# Physics Part3

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PHYSICS IIDr. Ing. Valerica D. Ninulescu2010Contents1 The experimental foundations of quantum mechanics 11.1 Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Photoelectric eect . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Compton eect . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Atomic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Bohr model of the hydrogen atom . . . . . . . . . . . . . . . 61.6 Experimental conrmation of stationary states . . . . . . . . 101.7 Einsteins phenomenological theory of radiation processes . . 111.7.1 Relations between Einstein coecients . . . . . . . . . 131.7.2 Spontaneous emission and stimulated emission as com-peting processes . . . . . . . . . . . . . . . . . . . . . 151.8 Correspondence principle . . . . . . . . . . . . . . . . . . . . 161.9 Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . 171.10 Heisenberg uncertainty principle . . . . . . . . . . . . . . . . 171.10.1 Uncertainty relation and the Bohr orbits . . . . . . . . 181.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Principles of quantum mechanics and applications 212.1 First postulate . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . . 212.3 Probability conservation . . . . . . . . . . . . . . . . . . . . . 232.4 Constraints on the wavefunction . . . . . . . . . . . . . . . . 252.5 Time-independent Schrodinger equation . . . . . . . . . . . . 252.6 Potential wells . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 The one-dimensional innite well . . . . . . . . . . . . . . . . 282.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 282.7.2 Energy eigenvalues and eigenfunctions . . . . . . . . . 292.7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 The rectangular potential barrier. Potential barrier penetration 34iv CONTENTS2.8.1 Case 0 < E < V0 . . . . . . . . . . . . . . . . . . . . . 342.8.2 Case E > V0 . . . . . . . . . . . . . . . . . . . . . . . 372.8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 372.9 Non-rectangular potential barriers . . . . . . . . . . . . . . . 382.10 Applications of tunneling . . . . . . . . . . . . . . . . . . . . 392.10.1 Field emission of electrons . . . . . . . . . . . . . . . . 392.10.2 Alpha-particle emission . . . . . . . . . . . . . . . . . 402.11 The quantum harmonic oscillator . . . . . . . . . . . . . . . . 412.12 Three-dimensional Schrodinger equation . . . . . . . . . . . . 472.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A Fundamental physical constants 51B Greek letters used in mathematics, science, and engineering 55Glossary of notationse exponential function, ez= exp(z)i complex unity, i2= 1Im(.) imaginary part of a complex numberR the set of real numberszcomplex conjugated of z. average_rintegral over the whole three-dimensional spaceAadjoint of Av vector roughly similar; poorly approximateskB Boltzmann constantCM centre of massTDSE time-dependent Schrodinger equationTISE time-independent Schrodinger equationChapter 1The experimentalfoundations of quantummechanicsAt the end of the 19th century, it seemed that physics was able to explainall physical processes. According to the ideas of that time, the Universe wascomposed of matter and radiation; the matter motion could be studied byNewtons laws and the radiation was described by Maxwells equations. Thiscondence began to disintegrate.1.1 Thermal radiationIt is well known that every body with non-zero temperature emits elec-tromagnetic radiation with a continuous spectrum that contains all wave-lengths.Examples: the infrared radiation of a household radiator solar radiationThe radiation incident on the surface of a body is partially reected andthe other part is absorbed. For example, dark bodies absorb most of the inci-dent radiation and light bodies reect most of the radiation. The absorptioncoecient of a surface is dened as the fraction of the incident radiationenergy that is absorbed; this coecient is dependent on wavelength.2 1 The experimental foundations of quantum mechanicsSuppose a body at thermal equilibrium with its surroundings. Such abody emits and absorbs the same energy in unit time, otherwise its temera-ture can not remain constant. The radiation emitted by a body at thermalequilibrium is termed thermal radiation.A blackbody is an object that absorbs all electromagnetic radiation fallingon it.In 1859 G. R. Kirchho stated that the ratio of the emissivity power(power emitted by unit area at a given wavelength) and the absorption coe-cient at that wavelength is the same for all bodies at a given temperature.It follows that: the blackbody is not only the best absorber, but it is also the bestemitter the emissivity power of a blackbody is a universal functionA good approximation of a blackbody can be done as follows. Consider acavity with a small entrance hole and maintained at a constant temperature.For an observer placed outside, a ray that enters the cavity is absorbed as itis scattered by the interior walls. The entrance hole behaves as a blackbody.Blackbody radiation laws: The blackbody radiation is isotrope and nonpolarized. The emissive power (energy radiated from a body per unit area perunit time), P, of a blackbody at temperature T grows as T4:P = T4, (1.1)where 5.67108Wm2K4is StefanBoltzmann constant. Thisresult is known as StefanBoltzmann law. The wavelength max for maximum emissive power from a blackbodyis inversely proportional to the absolute temperature,maxT = b , (1.2)which is called Wiens displacement law. The constant b is calledWiens displacement constant and its value is b 2.898 103mK.max shifts with the temperature, this is why it is called a displace-ment law.For example, when iron is heated up in a re, the rst visible radiationis red. Further increase in temperature causes the colour to change toorange, then yellow, and white at very high temperatures, signifyingthat all the visible frequencies are being emitted equally.1.1 Thermal radiation 3 For a complete quantitative characterization of the blackbody radia-tion, we should give the thermal radiation power per unit of area andunit of wavelength, denoted here R(, T) . Classical physics furnisheda result in good agreement with the experiment only at small frequen-cies. In 1900 Max Planck determined a formula which agrees with theexperiment at whatever temperature based on a new and revolution-ary idea: the exchange of energy between a body and its surroundingscan be performed only in discrete portions, the minimum energy im-plied in the exchange being proportional to the frequency, h. Planckradiation law readsR(, T) = 2 hc251exp(hc/kBT) 1 . (1.3)The constant h is called Plancks constant and has the value h 6.626 1034J s .Plancks radiation formula contains all the information previously ob-tained.Another form of the Plancks law gives the spectral energy density(, T) of a blackbody. By use of the relationship(, T) = (4/c)R(, T) (1.4)we nd(, T) = 8 hc51exp(hc/kBT) 1 . (1.5)Figure 1.1 presents the graph of the spectral energy density (, T)given by Eq. (1.5).Planck radiation formula can be expressed in terms of angular frequencyor the frequency. A straightforward calculation gives for the spectral energydensity the expressions(, T) = 22c3exp(/kBT) 1 (1.6)and(, T) = 8 2c3hexp(h/kBT) 1 . (1.7)4 1 The experimental foundations of quantum mechanics0 1 2 3 4 50123456, m(,T), 103J m41000 K1500 K1750 K2000 KFig. 1.1 Spectral energy density (, T) of a blackbody at a few temperatures.Applications Pyrometer: device for measuring relatively high temperatures by mea-suring radiation from the body whose temperature is to be measured. Infrared thermography: a fast non