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Towards the end of the 19th century, many physicists felt that all principles of physics had been discovered and little remained but to clear up a few minor problems and to improve experimental methods in order to investigate the next decimal place. Newton’s mechanics had been brought to a high degree of sophistication. The Newton’s laws could explain the motion of everyday objects and planets. It was thought that Newtonian mechanics (now called as classical mechanics) could explain event the motion of atoms and subatomic particles. However, towards the end of 19th century, experimental evidence accumulated showing that classical mechanics failed when it was applied to very small particles and it took until 1920’s to discover the appropriate concepts and equations for describing them. This new mechanics is called quantum mechanics. Classical physics predicts a precise trajectory for particles, with precisely specified locations and momenta at each instant and allows translational, rotational and vibrational modes of motion to be excited to any energy simply by controlling the forces that are applied. Failures of classical physics Let us review some experimental evidences which showed that several concepts of classical physics are untenable. 1. Black body radiation A hot object emits electromagnetic radiation. We all know, for instance that when an iron bar is heated, it first becomes red and then more and more red when heated further. On further heating the radiation becomes white and then blue. The wavelength shifts from higher to lower and the frequency shifts from lower to higher as the body is heated. The exact frequency spectrum depends on the body itself but an ideal body, called as a black body which absorbs and emits all frequencies, serves as an idealization for any radiating material. A good approximation to a black body is a pin hole in an empty container maintained at a constant temperature because any radiation leaking out of the hole has been absorbed and re-emitted inside so many times that it has come to thermal equilibrium with the walls. Fig. 1 shows the variation of energy output as a function of frequency.
The figure shows that the maximum of the peak shifts to longer frequency or shorter wavelength as the temperature is raised. An analysis of the data led Wien to formulate Wien’s displacement law:
C2=1.44cm K where λmax is the wavelength corresponding to the maximum of the distribution at a temperature T. The constant C2 is called the second radiation constant. Using its value we can predict that λmax = 2900 nm at 1000K A second feature of the black body radiation has been noticed in 1879 by Joseph Stefan who considered the total energy density ε, the total electromagnetic energy in a region divided by the volume of the region (ε =E/V). The energy density increases as the temperature is increased and Stefan-Boltzmann law states that
An alternative form is in terms of excitance M, the power emitted by a region of surface divided by the area of the surface
; σ = 5.67 x 10-8 Wm-2 K-4
σ is called Stefan-Boltzmann constant. This law implies that 1 cm2 of the surface of a black body at 1000 K radiates about 6W when all wavelengths of the emitted radiation are taken into account. Rayleigh studied the black body radiation theoretically and thought of the electromagnetic field as a collection of oscilllators of all possible frequencies. He arrived at Rayleigh-Jeans law
ρ is the proportionality constant between d λ and the energy density in that range of wavelengths; k= Boltzmann constant = 1.381 x 10-23 JK-1. Although Rayleigh-Jeans law is quite successful at long wavelengths it fails at short wavelengths. The equation predicts that oscillators of very short wavelength (corresponding to uv, X rays, gamma rays etc) are strongly excited even at room temperature. This absurd result which implies large amount of energy is radiated in the high frequency region of the electromagnetic spectrum is called ultraviolet catastrophe. According to classical physics even cool objects should glow in darkness, there should in fact be no darkness. 2.Planck distribution The first person to offer a successful explanation of blackbody radiation was the German physicist Max Planck in 1900. Like Rayleigh and Jeans before him, Planck assumed that the radiation emitted by the black body was caused by the oscillations of the electrons in the constituent particles of the material body. These electrons were pictured as oscillating in an atom much like electrons oscillate in an antenna to give off radio waves. In these “atomic antennae”, however, the oscilllations occur at much higher frequency i.e., uv, vis, IR rather than radio wave region. Implicit in the derivation of R-J is the assumption that the energies of the electronic oscillators responsible for the emission of the radiation could have any value whatsoever; this is one of the basic assumptions of classical physics. In classical physics, the variables that represent observables (such as position, momentum and energy) can take on a continuum of values. Planck thought that he has to break away from this assumption to produce expressions that would reproduce experimental data. He made the revolutionary assumption that the energies of the oscillators were discrete and had to be proportional to an integral multiple of the frequency or in equation form E = nhν where E is the energy of an oscillator, n is an integer, h is proportionality constant called Planck’s constant; h = 6.626 x 10 –34Js. Therefore, a ray of light of a certain frequency can be thought of consisting of a stream of particles each one having an energy hν. These particles are called photons. The implications of Planck’s hypothesis for blackbody radiation are as follows. The particles in the walls of the body are in thermal motion and this motion excites the oscillators of the electromagnetic field. At equilibrium, there is no net flow of energy between the walls and the field. According to classical theory all the oscillators of the field share equally the energy supplied by the walls and so even the highest frequencies are excited. In quantum theory, however, the oscillators are excited only if they can acquire an energy of at least hν. This is too large for the walls to supply in case of high frequency oscillators and so they remain unexcited. The effect of quantization is to eliminate the contribution from the high frequency oscillators. Detailed calculations resulted in Planck’s distribution law for black body radiation. The energy density in the range λ to λ + d λ is given by
where The Planck’s density of states is similar to R-J expression apart from all important factors involving exponentials. When the λ is short, the term (hc/ λkT) >>1 and e(hc/ λkT) → faster than λ5 →0; therefore,ρ→0 as λ→0 or ν→ . Hence energy density approaches zero at high frequencies, in agreement with the observation. For long wavelengths hc/ λkT <<1, and the denominator in the Planck distribution can be replaced by
When this approximation is substituted in equation 5 we find that the Planck distribution reduces to the R-J law. The Planck distribution also accounts for the Stefan-Boltzmann and Wien laws. Stefan-Boltzmann law can be obtained by integrating the energy density over all wavelengths from λ=0 to λ= , which gives
Substitution of the values of the fundamental constants gives σ = 56.704 nW m-2 K-4 in accord with the experimental value. Wien law: It can be obtained by looking for the wavelength at which (d ρ /d λ)=0. When we take the derivative, set it equal to zero and make the approximation that the wavelength is so short that λ<<hc/kT, we obtain
This result lets us identify the second radiation constant as c2= hc/k = 1.439 cm K, which is also in agreement with the experiment. In essence, Planck found that oscillators cannot possess arbitrary amounts of energy. The discovery of quantisation (Latin quantum – amount) led to quantum mechanics. The whole structure of physics had to be revised. 3. Heat capacities: Accounting for black body radiation involves examining how energy is taken up by the electromagnetic field. Accounting for the heat capacities of solids involves examining how energy is taken up by the vibrations of particles. Therefore, a study of heat capacities can also be expected to show evidence of quantization. If classical physics were valid, the mean energy of vibration of an atom oscillating in one dimension in a solid should be kT (acc to equipartition principle). Each of the N atoms in a block is free to vibrate in 3 dimensions and so the total vib energy of a block is expected to be 3NkT. The molar vib energy is 3NAkT = 3RT (NA =Avogadro No.) The molar heat capacity at constant volume is and so according to classical physics Cv,m = 3R, independent of the temperature. This result if known as Dulong and Petit’s law. They proposed it on the basis of some experimental evidence. Einsten tested Dulong and Petit’s law at low temperatures when technological advances were available. He found significant deviations and all metals were found to have molar heat capacities lower than 3R at low temperatures and the values approached to zero as T→0
Einstein assumed that each atom could vibrate about its equilibrium position with a single frequency ν and then used Planck’s hypothesis to assert that the energy of any oscillation is nh ν, n being an integer. First he calculated the molar vibrational energy and then differentiated it w.r.t. T to get Cv,m
At high temperatures (kT>>h ν) the exponentials can be expanded as 1-(h ν/kT) +…… and higher terms ignored. The result is
in agreement with the classical result. At low temperatures e(-h ν / kT ) →0 and so Einstein’s formula accounts for the decrease of heat capacity at low temps. The physical reason for this success is that, as in the Planck’s calculation, at low temps only a few oscillators possess enough energy to begin oscillating. At higher temps there is enough energy available for all the oscillators to become active; all 3N oscillators contribute and heat capacity approaches its classical value. Even though Einstein’s relation gives a curve similar in shape to the experimental curve, there is poor numerical agreement. This arises from the fact that Einstein assumed that all atoms oscillate with the same frequency. In fact, they oscillate with different frequencies. Debye improved the relation further by averaging over all the frequencies present and the formula is known as Debye formula
4. Atomic and molecular spectra The most compelling evidence for the quantization of energy comes from the observation of the frequencies of radiation absorbed and emitted by atoms and molecules.
The obvious feature of both is that radiation is emitted or absorbed at a series of discrete frequencies. This observation can be understood if the energy o the atoms or molecules is also confined to discrete values, for then energy can be discarded or absorbed only in discrete amounts. Then if the energy of an atom decreases by ΔE, the energy is carried away as radiation of freq ν= ΔE/h, and a line appears in the spectrum. Wave-particle duality At this stage we have established that the energies of the electromagnetic field and of oscillating atoms are quantized. Now we shall see the exptal evidence that led to the revision to two other concepts concerning natural phenomena. One experiment shows that electromagnetic radiation – which classical physics treats as wave-like – actually also displays the characteristics of particles. Another expt shows that electrons – which classical physics treats as particles – also display the characteristics of waves. (a) Particle character of e.m radiation The observation that em radiation of freq ν can possess only the energies 0,h ν, 2 h ν, … suggests that it can be thought of as consisting of 0,1,2,… particles, each particle having an energy h ν.These particles are called photons. Further evidence for the particle-like character of radiation comes from the measurement of the energies of electrons produced in the photoelectric effect. This effect is the ejection of electrons from metals when they are exposed to uv radiation. The exptal characteristics of the p.e. effect are as follows: i. No electrons are ejected regardless of the intensity of the light, unless its freq exceeds a threshold value characteristic of the metal./ ii. K.E. of ejected electrosn is linearly proportional to the freq of the incident light. iii. Even at low intensities, electrons are ejected immediately if the freq is above threshold. According to classical physics, electromagnetic radiation in an electric field oscillating perpendicular to its direction of propagation and the intensity of radiation is proportional to the square of electric field. As the intensity increases, the electrons eventually break away with a threshold freq ν0. However, the above 3 observations regarding p.e. effect suggest that it depends on an electron being ejected if it is involved in a collision with a particle like projectile carrying enough energy to knock it out of the metal. If we suppose that the projectile is a photon of energy h ν, ν being the freq of light, then the conservation of energy requires that the K.E. of electron should obey ½ mv2=h ν – φ φ is the work function of the metal; the energy required to remove an electron. If h ν is less than φ, photoejection cannot occur – this accounts for observation 1. K.E. is proportional to freq – this for observation 2. When a photon collides with an electron it gives up al its energy and we expect electrons to appear as soon as collision occurs – observation 3. Hence a corpuscular theory of light (light consisting of particles) must be brought in. b. Wave character of particles: Diffraction of electrons: In 1925 Clinton Davisson and Lester Germer, performed a crucial experiment by which people were forced to consider the wave nature of particles. They observed diffraction of electrons by a Ni crystal. Diffraction is a characteristic properties of waves because it occurs when there is interference between their peaks and troughs. Depending on whether the interference is constructive or destructive the result is a region of enhanced or diminished intensity. Accidentally, there was a rise in temperature which caused their polycrystalline sample to anneal and the ordered planes of atoms then acted as a diffraction grating. At the same time, G.P. Thomson, working in Scotland, showed that a beam of electrons was diffracted when passed through a thin gold foil. The Davisson-Germer expt, which has since been repeated with other particles (including molecular hydrogen), shows clearly that particles have wave-like properties. We have also seen that waves of electromagnetic radiation have particle like properties. Thus we are brought to the heart of modern physics. When examined on an atomic scale, the classical concepts of particle and wave melt together, particles taking on the characteristics of waves and waves the characteristics of particles.
[math]½ mv\ltsup\gt2\lt/sup\gt=h*ν – φ[/math]