Model of Hydrogen Atom
* To prove L=angular momentum=mvr=nh/2π mv^{2}/r=kze^{2}/r^{2} or (mvr)v=kze^{2} or Lv=kze^{2} E_{k }=mv^{2} /2 =kze^{2} /2r E_{p }=-kze^{2} /r =-mv^{2} TE =-kze^{2} /2r =-mv^{2} /2 =-mk^{2}z^{2}e^{4} /2L^{2} . Differentiating, d(TE) =(mk^{2}z^{2}e^{4} /L^{3} )dL=(mv^{2}L^{2} /L^{3})dL=mv^{2}dL/L =hν=h/T=hv/2πr or mvr*dL/L = h/2π or dL=h/2π or L=nh/2π The essential part of derivation was d(TE)/dL =2πν=2π / T = ω = angular velocity and TE/L =-(1/2) v/r =-ω /2 We can also put 2TE*L^{2} =constant, differentiating and equating to zero, we get ΔTE*L^{2} +2L ΔL *TE=0 or ΔTE / ΔL =-2TE/L=v/r =√(kze^{2}/m)*r^{ } - 3/2 Mass, charge being known, out of (L , TE, v, r ) you fix the value of any one, and can get value of other 3 by using 2 blue equations) We find that quantum mechanically TE/L=v/r and not v/2r,
TE=hν =h/T =hv/2π r=h/2π*(v/r)= or TE/L =v/r v^{2}r =kQq/m_{e} or ν^{2}r^{3} =kQq/4π^{2}m_{e} or ν^{2}V =(1/3)kQq/πm_{e} where V is the volume of sphere with radius r and ν is frequency of revolution. These are classically derived results. Taking Q=ze=1*e, ν^{2}V =(1/3π)*K*e*(e/m_{e}) =26.83 We also get L=√(mKZ) √r *e Another way to look at the above problem :
In gravitation as well as in Bohr orbits, the treatment is similar
and classical. In gravitational orbits, we know the radius r and hence
calculate the total energy and angular momentum which are constants
since there is conservation law for both. But in Bohr orbit , we go
the other way round and first fix the angular momentum and then
calculate speed from LV formula and radius etc and finally
total energy. We invoke the
Heisenberg uncertainty principle Δp Δx = Bijan's Derivation: In the above derivation, we have equated centripetal force with electrostatic force. Actually, the force exerted on a moving electron is not the coulomb force and moreover electro dynamic force is not a conservative force where as Coulomb force is conservative. Immediate consequence of this is non conservation of angular momentum which is contrary to our assumption. Moreover, both charge and mass are mixed up . We adopt a pure gravitational approach here. In gravitation, when the planet revolves in a closed circular orbit (which is a special case of elliptical orbits and the same is taken as per Bohr assumption), the force is conservative, KE is equal to the TE with sign changed, and Potential energy is numerically 2 times the kinetic energy . So KE= constant PE=-2KE =-mv^{2} =-mv*v=mv*(2πr/T)=-(mvr)*(2π/T) =-(mvr)*(2πν) Since PE is constant times the TE in a circular orbit, conservation of TE implies conservation of PE. So PE=const and (mvr)*(2πν) =const . mvr=(const/2π)*(1/ν)=C*(1/ν). The constant ,say C should have the dimension of energy or negative energy if we define -sign suitably. If const=C=(nhν/2π)=E/2π, mvr=constant=nh/2π Here conservation of Angular momentum follows from conservation of potential energy PE=-2nhν=constant and conservation of TE=- nhν In a closed circular orbit, there are 4 parameters TE, L , r , v . At every point in the orbit, TE and L are conserved. Also r, v are conserved in magnitude but not in direction, However rv is conserved both in magnitude as well as in direction which implies that r, v are in a fixed plane. Out of the 4 parameters, we can assign (classically) any arbitrary value to any one of the parameters and the other 3 can be expressed in terms of this parameter. case 1: Assign any value to L v=kze^{2}/L r=L^{2}/kze^{2}m E=-(kze^{2})^{2 }m /(2L^{2})
acceleration=a=v^{2}/r=(kze^{2})^{3}m /L^{4}
;Putting L= case 2: Assign any value to E L=√m*kze^{2}/√(-2E) v= √(-2E) /√m r=(kze^{2})^{ } /(-2E)
acceleration=a=v^{2}/r=4E^{2} /kze^{2}m ;
Putting -2E=(kze^{2})^{2}m /L^{2} = (kze^{2})^{2}m
/ case 3: Assign any value to v L=kze^{2}/v r = (kze^{2}) /(mv^{2}) E=- mv^{2}/2
acceleration=a=v^{2}/r=mv^{4} /kze^{2};
Putting v=(kze^{2}) /L = (kze^{2})/ case 4: Assign any value to r v= √(kze^{2}) /√(mr) L=√(kze^{2}mr) E=kze^{2}/2r
acceleration=a=v^{2}/r=kze^{2} /mr^{2};
Putting r=L^{2}/(kze^{2})m = (kze^{2})/ Since circle is a special case of ellipse, In elliptical orbits which are generalized cases, case 3and 4 are not applicable since v,r change from point to point. However, in elliptical orbits, conservation of E does not automatically ensure conservation of L or vice versa. A third component is to conserved for conservation of both E and L If we start from E, conservation of eccentricity e is essential for conservation of L. If we start from L, we observe that a , e need not be individually conserved for L bur a*√(1-e^{2}) =b should be conserved. But for conservation of E, a only should be conserved. Hence for both, a and e should be individually conserved. * To find out value of r, v, TE,λ Lv=kze^{2} or (nh/2π)v= kze^{2} or v=2πkze^{2} /nh = z*( 2πkze^{2}/h)*(1/n)=z*α /n r= kze^{2 } / mv^{2} or r =n^{2}h^{2} / 4π^{2}mkze^{2} TE=-mv^{2} /2 =-2π^{2}mk^{2}z^{2}e^{4} /n^{2}h^{2} ΔTE=hc/ λ =(2π^{2}mk^{2}z^{2}e^{4}/h^{2}) [1/n_{1}^{2} - 1/n_{2}^{2}] or 1/λ =(2π^{2}mk^{2}z^{2}e^{4}/ch^{3}) [1/n_{1}^{2} - 1/n_{2}^{2}]=(2π^{2}mk^{2}e^{4}/ch^{3}) [1/n_{1}^{2} - 1/n_{2}^{2}] when z=1 or 1/λ =R [1/n_{1}^{2} - 1/n_{2}^{2}] where R=(2π^{2}mk^{2}z^{2}e^{4}/ch^{3}) Comparison between Circular Orbit in Gravitation & Bohr Orbit : Gravitation Circular Orbit Bohr Orbit (1) TE=- GMm/2r TE=-KQq/2r (2)PE= - GMm/r PE=-KQq/r (3)KE= GMm/2r KE=KQq/2r (4)L =m √(GM/r) r L =m √(KQq/mr) r =√m √m √(GM) √r =√m √q √(KQ) √r (5)P = m √(GM/r) P = m √(KQq/mr) = √m √m √(GM/r) = √m √q √(KQ/r) (6) (1/2)mv^{2} gives unit of KE (1/2)qv^{2} gives unit of KE (7)Dimensionally G=L^{3}T^{-2}/M = L^{2}T^{-2}*(L/M) K=ML^{3}T^{-2} / Q^{2} ={L/(Q^{2}/M)} L^{2}T^{-2} Or G/c^{2} =L/M K/c^{2} =L/(Q^{2}/M) LHS of the order of 10^{-27} LHS of the order of 10^{-7} ;
(G/c^{2}) *(K/c^{2}) is of the order of 10^{-34}
which is same as h and if we further multiply by √2,
becomes almost equal to (8)TE * L^{2} =-(1/2)mG^{2}Z1^{2}m^{4} TE * L^{2} =-(1/2)mK^{2}Z^{2}e^{4} (Z,Z1 are integers) (9) We observe that the energy formulation is similar for gravitation and bohr orbit. But there is a difference in the formulation of angular and linear momentum marked by red. In bohr orbit, one mass is substituted by charge.
* To find dimension of the atom From de broglie hypothesis, λ = h/p = h/mv We know that when a string of length l is tied to a rigid support at both ends and is stretched, it vibrates in such a manner so that the nodes are at both ends and the condition that is satisfied is sin (2πl/λ) =0 or 2πl/λ =nπ where n is any integer for standing wave to be formed. λ = 2l/n or l=n (λ/2) Hence, 2l/n=h/mv or p= nh/2l Taking l=2πr, we get 4πr =nh/mv or mvr=nh/4π E=p^{2}/2m = n^{2}h^{2}/8ml^{2} .....(a). This is the total energy of a vibrating particle trapped in a box of length l. But E=2π^{2}mk^{2}z^{2}e^{4} /n^{2}h^{2} ....(b) ; So equating (a) & (b), we get l=n^{2}h^{2} /4πmkze^{2} ^{}whose dimension is of the order of 10^{-10 }m.If in the Bohr equation, we substitute mass of electron by mass of neutron, the dimension comes to the order of 10^{-14}m which is roughly the size of nucleus. * To prove mvr=nh/2π from de broglie wavelength We already know that the condition for standing wave in case of a stretched string of length l tied rigidly to both ends is l=n (λ/2) de Broglie considered the circumference of Bohr's circular orbit as a string of length l=2πr where standing waves are formed. He assumed that these are not ordinary waves but matter waves of energy E=pc where p is the momentum & E = hc/λ . This follows from Einstein's Special theory of relativity E=√(m_{0}^{2}c^{4} +p^{2}c^{2}). For entities with zero rest mass E=pc Equating E from both cases, λ=h/p Here de broglie enters the scenrio , where he postulated the symmetry between mass less entities and entities having non-zero rest mass and extended the concept to λ=h/p where p=mv for these entities. Hence , this was a postulate and conclusion was not reached through derivation. Now, l= nh/(2mv) or 2πr= nh/(2mv) or mvr=nh / 4π. or angular momentum = h/4π, 2h/4π,3h/4π, ....... de Broglie assumed l=n λ . Why ??? possible explanation is that when matter is converted to energy, there is equal contribution of matter and anti-matter. Hence anti-matter also contributes anti-matter angular momentum= h/4π, 2h/4π,3h/4π, ....... Hence total angular momentum of matter wave= h/2π, 2h/2π,3h/2π, .......nh/2π or mvr=nh/2π (this is not in any text book but my assumption)
* de broglie wavelength is velocity dependant whereas compton wavelength is not velocity dependant. |
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*Electric Field outside a current carrying Conductor : Why an electric field is not produced outside a current carrying conductor ? It depends upon what type of current is flowing through the conductor. Suppose a constant current is flowing. Take any cross section of the wire. the amount of charge leaving the cross section=the amount of charge entering the cross section. So the whole wire is electrically neutral and no electric field is produced. But consider a circuit with AC source. that variable current will produce variable magnetic field. If you will examine it carefully, electric field and magnetic field will be produced perpendicularly to each other and will be variable of course. So, now this will become the source of Electromagnetic Wave. Because when changing electric field and changing magnetic field is produced at a point simultaneously and perpendicular to each other, the electromagnetic wave is generated which will move in the direction perpendicular to Electric and magnetic field. To be specific, the direction of wave is given by the cross product of E and B. |
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