Towards Quantum Mechanical Model of Atom
Sohail Awan
Greetings! I am Sohail, a dedicated educator hailing from the vibrant and culturally diverse land of Pakistan.
In view of the shortcoming of the Bohr’s model, attempts were made to develop a more suitable and general model for atoms. Two important developments which contributed significantly in the formulation of Quantum Mechanical Model were :?
1. Dual behaviour of matter,?
2. Heisenberg uncertainty principle.?
Louis de Broglie (1892 – 1987)
Louis de Broglie, a French physicist, studied history as an undergraduate in the early 1910’s. His interest turned to science as a result of his assignment to radio communications in World War I. He received his Dr. Sc. from the University of Paris in 1924. He was professor of theoretical physics at the University of Paris from 1932 untill his retirement in 1962. He was awarded the Nobel Prize in Physics in 1929
Dual Behaviour of Matter (Quantum Mechanical Model)
The first step towards Quantum Mechanical Modelis take French physicist, de Broglie, in 1924 proposed that matter, like radiation, should also exhibit dual behaviour i.e., both particle and wavelike properties. This means that just as the photon has momentum as well as wavelength, electrons should also have momentum as well as wavelength, de Broglie, from this analogy, gave the following relation between wavelength (λ) and momentum (p) of a material particle.?
where m is the mass of the particle, v its velocity and p itsmomentum. de Broglie’s prediction was confirmed experimentally when it was found that an electron beamundergoesdiffraction, a phenomenoncharacteristicof waves. This fact has been put to use in making an electronmicroscope, which is based on the wavelike behaviour of electrons just as an ordinary microscope utilises the wave nature of light. An electronmicroscopeis a powerful tool in modern scientific research because itachievesa magnification of about 15 million times.
It needs to be noted that according to de Broglie, every object in motion has a wave character. The wavelengths associated with ordinary objects are so short (because of their large masses) that their wave properties cannot be detected. The wavelengths associated with electrons and other subatomic particles (with very small mass) can however be detected experimentally.
Werner Heisenberg (1901 – 1976)
Werner Heisenberg (1901 – 1976) received his Ph.D. in physics from the University of Munich in 1923. He then spent a year working with Max Born at Gottingen and three years with Niels Bohr in Copenhagen. He was professor of physics at the University of Leipzig from 1927 to 1941. During World War II, Heisenberg was in charge of German research on the atomic bomb. After the war he was named director of Max Planck Institute for physics in Gottingen. He was also accomplished mountain climber. Heisenberg was awarded the Nobel Prize in Physics in 1932.
Heisenberg’s Uncertainty Principle
The first step towards Quantum Mechanical Modelis take?Werner Heisenberg a German physicist in 1927, stated uncertainty principle which is the consequence of dual behaviour of matter and radiation. It states that it is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron.
Mathematically, it can be given as in equation:
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where ?x is the uncertainty in position and ?px (or ?vx ) is the uncertainty in momentum (or velocity) of the particle. If the position of the electron is known with high degree of accuracy (?x is small), then the velocity of the electron will be uncertain [?(vx ) is large]. On the other hand, if the velocity of the electron is known precisely (?(vx ) is small), then the position of the electron will be uncertain (?x will be large). Thus, if we carry out some physical measurements on the electron’s position or velocity, the outcome will always depict a fuzzy or blur picture.
The uncertainty principle can be best understood with the help of an example. Suppose you are asked to measure the thickness of a sheet of paper with an unmarked metrestick. Obviously, the results obtained would be extremely inaccurate and meaningless, In order to obtain any accuracy, you should use an instrument graduated in units smaller than the thickness of a sheet of the paper. Analogously, in order to determine the position of an electron, we must use a meterstick calibrated in units of smaller than the dimensions of electron (keep in mind that an electron is considered as a point charge and is therefore, dimensionless). To observe an electron, we can illuminate it with “light” or electromagnetic radiation. The “light” used must have a wavelength smaller than the dimensions of an electron. The high?momentum photons of such light p=h/ λ
would change the energy of electrons by collisions. In this process we, no doubt, would be able to calculate the position of the electron, but we would know very little about the velocity of the electron after the collision.
Significance of Uncertainty Principle in?Quantum Mechanical Model
Quantum Mechanical Model is more Significancefor us. One of the important implications of the Heisenberg Uncertainty Principle is that it rules out existence of definite paths or trajectories of electrons and other similar particles. The trajectory of an object is determined by its location and velocity at various moments. If we know where a body is at a particular instant and if we also know its velocity and the forces acting on it at that instant, we can tell where the body would be sometime later. We, therefore, conclude that the position of an object and its velocity fix its trajectory. Since for a sub-atomic object such as an electron, it is not possible simultaneously to determine the position and velocity at any given instant to an arbitrary degree of precision, it is not possible to talk of the trajectory of an electron.?
The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic objects and is negligible for that of macroscopic objects. This can be seen from the following examples. If uncertainty principle is applied to an object of mass, say about a milligram (10–6 kg), then
The value of ?v?x obtained is extremely small and is insignificant. Therefore, one may say that in dealing with milligram-sized or heavier objects, the associated uncertainties are hardly of any real consequence.?
In the case of a microscopic object like an electron on the other hand. ?v.?x obtained is much larger and such uncertainties are of real consequence. For example, for an electron whose mass is 9.11×10–31 kg., according to Heisenberg uncertainty princi
le
It, therefore, means that if one tries to find the exact location of the electron, say to an uncertainty of only 10–8 m, then the uncertainty ?v in velocity would be
?
which is so large that the classical picture of electrons moving in Bohr’s orbits (fixed) cannot hold good. It, therefore, means that the precise statements of the position and momentum of electrons have to be replaced by the statements of probability, that the electron has at a given position and momentum. This is what happens in the quantum mechanical model of atom.?
Reasons for the Failure of the Bohr Model
One can now understand the reasons for the failure of the Bohr model. In Bohr model, an electron is regarded as a charged particle moving in well defined circular orbits about the nucleus. The wave character of the electron is not considered in Bohr model. Further, an orbit is a clearly defined path and this path can completely be defined only if both the position and the velocity of the electron are known exactly at the same time.?
This is not possible according to the Heisenberg uncertainty principle. Bohr model of the hydrogen atom, therefore, not only ignores dual behaviour of matter but also contradicts Heisenberg uncertainty principle. In view of?these inherent weaknesses in the Bohr model, there was no point in extending Bohr model to other atoms. In fact an insight into the structure of the atom was needed which could account for wave-particle duality of matter and be consistent with Heisenberg uncertainty principle. This came with the advent of quantum mechanics.