Quantum Mechanical Model of an Atom |ChemFam #09|

Quantum mechanical model of atom was established with the help of dual nature of matter and Heisenberg’s uncertainty principle. Classical mechanics based on Newton’s laws of motion could not be applied to the microscopic objects like electrons. It was successful in explaining the motion of macroscopic objects like the fall of apple from the tree etc which has just particle character but wave character is of no significance for it. But this mechanics could not explain the motion of microscopic particles which exhibit both wave-particle duality. Therefore quantum mechanics came into picture as it is a branch of science that takes into account the dual behaviour of matter (like electrons).

Quantum mechanics was developed independently in 1926 by Warner Heisenberg and Erwin Schrödinger. The model describes the position of an electron in a three dimensional space by using some mathematical functions called as wave functions in a probabilistic manner. The wave functions are often denoted as Ψ.The atomic wavefunctions are also called as orbitals.

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The model somewhat gets an update from the previously explored Bohr’s model of an atom. Neil Bohr was the first to explain quantitatively the general features of hydrogen atom i.e., structure and its spectrum. He extended the work of Rutherford to give the structure of atom. Once, the existence of the nucleus was proved through Rutherford’s model of atom, now the work left was to concentrate on the arrangement of electrons around the nucleus. Neil Bohr was successful in answering those questions. Bohr suggested that the electron around the nucleus revolves only in certain fixed circular path of fixed radius and energy. These circular paths are also known as orbits, stationary states or allowed energy states or energy levels. As the energy of an electron revolving in a fixed circular path called orbit is fixed. Therefore orbits are called energy states or energy shells. Hence, electrons when absorbs energy , they move from lower to higher energy levels. To its contrary electron moves from higher energy states to a lower energy state by emitting energy. Thus, we can say that energy of the electron only changes when it undergoes from one energy level to the other. But while revolving in a fixed orbit, they do not radiate or lose energy. In this manner, Bohr overcome Rutherford’s difficulty to account for the stability of atom. That is why electrons do not lose energy while revolving and fall into the nucleus.

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Applying wave mathematics to the electron wave, Erwin Schrödinger formulated an energy equation which is the basis for quantum mechanical model of an atom. The equation is quite complex, so we will not study in detail the equation here. Schrödinger equation is shortly written as,

ĤΨ = EΨ

It is for an atom or a molecule whose energy does not change with time. Here, Ĥ is a mathematical operator called as Hamiltonian operator. It is a set of mathematical operations which represents the total energy’s of the quantum particle( such as an electron in an atom). Thus Ĥ can be constructed by knowing the total energy of the system. The total energy of the system takes into account the kinetic energies of all subatomic particles( electrons, nuclei), attractive potential between the electrons and nuclei individually. The solution of this equation gives E and Ψ.

Bohr’s model of atom had limitations. The quantum mechanical model of atom was developed to eliminate those limitations. Towards the development of the model, the dual nature of matter proposed by de-Broglie and the Heisenberg’s uncertainty principle was of utmost importance. Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but unlike de-Broglie, Schrödinger properly thought of electrons in terms of three-dimensional stationary waves instead of circular stationary waves. The three-dimensional stationary waves, or wavefunctions are represented by the Greek letter psi, ψ.

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Hydrogen Atom and the Schrödinger Equation

The solution of Schrödinger equation for hydrogen give certain values, wave function(s) (ψ) of the electron which are associated with energy levels called orbitals.

Wave function: It is a mathematical function whose value depends upon the coordinates of the electron in the atom and does not carry any physical meaning. When an electron is in any energy level, the function correspond to that energy state. The wave function in case of hydrogen or hydrogen like species with one electron corresponds to atomic orbitals. The wave functions pertaining to one electron species are called one electron system. Schrödinger equation does not talk about the presence of an electron in a specified orbit, it just talks about the probability of finding the electron. The probability of finding the electron in at a point within an atom is proportional to the square of the wave function i.e., |ψ|² at that point. So, quantum mechanics when applied to hydrogen atom answers all the questions regarding the spectrum of hydrogen and others even Bohr failed to justify.

Multi Electron Atoms: Schrödinger equation when applied to multi electron atoms, the result obtained are not correct. So, usually for multi-electron atoms, some approximate methods are used. These methods are beyond our discussions as of now. These methods are purely based on mathematical formulations. We could consider multi electron atom such as He or Li etc. The difference lies in the consequence of increased nuclear charge. This results in contraction of the orbitals. The energy of hydrogen-like species depend only on the proncipal quantum number (n). Whereas the energies of the orbitals in multi-electron atom depends on both principal quantum number (n) and azimuthal quantum number (l).

Salient Features of Quantum Mechanical Model of an Atom:

Quantum mechanical model gives the successful picture of the atom. The following are some features of quantum mechanical model -

The energy of electrons is quantised that means that a certain specific amount of energy is associated with an electron present around the nucleus.
The electrons are only present in quantised energy levels which are a direct result of the wave-like properties of electrons and are permissible solutions of Schrödinger wave equation.
It replaces the concept of fixed circular orbits or trajectories by the probability of finding the electron at different points in an atom.
Each orbital is specified by a definite energy. One orbital can contain a maximum of two electrons. In multi-electron atoms, the electrons are filled in various orbitals of increasing energies. Therefore, for each electron there is a specified orbital wave function characteristic of the orbital that it occupies.
Wavefunction (ψ) has as such no physical significance but the square of wavefunction i.e., |ψ|² known as probability density which gives the probability of finding an electron at a point within an atom. From the value of |ψ|² at different points within an atom it is possible to predict the region around the nucleus where electron can most probably be found. The maximum probability region is actually what is atomic orbital.