HEISENBERG PRINCIPLE OF INDETERMINANCY
A finite wave packet ( or train of waves ) with a length D x along the x-axis
cannot be monochromatic . Instead of a frequency range D w, we may use a range of wave parameters , D k. The wave parameter ( or wave number ) is k = 2 Õ/ l. D x and D k are connected as D x D k ³ 1. This equation is valid for any wave process. Let us apply it to the de-broglie wave associated with a particle moving along the x-axis and having a momentum px = p. It follows from de broglie’s hypothesis , p = h / l.
A similar relation should exist between D p and D k, incremental changes in p and k :
D p = D k h
and so,
D k = D p/ h = D px/h
it follows
D x D px ³ h ………………………………. (1)
If a particle is moving along the y or z axis so that the rectangular components of its momentum are py and pz, it follows from above :
D y D py ³ h ………………………………………(2)
D z D pz ³ h ………………………………………(3)
(1) to (3) are called Heisenberg’s indeterminacy or uncertainty relations.
The uncertainty relations show that the x, y, z co ordinates of a particle and the respective rectangular components of its momentum , px, py and pz cannot simultaneously be equal to x and px, y and py, z and pz or that the quantities connected by the uncertainty relations cannot be zero simultaneously.
Apart from the uncertainty relations for position and momentum, there is another relation of a similar kind for energy and time.
This relation is of special importance to atomic and nuclear physics.
BOHR’S THEORY OF ATOMIC SPECTRA
In 1913, Bohr came out with the first non classical theory of the atom. Bohr based his theory on three postulates, known as Bohr postulates. Bohr advanced his theory to explain the structure of the hydrogen atom,and generally similar systems made of a nucleus of charge Ze and one
electron in motion around this nucleus. Exampls of such systems are singly ionized helium, doubly ionized lithium etc.
QUANTUM THEORETICAL INTERPRETATION OF BOHR’S POSTULATES
1) Using Bohr’s postulates as a basis , the emission of spectral lines by an excited atom and also the absorption of light by atoms was examined. The emission and absorption of radiation has been fully explained in conformity with observations, and Bohr’s postulates have been interpreted by quantum mechanics.
2) We consider the electron in a hydrogen atom ( or in a one-electron ion ) in a certain energy state. Let n be the principal quantum number characterizing this state and determining the electron energy en. The state of an electron in quantum mechanics is uniquely specified by giving its wave function . Let yn (x,y,z,t) be the wave function of an electron in a state of energy en . The probability that the electron will be in a particular element of volume D v at a particular time is proportional to the square of its wave function, and so we get ½ yn ½2 D v . The most important result obtained by quantum mechanics is that if an electron is in an energy state characterized by the principal quantum number,n, the probability that the electron will be found in a particular element of volume in the atom is independent of time, that is, time – invariant. From the view point of classical theory, an electron in such a state is not oscillating in the atom and cannot radiate, that is, its energy remains unchanged. The state of an electron characterized by a particular energy en is called stationary, that is time invariant. This is Bohr’s first postulate.
3) Bohr’s second postulate has likewise been explained by quantum mechanics in simple terms. This is also true of his third postulate ( the rule of frequencies). If external influences affect the state of an electron so that it is forced to jump from state n to state m, the probability that the electron will be in an element of volume D v in the atom is no longer given by ½ yn ½2 D v . If the electron undergoes a quantum transition between states n and m, it will reside part of the time in state n and part of the time, in state m. So we may say that the probability that the electron will be in a particular element of volume D v is now given by yn ym D v . It is proved in quantum mechanics , that under the circumstances , the electron has a dipole electric moment which varies periodically with time. The frequency , w , at which the dipole moment changes is the same as the frequency of radiation emitted when the electron jumps from state n to state m and is given by w = ( en - em )/ h, in precise agreement with Bohr’s third postulate.