Energy Quantization

QUANTUM MECHANICS

1. QUANTUM THEORY Frederick Nti Energy Quantization

2. It was once thought that the motion of atoms and subatomic particles could be expressed using classical mechanics T he laws of motion introduced by Isaac Newton these laws were very successful at explaining the motion of everyday objects and planets. However, experimental evidence showed that classical mechanics failed when it was applied to particles as small as electrons I t took until the 1920s to discover the appropriate concepts and equations for describing them. The concepts of this new mechanics are described in quantum mechanics The Origins of Quantum M echanics

3. Energy quantization The  quantization  of  energy  refers to the absorption or emission of  energy  in discreet packets, or quanta. As the intensity of electromagnetic  energy  increases or decreases, it steps up or down from one  quantized  level to another, rather than follow a smooth and continuous curve . The establishment of energy quantization called for the replacement of classical mechanics Energy quantization became evident under three main studies The black-body radiation Heat capacities Atomic and molecular spectra

4. Energy quantization Black body is a material capable of emitting and absorbing all wavelengths of radiations uniformly. The classical approach to the description of black-body radiation results in the ultraviolet catastrophe. The prediction of classical physics  that an ideal black body at thermal equilibrium will emit radiation  in all frequency ranges, emitting more energy as the frequency increases . T he sum of emissions in all frequency ranges suggest that a blackbody would release an infinite amount of energy, contradicting the principles of   conservation of energy   This drew attention to the need of a new model for the behavior of blackbodies Black Body Radiation

5. Energy quantization Black Body Radiation A good approximation to a blackbody is a pinhole in an empty container maintained at a constant temperature A ny radiation leaking out of the hole has been absorbed and re-emitted inside so many times as it reflected around inside the container that it has come to thermal equilibrium with the walls The 19 th century approach adopted to explain black-body radiation was to calculate the energy density , d E Rayleigh–Jeans law for the density of states

6. Energy quantization Black Body Radiation Rayleigh–Jeans law is quite successful at long wavelengths (low frequencies ) But fails badly at short wavelengths (high frequencies ). The equation therefore predicts that oscillators of very short wavelength (corresponding to ultraviolet radiation, X-rays , and even gamma rays ) are strongly excited even at room temperature. This absurd result , implies that a large amount of energy is radiated in the high-frequency region of the electromagnetic spectrum This is called the ultraviolet catastrophe . According to classical physics, even cool objects should radiate in the visible and ultraviolet regions, so objects should glow in the dark; there should in fact be no darkness

7. Energy quantization To avoid this catastrophe, Max Planck proposed that the electromagnetic field could take up energy only in discrete amounts This is called quantization of energy His theory is expressed in the relation below: This fits the experimental curve at all wavelengths For long wavelengths, and the denominator in the Plank distribution can be replaced by Black Body Radiation

8. Energy quantization T he French scientists Pierre-Louis Dulong and Alexis- Thérèse Petit determined the heat capacities, CV = (∂ U /∂ T ) V of a number of monatomic solids. T hey proposed that the molar heat capacities of all monatomic solids are the same and close to 25 J K −1 mol − 1 . Dulong and Petit’s law is easy to justify in terms of classical physics in much the same way as Rayleigh attempted to explain black-body radiation. Unfortunately, significant deviations from their law were observed when advances in refrigeration techniques made it possible to measure heat capacities at low temperatures. Heat Capacities

9. Energy quantization It was found that the molar heat capacities of all monatomic solids are lower than 3 R at low temperatures, and that the values approach zero as T →0. To account for these observations, Einstein assumed that each atom oscillated about its equilibrium position with a single frequency ν. He then invoked Planck’s hypothesis to assert that the energy of oscillation is confined to discrete values, and specifically to nh ν , where n is an integer. Einstein discarded the equipartition result, calculated the vibrational contribution of the atoms to the total molar internal energy of the solid and obtained the expression known as the Einstein formula : Heat Capacities

10. Energy quantization The most compelling and direct evidence for the quantization of energy comes from spectroscopy Spectroscopy is the detection and analysis of the electromagnetic radiation absorbed, emitted , or scattered by a substance. The obvious feature of both is that radiation is emitted or absorbed at a series of discrete frequencies. I f the energy of an atom decreases by Δ E , the energy is carried away as radiation of frequency ν, and an emission ‘line’, a sharply defined peak, appears in the spectrum. We say that a molecule undergoes a spectroscopic transition , a change of state, when the Bohr frequency condition Δ E = h ν Atomic & Molecular Spectra

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