Quantum Mechanical Model of the Atom

The wave equation of Schrodinger's and its solutions serve as the foundation for the quantum mechanical model. The concept of shells, subshells, and orbitals is introduced by the wave equation solution. Newton's laws of motion, the foundation of conventional mechanics, can be used to explain the movement of entities that people encounter daily accurately. 

According to classical mechanics, a particle's location and momentum determine its physical nature. If one is aware of both of these characteristics, one can use classic mechanics to forecast the system's prospective state in relation to the force operating on it. The dual nature of matter, which is important for microscopic particles, is ignored by classic mechanics. As a result, it cannot account for the mobility of small particles. 

A novel mechanical model termed a quantum mechanical model of atom was established on the basis of the dual nature of microscopic particles and Heisenberg's principle. Thus this article provides more detailed knowledge of the quantum mechanical model of the atom and its important features.

What is the Quantum Mechanical Model of an Atom?

The model that underpins contemporary knowledge of the atom is regarded as the quantum mechanical model of atom. Schrodinger's wave equation cannot be precisely solved for a multi-electron atom, making its application to multi-electron atoms challenging. Approximate techniques were used to get around this problem. The Schrodinger wave equation was used to determine the structure of an atom, which resulted in the development of the quantum mechanical model of atom.

Schrodinger Wave Equation

The Schrodinger wave equation describes how the wave function of space changes based on the field of force that an electron is moving through. The time-independent Schrodinger equation is,

^{\[\hat H\Psi  = E\Psi \]} 

Here H^ is known as the Hamiltonian operator.

The wave function, which is represented by the symbol Ψ, is a function of the particle's location coordinates Ψ(x, y, z) E is the system's energy.

^{\[\hat H = \left[ {\frac{{ - {h^2}}}{{8{\pi ^2}m}}\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}} \right) + V} \right]\]} 

The time-independent Schrodinger equation can also be written as,

^{\[\left[ {\frac{{ - {h^2}}}{{8{\pi ^2}m}}\left( {\frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {z^2}}}} \right) + V\psi } \right] = E\psi \]} 

Multiplying -8π²m/h² and rearranging,

^{\[\frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {z^2}}} + \frac{{8{\pi ^2}m}}{{{h^2}}}\left( {E - V} \right)\psi  + 0\]} 

The aforementioned Schrodinger wave equation is considered a Schrodinger wave equation of time independent since it excludes time as a variable. The particular values of E, the overall energy, allow for this equation's solution. i.e., the system's energy is quantised. The associated wave functions represent the atomic orbitals, which correspond to the permissible overall energy values known as Eigenvalues.

What are the Main Features of a Quantum Mechanical Model of an Atom?

• The electrons in atoms possess quantified energy.

• The wave-like characteristics of electrons directly contribute to the presence of quantified electronic energy levels. Schrodinger wave equation results provide the permitted levels of energy.

• Heisenberg's uncertainty principle states that it is impossible to identify an electron's precise location and momentum using absolute accuracy. As a result, the orbital was first established by quantum mechanics. The probability of discovering an electron is higher in the three-dimensional space known as orbital.

• The wave function Ψ, which depicts an atomic orbital, is obtained from the Schrodinger wave equation's result for the permitted energies of an atom. The wave function Ψ is an effective way to describe the wave behaviour of an electron in an orbital.

• There is no physical significance to the wave function on its own. Nevertheless, the probability of detecting the electron in a tiny volume dxdydz surrounding a point (x, y, z) is x proportional to |Ψ(x, y, z) |². This dxdydz |Ψ (x, y, z) |² is called probability density and is constantly positive.

Interesting Facts

• The area that an atom's electron will most probably be in for 90 per cent of the time is known as its atomic orbital.

• The electron could only take highly precise circular orbits surrounding the nuclei in the Bohr model. A radical divergence from that is the quantum mechanical model.

• The Schrodinger wave equation's solutions, known as wave functions, only provide the probability that an electron will be found at a specific location within the nucleus. 

• The term "electron cloud" is frequently used to describe where the electrons are located in the quantum mechanical model of an atom.

Key Features to Remember

• Bohr's theories regarding the position of the electron were replaced with an uncertainty parameter in the Schrodinger wave equation.

• The quantum mechanical model of the atom is recognised as the theoretical foundation for the current study of the atom.

• The only thing that can be said about the electron's position is the probability that it is someplace in a particular region.

• The probability of discovering an electron in a particular area of the atom is represented by the square of the wave function Ψ².