Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated at a number of discrete waveengths. Each kind of atom has its own characteristic spectrum i.e. a different set of waveengths at which the ines of the spectrum are found. To expain the discrete nature of the spectra ines, Bohr postuated (1913) 1. Eectron in an atom moves in circuar orbits, obeying cassica mechanics, about the nuceus under Couomb interaction thus obeying cassica eectromagnetic theory.. Of a the infinite set of orbits possibe in cassica mechanics, eectron moves ony in those for which its orbita anguar momentum is an integra mutipe of h/π, i.e. L = n, which is a departure from cassica mechanics. Further, eectron moving in such an orbit does not oose energy by eectromagnetic radiation, though it is in acceerated motion, thereby defying cassica eectromagnetic theory on the way. 3. Eectromagnetic radiation is emitted if eectron, initiay moving in an orbit with energy E i, jumps discontinuousy to another orbit of energy E f and the frequency of the emitted radiation is ν = (E i E f )/h. Based on these postuates of Bohr, cacuation of the energy of an atomic eectron moving in one of the aowed orbits is straight forward. From postuate 1, an eectron of charge e and mass m moving round a nuceus of charge Ze and mass M in an orbit of radius r with veocity v, we get 1 Ze 4πɛ r = mv r. (33) From postuate, we get These two together give us L = mvr = n, n = 1,, 3,... (34) r = 4πɛ me n v = n Z (35) Z = a 1 Ze 4πɛ n, (36) where, a 5.3nm is known as Bohr radius. The ratio between the speed of eectron in the first Bohr orbit (n = 1) and the speed of ight is equa to a dimensioness constant α, known as fine structure constant: α = v 1 c = 1 4πɛ e c = 1 137, (37) 1
which actuay gives the strength of eectromagnetic couping. Now the kinetic and potentia energy of this eectron, using (33), are, K = mv V = r = 1 Ze 4πɛ r Ze 4πɛ r dr = 1 Ze 4πɛ r. Therefore, the tota energy of the eectron, E = K + V, moving in an orbit specified by n of equation (34) is, E = mz e 4 (4πɛ ) 1 n = R Z, n = 1,, 3,... (38) n i.e. the quantization of orbita anguar momentum eads to quantization of orbita energy. R = 13.6eV is caed Rydberg constant. From postuate 3, the frequency of eectromagnetic radiation emitted when the eectron makes transition from quantum state (E i, n i ) to the quantum state (E f, n f ) is given by, ( ) ν = E i E f h = ( 1 4πɛ ) mz e 4 4π 3 1 n f 1 n i. (39) The above expression (39) expains the discrete atomic spectra ines i.e. discrete frequency of eectromagnetic radiation when eectron jumps around among the discrete energy eves of the atom. Direct confirmation that the energy states of an atom are quantized came from the experiment of Frank and Hertz (1914). The experiment provided not ony the evidence of discrete energy states of Hg atom but actuay measured the energy differences between its quantum states. Wison-Sommerfed quantization rues To interpret two rather successfu quantization rues Panck s quantization of energy (1), ɛ = nhν and Bohr s quantization of anguar momentum (34), L = n of an eectron moving in a circuar orbit in a consistent way, Wison and Sommerfed (1916) offered a scheme that incuded both Panck and Bohr quantization rues as specia cases. Wison-Somerfed quantization rue states that for any physica system in which the coordinates are periodic functions of time, the quantum condition for each coordinate is p q dq = n q h (4) where q is one of the coordinates, p q is the momentum associated with that coordinate, n q is a quantum number which takes on integra vaues and integration is taken over one period of the coordinate q. For instance, (q, p q ) for one dimensiona simpe harmonic osciator is (x, p x ), dispacement from center and inear momentum, for eectron moving in an orbit it is ange and orbita anguar momentum (θ, L θ ). Most
of the time, equation (4) is refered as quantization of action variabe, J = p q dq = n q h, where J is the action. The rue for quantization of cavity radiation by Panck (1) can be derived from action quantization (4). Considering an one-dimensiona simpe harmonic osciator (a whoe ot of which made up the cavity radiation), the task is to cacuate the quantization integra p x dx, where the instantaneous position x and momentum p x are reated by the tota energy E, E = p x m + kx p x me + x E/k = 1. (41) A pot in two-dimensiona space having having coordinates x and p x, caed phase space, the motion of simpe harmonic osciator traces out an eipse, x /a +p x/b = 1, semi-axes being E a = and b = me. (4) k The vaue of p x dx is just equa to the area encosed by the above eipse, thus p x dx = π a b = π E k mei = πe k/m. (43) Since, for an inear osciator, the frequency of osciation is reated to k and m as k/m = πν, Wison - Sommerfed quantization rue (4) eads to Panck s quantization (1), p x dx = E ν = n xh E = n x hν. (44) That is, the ony aowed states of osciation are those represented in the phase space by a series of eipse with quantized area p x dx = nh. The area between two successive eipses, i.e. states of osciation, is equa to h. Hence, Panck s energy quantization is equivaent to the quantization of action. Simiary, Bohr s quantization of orbita anguar momentum foows from action quantization. For an eectron moving in a circuar orbit, the appropriate coordinate is the ange θ, which is a periodic function of time, and corresponding momentum is orbita anguar momentum L, which is a constant (anguar momentum is conserved for inverse-square aws). From (4) it foows that, p θ dθ = n θ h L dθ = nh. (45) Since L is constant, we have Ldθ = L π dθ = πl and thus πl = nh L = n (46) which is Bohr s quantization condition. A more physica interpretation of the Bohr s orbita anguar momentum quantization comes from de Brogie s hypothesis of wavepartice duaity. 3
Motivation for action discretization cassica adiabatic invariants Penduum with variabe string ength: Consider an idea penduum consisting of a massess string of ength with a mass m attached and undergoing simpe harmonic motion of sma ampitude A i.e. sma ange θ. The equaiton of motion of the penduum, for sma θ, is mg sin θ = F = m d x dt d θ dt = g θ. (47) The soution of the above differentia equation is θ = A cos(πνt + α), where ν = 1 π g. (48) The force acting on the mass m when the penduum makes an instanteneous ange θ is, ) F = mg cos θ + mv mg (1 θ + m θ (49) ] = mg [1 A cos (πνt + α) + ma (πν) sin (πνt + α) (5) Ceary, the force F is time-dependent and maximum when θ =. Let us cacuate the average force F over one compete time period of motion, F = 1 T = mg 1 T F dt dt mga T cos (πνt + α) dt + ma 4π ν T sin (πνt + α) dt = mg + 1 4 mga. (51) The kinetic energy of the penduum is, KE = 1 mv = 1 m θ = 1 m A 4π ν sin (πνt + α). (5) Therefore the tota energy of the penduum, using (48) is, The action J = p θ dθ being, J = m E = KE max = 1 ma 4π ν = 1 mga. (53) θdθ = m A 4π ν sin (πνt + α) dt = m A 4π ν T = E ν, (54) which is a constant. Next consider an externa agency to shorten ength of the string sowy by puing it smoothy the string is shorten by over a time spanning many periods such that change of ength over one period δ satisfies δ/ 1. Such a sow change in a parameter of the system is termed as adiabatic whie the system itsef is undergoing harmonic motion throughout the time. Work done by externa agency over one period is δw = F δ = mg δ + 1 4 mga δ (55) 4
where the first term is work done against gravity and second term is the increase in kinetic energy δe as the mass goes up. Energy of the penduum for fixed ength is E max = mga /, therefore, δe E = 1 δ. (56) Shortening the ength by δ resuts in increase of frequency δν, ( ) 1 g δν = δ π = 1 δ ν δν ν Putting the above two equations (56, 57) together we obtain, = 1 δ (57) δe E = δν ν which upon integration yied adiabatic invariant (an adiabatic invariant is a property of a physica system which stays constant when changes are made sowy) E ν constant. (58) The action for the system, where the anguar momentum is given by p θ = m θ, is p θ dθ = E ν (59) The motion of the penduum when its ength changes sowy over time is not quite periodic over each cyce. However, the cassica action remains approximatey constant as ong the change is very sow the cassica action is an adiabatic invariant. Wison and Sommerfed formuated action quantization by equating the quantum number of a system with its cassica adiabatic invariant. 5