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## A First Look at Quantum Physics

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**A First Look at Quantum Physics**A First Look at Quantum Physics 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Historical Note • at the end of the 19th century, the overwhelming success of classical physics – CM, EM, TD made people believe the ultimate description of nature has been achieved. • at the turn of the 20th century, classical physics was challenged by Relativity & microphysics. • the series of breakthroughs： (1) Max Planck → the energy of a quantum：the energy exchange between an EM wave & matter occurs only in integer multiples of 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Historical Note (2) Einstein → photon：light itself is made of discrete bits of energy; an explanation to the photoelectric problem. (3) Neils Bohr → model of hydrogen atom：atoms can be found only in discrete states of energy & atoms with radiation takes place only in discrete amounts of ν. → Bohr’s model Rutherford’s model 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Historical Note (4) Compton → scattering X-rays with e-：the X-ray photons behave like particles with momenta 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Essential Relativity • for a free particle of rest mass m moving at speed υ, the total energy E, momentum p, and kinetic energy T can be written in the relativistically correct forms where • using → & 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Essential Relativity • in QM the momentum is a more natural variable than γ, a useful relation can be given by , the rest energies of various atomic particles will often be quoted in energy units; for the electron and proton the rest energies are given by • the non-relativistic limit of E.g. , where , is easily seen to be 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Essential Relativity • the ultra-relativistic limit when , can be approximated to be , which is also seen to be consistent with the energy-momentum relation for photons, namely • (i) e- in atoms：when the relativistic effects become non-negligible. (ii) deuteron： for the simplest nuclear system; compared with →deuteron can be considered as non-relativistic system 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • spectral energy density of blackbody radiation at different temp. the peak of the radiation spectrum occurs at freq that is proportional to the temp. • Wien’s displacement law： spectral distribution only depends on temperature ideal blackbody 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • blackbody radiation： (1) Rayleigh’s energy density distribution： when the cavity is in thermal equilibrium, the EM energy density in to is given by according to the equipartition theorem of classical thermodynamics, all oscillators in the cavity have the same mean energy： → is integrate over all freq, the integral diverges → this result is absurd→ called the ultraviolet catastrophe 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • blackbody radiation： (2) Plank’s energy density distribution： avoiding the ultraviolet catastrophe, Planck considered that the energy exchange between radiation & matter must be discrete： → → the spectrum of the blackbody radiation reveals the quantization of radiation, notably the particle behavior of EM waves 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • photoelectric effect： (1) 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • photoelectric effect： (2) when a metal is irradiation with light, electrons may get emitted (3) it was fond that the magnitude of the photoelectric current thus generated is proportional to the intensity of the incident radiation, yet the speed of the electrons does not depend on the radiation’s intensity, but on its frequency. → the photoelectric effect provides compelling evidence for the corpuscular nature of the EM radiation 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • Compton effects： • Compton treated the incident radiation as a stream of particles-photons-colliding elastically with individual e- 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • Compton effects： by momentum conservation & energy conservation → → the Compton effect confirms that photons behave like particles; they collide with e- like material particles 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • wave aspect of particles： de Broglie → the wave-particle duality is not restricted to radiation, but must be universal: all material particles should also display a dual wave-particle behavior： known as the de Broglie relation, connects the momentum of a particle with the wavelength & wave vector of the wave corresponding to this particle 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • wave aspect of particles： Davission-Germer exp. confirmation of de Broglie’s hypothesis： the intensity max of the scattered e- corresponds to the Bragg formula 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantum Physics： as a Fundamental Constant • wave aspect of particles： de Broglie’s wavelength： For an Ni crystal, , 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • Bohr’s assumption： (1) only a discrete set of circular stable orbit are allowed (2) the orbital angular momentum of the electron is an integer multiple of → (3-a) 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • (3-b) 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • (3-c) • (3-d) 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • discussion (i) classically, (ii) for a circular orbit, the attractive force = centrifugal force (iii) with , (iv) considering a transition from to , according to Einstein’s relation, , . & the fractional of angular momentum is so small → with → from (ii), 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • Bohr suggested that hold even for energy small quantum number. The allowed value of is the same for positive & negative values, this means that if a given value of the angular momentum is allowed, its negative must also be allowed. (a) if , then this criterion is satisfied, for (b) if , the allowed values are (c) with any other value of , however, this condition cannot be met. 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • correspondence principle first given by Bohr： Bohr noted that the photons emitted in transitions between the quantized energy levels satisfy the Balmer formula, written is the form a classical particle undergoing circular acceleration would emit radiation at its orbital freq., which is given by “the connections & interpolations between the QM & classical description of physical are stressed in this course.” 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – Bohr Model of H Atom • correspondence principle & the classical period： (a) show that the correspondence principle can be generalized to show that the classical periodicity, , of a quantum system in the large limit is given by (b) using the expression for the quantized energies of a particle in a box length , find the classical period in state & compare it to the expectations based on the classical motion 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – CM & QM • (1) the relationship between CM & QM certain sense is similar to that which exist between geometric & wave optics (2) in QM the wave function of quasi-classical form; where is called action (3) the small parameter have is the ratio • transition from QM to CM formally is described by the WKB-method at tends to 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – CM & QM • the analogy between Optics & Mechanics showed to be vary fruitful to produce very important physical insight • the 1st analogy put geometrical optics in correspondence with CM the development of this analogy was the formulation of electron optics • the formulation of electron optics is similar to EM geometrical optics provided to replace the motion of light rays & refractive index with electron rays and potential, respectively 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Semi-classical model – CM & QM • the 2nd analogy is extended to the wave level, going from Optics to Mechanics by de Broglie & Schrodinger, obtaining the wave mechanics & subsequently the QM • from CM to QM：Schrodinger eq. has been recognized as the non-relativistic limit of a more general wave mechanical formulation induced by the correspondence with optics. The non-relativistic limit of Klein-Gordon eq. is just the Schrodinger eq 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantization Rules • Wilson & Sommerfeld offered a scheme that included , & as special cases • in essence their scheme consists in quantizing the action variable of classical mechanics phase integral • for 1D, , since the particle goes from one limit of oscillation to the other and back 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantization Rules • → the limit of oscillation are determined by . thus the quantities in the brackets vanish & → so if , then • the quantization of the action, J, is usually referred to as “the Bohr-Sommerfeld quantum condition.” 2006 Quantum Mechanics Prof. Y. F. Chen**A First Look at Quantum Physics**Quantization Rules • Ex： Harmonic oscillator , if , then , Plank Quantization rule • Ex： for an electron moving in a circular orbit of radius r. 2006 Quantum Mechanics Prof. Y. F. Chen