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- Particle nature could explain the black body radiation and photoelectric effect satisfactorily but on the other hand, it was

not consistent with the known wave behaviour of light which could account for the phenomena of interference and diffraction. - The only way to resolve the dilemma was to accept the idea that light possesses both particle and wave-like properties, i.e., light

has dual behaviour. - Depending on the experiment, we find that light behaves either as a wave or as a stream of particles. Whenever radiation interacts

with the matter, it displays particle like properties in contrast to the wavelike properties (interference and diffraction), which it exhibits when it propagates. Thus light has dual nature. Some microscopic particles like electrons also exhibit this wave-particle duality

### De-Broglie Equation:

- 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 the wavelength (λ) and momentum (p) of a material particle.

where m is the mass of the particle, v its velocity and p its momentum.

#### Notes:

- De Broglie’s prediction was confirmed experimentally when it was found that an electron beam undergoes diffraction, a

phenomenon characteristic of waves. This fact has been put to use in making an electron microscope, which is based on the

wave like the behaviour of electrons just as an ordinary microscope utilizes the wave nature of light. An electron microscope is

a powerful tool in modern scientific research because it achieves a 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.

### Heisenberg’s Uncertainty Principle:

- 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

- Where Δx is the uncertainty in position, Δp is the uncertainty in momentum and Δv is the uncertainty in velocity of the particle.
- If the position of the electron is known with a high degree of accuracy (Δx is small), then the velocity of the electron will be

uncertain [Δ(vx) is large]. On the other hand, if the velocity of the electron is known precisely (Δ(vx ) is small), then the

position of the electron will be uncertain (Δx 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.

#### Notes:

- 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 metre scale. 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 scale calibrated in units of smaller

than the dimensions of electron ( the electron is considered as a point charge). - 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 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:

- 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 subatomic 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 about the trajectory of an electron. - The effect of Heisenberg Uncertainty Principle is significant only for the motion of microscopic objects and is negligible for that of macroscopic objects.
- Thus 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 an atom.

### Schrodinger’s Wave Equation:

- Schrodinger independently studied the nature of electron and gave equation which is known as Schrodinger wave equation.

E = Total energy of the system

V = Potential energy of the system

m = Mass of electron

h = Planck’s constant

∇ = Operator

ψ = Wave function and is the amplitude of the electron wave.

- ψ
^{2}gives the probability of finding the electron at various places in a given region of space to another. Thus probabilities of finding the electron in different regions are different. This is in accordance with uncertainity principle. - The acceptable values of wave functions provide the regions around the nucleus in which probability of finding the electron is

maximum. These regions are called orbitals. Thus solving Schrodinger wave equation we can get the shape of the orbitals.This equation also gives certain specific numbers called quantum numbers which specify the location of an electron in an atom.

### Concept of Orbitals:

- 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 an atom.
- After solving Schrodinger wave equation, the acceptable values of wave functions provide the regions around the nucleus in which probability of finding the electron is maximum. These regions are called orbitals. Thus solving Schrodinger wave equation we can get the shape of the orbitals.
- Thus electron is a cloud of negative charge with different shapes.
- The three-dimensional region in the space around the nucleus in which the probability of finding the electron is maximum is called orbital.
- In each orbital, the electron has a definite energy. The energy of an orbital is lower if it is concentrated near the nucleus.

- It is to be noted that orbital is not same as that of orbit.
- There are four types of orbitals known to us till now, they are s, p, d and f orbitals

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