Nucleons in nucleus behave not like particles, moving with momenta p but like waves with de Broglie...

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  • Nucleons in nucleus behave not like particles, moving with momenta p but like waves with de Broglie wavelength B~h/p: Because p ~250 MeV/c < E =939 MeV its behaviour is described by non-relativistic Quantum Mechanics (NR QM). 1900 M. Planck blackbody radiation: emission of el-magn waves by the heated bodies: light is not only a wave but also a particle (-particle) carried a portion of energy E=h 1905 A. Einstein photoelectric effect 1924 De Broglie - particles have also a wave origine - B~h/p (based on Einstein photoeffect and Compton scattering) 1927 Thompson , Davisson and Germer experiment showed the diffraction behaiviour of the electrons with wavelength B~h/p

    M Born - the state of Quantum System is described by the wave function (x,t) (or probability amplitude) which connected with probability P to find the system in volume dV as dP = | |2 dV where | |2 is probability density P= V dp = V | |2 dV total propability with normalization to unity: V | |2 dV=1 The wave function (x,t) is the state vector similar to the r(t) in classical mechanics

    Superposition principle: the difference between bullets passed through two slits and electrons scattered on two small slits is that one bullet always fly through only one slit wih probability P1 or P2 and in the case of two opened slits the detector will detect P12=P1+P2 while electron behaves as a wave so the probability that two electrons will pass though two slits is the same as in the case of waves with interferomety term P12=| 12 |2=|1+2|2 P1+P2=|1|2+ |2|2 i.e. In the case of microworld the superposition principle is valid not for probabilities but for probability amplitudes: 12 =1+2 here P1 =|1|2 (P2=|2|2 ) is the probability to find electron in the detector if only 1st (2nd) slit is open. It means that each electron feels the existance of two slits, and the wave of one electron from 1st slit is interfering with the wave of the same electron of the other slit. Superposition principle is valid not only for 2 but for many states =1+2 +3 ++N and the particle can be described by the electro-magnetic wave packat with group velocity vgr=p/m=vpart => 1925 E Schroedinger QM : particle is not classical : its momenta and position cannot be simultaneously measured exactly but with some accuracy : 1927 Heisenberg uncertainty principle : for position x and momentum p : xpx< /2 , or for energy E and time t Et < /2 (E=h) where =h/2 Plancks constant or for angular momentum L and its projection on z lz and on xy : lz < /2

    B.R. Martin. Nuclear and Particle Physics. Appendix A. Some results in Quantum mechanics.A.1. Barrier penetration.

  • QM: Schrodinger equationThe Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction (x,p,t) which predicts analytically and precisely the probability P= |(x,p,t) |2 of the event. The kinetic (T) and potential (V) energies are transformed into the Hamiltonian which acts upon the wave function to generate the evolution of the wavefunction in time and space. T + V = E

    The Schrodinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

  • Operators in Quantum Mechanics Associated with each measurable parameter in a physical system is a quantum mechanical operator. Such operators arise because in quantum mechanics you are describing nature with waves (the wavefunction) rather than with discrete particles whose motion and dymamics can be described with the deterministic equations of Newtonian physics. Part of the development of quantum mechanics is the establishment of the operators associated with the parameters needed to describe the system. Some of those operators are listed below.

    It is part of the basic structure of quantum mechanics that functions of position are unchanged in the Schrodinger equation, while momenta take the form of spatial derivatives. The Hamiltonian operator contains both time and space derivatives.

  • One dimensional Schrodinger equation

  • Free-particle in one dimension

  • Infinite potential cell

  • Potential Barier, E>V0 and (tunnelling effect) E
  • A.2.Density of the states and Three dimensional boxIn the case of the well with infinite potential barrierthe solution of the Schrodinger eq.

    are the standing waves vanishing at the ends of the well

    corresponding to discrete energy levels always > 0because of uncertainty principle:p1/L -> (E=p2/2m)1/L2 .For 3 dimensional well the energy orand the wavefunction are where nx>0, ny>0, nz>0

    The number of the states is proportional to the number of lattice points dinstanced from each other by (/L): (L/)3 Or in the Fermi sphere it will be 1/8 of the sphere (because nxyz>0)n(k0)=1/8 4/3k03 (L/)3=V/(2)34/3 k03 - for all k

  • A.3 Perturbation theory and the Second Golden Rule In perturbation theory the Hamiltonian at any time t may be written as H(t)=H0+V(t), where H0 is unperturbed Hamiltonian and V(t) is small. The solution for eigenfunctions of H starts by expanding in the terms of the complete set of energy eigenfunctions |un> of H0 : H0 |un> =En |un> |(t)>=cn(t) |un> exp{-iEnt/}, where En are the corresponding energies.If |(t)> is normalized to unity =1 then |cn(t)|2 is probability that at time t the system will be in the state |un>. Substituting it in Schrodinger eq :i cf(t)/t=Vfn(t) cn(t) exp{-ifnt}, where matrix element Vfn(t)= and angular frequency fn=(Ef-En)/ .Initial state of the system |ui> at t=0 then cn(0)=ni and ci(t)=ci(0)+1/i0tVii(t)dtFinal state of the system fi cf(t)= 1/i0tVfi(t)ei fi tdt general, for V(t) For const V =0 and then V0 cf(t)= Vfi/( fi )(1-ei fi t) Probability for transition from state i to state f : Pfi=|cf (t)|2 =4|Vfi|2/2 [sin2(1/2 fi t)/2fi ]. At large t it is valid only if | fi |= |Ef-En|1/2 t(Ef-Ei).Pfi =2t/ |Vfi|2(Ef-Ei) probabilitydPfi /dt= 2/ |Vfi|2(Ef-Ei) transition probability per unit time valid for discrete final states.

    dTfi /dt= dPfi /dt (Ef)dEf = 2/ [|Vfi|2 (Ef)]Ef=Ei transition probability per unit time for continious spectra

  • Dirac Delta Function

  • Appendix C 1911 Rutherford scatteringIn Ernest Rutherford's laboratory, Hans Geiger and Ernest Marsden (a 20 yr old undergraduate student) carried out experiments to study the scattering of alpha particles by thin metal foils. In 1909 they observed that alpha particles from radioactive decays occasionally scatter at angles greater than 90, which is physically impossible unless they are scattering off something more massive than themselves. This led Rutherford to deduce that the positive charge in an atom is concentrated into a small compact nucleus. During the period 1911-1913 in a table-top apparatus, they bombarded the foils with high energy alpha particles and observed the number of scattered alpha particles as a function of angle. Based on the Thomson model of the atom, all of the alpha particles should have been found within a small fraction of a degree from the beam, but Geiger and Marsden found a few scattered alphas at angles over 140 degrees from the beam. Rutherford's remark "It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you." The scattering data was consistent with a small positive nucleus which repelled the incoming positively charged alpha particles. Rutherford worked out a detailed formula for the scattering (Rutherford formula), which matched the Geiger-Marsden data to high precision.Only if cos 90o

  • Appendix C 2. Rutherford scattering CM versus QMClassical mechanics : Rutherford's model enables us to derive a formula for the angular distribution of scattered a-particles. Basic Assumptions: a) Scattering is due to Coulomb interaction between a-particle and positively charged atomic nucleus. b) Target is thin enough to consider only single scattering (and no shadowing) c) The nucleus is massive and fixed. This simplifies the calculation, but it could be avoided by working in the centre of mass frame. d) Scattering is elastic

  • Finite potential cellDiscrete energy for the particle in the boxand continious energy for free particle