\) Hence, the uncertainty on position and velocity for this microscopic particle is expected to be very large-

\(\Delta x\,\Delta v \ge \frac{{6.626 \times {{10}^{ – 27}}{\rm{\;}}}}{{4 \times 3.14 \times 9.108{\rm{\;}} \times {{10}^{ – 28}}}} \ge 0.579{\rm{\;erg\;s\;}}{{\rm{g}}^{ – 1}}\)

Hence, it can be concluded that the position and velocity of an electron can never be determined simultaneously and accurately.

Bohr’s Model of an Atom

In \(1913,\) Neil Bohr explained the stability of the atom based on the observations of his experiments. According to him:

• 1. An atom consists of a very small central core called the nucleus. The nucleus carries all the positive charge and most of the mass of the atom.
• 2. Electrons revolve around the nucleus in circular paths. These circular paths are called orbits. The centripetal force required for the circular motion is provided by the electrostatic force of attraction between the electrons and the nucleus.
• 3. Electrons can revolve only in some permissible orbits called shells or energy levels, each associated with a fixed amount of energy.
• 4. While an electron is revolving in orbit, it neither gains nor loses energy. Therefore, these orbits are called stationary states or energy levels.
• 5. An electron may jump from an orbit of higher energy to lower energy, thereby emitting energy.

For convenience, these energy levels are labelled \(K, L, M, N,\) and so on. The orbit closest to the nucleus is the \(K\) shell and has the least amount of energy, and the electrons present in it are \(K\) electrons, and so on with the successive shells and their electrons.

Reason for Failure of Bohr’s Model Atom

According to Bohr’s Model, an electron is a charged particle moving in a well-defined circular orbit around the nucleus. Bohr believed the path of the electron could be traced only by considering its particle nature. He did not consider its wave nature. His model was in contradiction with the uncertainty principle, according to which the path traced by the electron is only probable and not exact. It was later established that an electron and other similar microscopic particles have dual nature, i.e., they can behave as a particle and can also have a wave nature. Ultimately, Bohr’s Model lost its significance and could not be extended to other atoms.

Summary

Through this article, we studied the quantum mechanical model of atom definition, the Dual nature of matter and Heisenberg’s uncertainty principle. Also, we learned about the reason for the failure of Bohr’s Model of Atom and how the Quantum Mechanical model explains the dual nature of an electron through Schrodinger’s equation. We discussed the quantum mechanical model atom orbital and the salient features of the model.

FAQs on Quantum Mechanical Model

Q.1. When was the quantum mechanical model of atom created?
Ans: In \(1926,\) Erwin Schrodinger developed the quantum mechanical model of the atom, considering both the wave and particle nature of the electron.

Q.2. What are the features of the quantum mechanical model of the atom?
Ans: The features of the Quantum Mechanical Model of an atom are as follows-
1. The energy of the electrons in an atom is quantized, i.e., it can have only certain specific values.
2. The quantized energy levels in which electrons can be present are obtained from the solutions of the Schrodinger wave equation.
3. The exact position and the exact velocity (or momentum) of an electron cannot be determined simultaneously. Therefore, the path of the electron is only probable and not exact. This aspect has ultimately led to the concept of atomic orbitals.
4. The atomic orbital is represented by the wave function which is also known as the orbital wave function. Since a number of such waves functions are possible for an electron, the corresponding atomic orbitals are also possible. The electron has definite energy in each orbital, and it cannot have more than two electrons.
5. The probability of finding an electron at a point in an atom is proportional to the square of the wave function, i.e., It is also called probability density and is positive. From the value of at different points within the atom, one can predict the region around the nucleus where the electron will be most probably found or located.

Q.3. What is the importance of the quantum mechanical model?
Ans: The importance of the quantum mechanical model is to understand chemistry because it explains how electrons exist in atoms and how they determine the chemical and physical properties of the elements. 

Q.4. What does the quantum mechanical view of the atom require?
Ans: The quantum mechanical model of the atom is based on the dual nature of the electron; that is, an electron behaves like a particle as well as a wave.

Q.5. What is the quantum mechanical model of the atom?
Ans: The quantum mechanical model of the atom describes the electron as a three-dimensional wave in the electronic field of the positively charged nucleus. The wave motion of the electron in this field is described with the help of a differential equation known as the Schrodinger wave equation.

Q.6. How is the quantum mechanical model different?
Ans: The quantum model is different because it shows the dual nature of the electron.

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