Quantum mechanics - Tran Thi Ngoc Dung

Nội dung text: Quantum mechanics - Tran Thi Ngoc Dung

1. QUANTUM MECHANICS Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
2. VOCABULARY • Wave - Particle Duality • Uncertainty • Heisenberg’s Uncertainty Principle • Wave Matter • De Broglie’s wavelength
3. CONTENTS • Wave – Particle Duality of Matter • De Broglie’s Hypothesis – Matter wave • Diffraction of electron wave by single slit • Heisenberg’s Uncertainty Principle
4. “„Quantum mechanics‟ is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.” Richard P. Feynman
5. Matter Waves electron gun  DeBroglie (1924) proposed that, like detector photons, particles have a wavelength: Inversely proportional to q l = h/p momentum.  In 1927-8, it was shown (Davisson- Germer) that, like x-rays, ELECTRONS Ni Crystal can also diffract off crystals ! Interference peak ! Electrons can act like waves!! ) q • We will see later that the discrete I( atomic emission lines also arise from o the wavelike properties of the 0 60 q electrons in the field of the nucleus: What does this mean? q Atomic In discussion section: hydrogen
6. “Double-slit” Experiment for Electrons  Electrons are accelerated to 50 keV l = 0.0055 nm  Central wire is positively charged bends electron paths so they overlap  A position-sensitive detector records where they appear.  << 1 electron in system at any time [A. TONOMURA (Hitachi) pioneered electron holography] Exposure time: 1 s 10 s 5 min 20 min
7. WAVE - PARTICLE DUALITY OF LIGHT Lưỡng tính sóng hạt của ánh sáng LIGHT has nature of Wave–Particle Duality • Wave: Electromagnetic Wave : Interference, Diffraction. - Particle: PHOTON: Photoelectric effect, Compton Effect photon energy : E h 2   h 2 photon momentum: p k l l wavenumber :  2 2 p k k v vT l  2 wavevector :k n l n : unit vector
8.  M (x,y,z) r Electromagnetic Wave: z k z Consider a light plane wave, O v propagating in the z direction: Wave function at point O: o = A cos (t) Wave function at point M: (M, t)=A cos ((t-z/v)) =Acos(t-kz) (M,t) Acos(t k.r) in complex description : Ψ(M,t) Ae-i(t-k.r)  wavenumber : : complex wavefunction i(t-k.r)   2 2 Ψ(M,t) Ae  k with theconvention : Ψ(M,t) Re Ψ(M,t) v vT l    2 real realpart of complex wave wavevector :k n wave function function l n : unit vector
9. de Broglie’s hypothesis 1924 Light is dualistic in nature, behaving in some situations like waves and in others like particles. If nature is symmetric, this duality should also hold for matter. Electrons and protons, which we usually think of as particles, may in some situations behave like waves. If a particle acts like a wave, it should have a wavelength and a frequency. De Broglie postulated that a free particle moving with speed v, having kinetic energy E, momentum p should have a wavelength l, frequency  related to its momentum p and its energy E by: E h  h p ;p k l
10. Wave function of a monochromatic wave Wave function of a electromagnetic monochromatic wave i(t k.r) (M,t) oe Wave function of a free particle of momentum p and kinetic energy E: i (Et p.r) i(t k.r)  (M,t) oe oe
11. DIFFRACTION OF LIGHT / Electron BY 1 SLIT Electron wave
12. DIFFRACTION OF ELECTRONS BY 2 SLITS
13. Example 39.1 An electron-diffraction experiment In an electron-diffraction experiment using an accelerating voltage of 54 V. The electrons have negligible kinetic energy before being accelerated. Find the electron wavelength. p2 KineticEnergyof e : E eV ; p 2meeV 2me h h 6.625 10 34 l 1.67 10 10 m 31 19 p 2meeV 2 9.1 10 1.6 10 54
14. The statistic meaning of the Wave Function of a particle: Intensity of Light is proportional to the square of the amplitude of the wave at that point: I=kA2 (W/m2) Intensity of Light is proportional to the photon density at that point. I=e.c =N. hf.c 2 1 2 1 B e: energy density of electromagnetic wave. (J/m3) e oE 2 2  N : photon density (photon/m3) o IA2 N The amplitude squared of the wave is proportional to the photon density => proportional to probability of finding the photon per unit volume. For the matter wave , the amplitude squared of the wave is the probability of finding the particle per unit volume= probabilty density 2 * 2 | (r,t) | (. ) o
15. The statistic meaning of de Broglie Wave of a particle Probability of finding the particle 2 2 * per unit volume= probabilty o | (r,t) | (. ) density . Probability of finding the particle dP | (r,t) |2 dV in a volume dV 2 Probability of finding the particle P | (r,t) | dV in a volume V V probability of finding the 2 particle over all space =1 P | (r,t) | dV 1 (the particle is certainly found) Normalized Condition of the wave function / Điều kiện chuẩn hóa của hàm sóng
16. Constraints on Wavefunction In order to represent a physically observable system, the wavefunction must satisfy certain constraints: (x,t) - Must be a single-valued function - Must be normalizable. This implies that the wavefunction approaches zero as x approaches infinity. - Must be a continuous function of x. - the first derivative of (x,t) must be continuous
17. The Heisenberg’s Uncertainty Principles Heisenberg’s uncertainty states that, it is impossible to simultaneously determine both the position and the momentum of a particle with arbitrarily great precision. The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa x. px h y. py h z. pz h
18. HEISENBERG’S UNCERTAINTY PRINCIPLE OF POSITION AND MOMENTUM The uncertainty in the x position x. p h x a a x x a Consider the diffraction of electrons by 2 2 a single slit. The uncertainty in the x-component of the momentum : x 0 px psin q px psin q l è sin q a q x a h l h p p psin q . x x l a a h x. p a. x a x. px h
19. HEISENBERG’S UNCERTAINTY PRINCIPLE FOR ENERGY AND TIME  E. t 2 The uncertainty principle for energy and time interval has a direct application to energy levels. We have assumed that each energy level in an atom has a very definite energy. However, this is not true for all energy levels. A system that remains in a metastable state for a very long time (large t ) can have a very well-defined energy (small E ), but if it remains in a state for only a short time (small t ) the uncertainty in energy must be correspondingly greater (large E ).
20. x 5 10 11m   x. p p x 2 x 2 x  p 1.055 10 24(kg.m / s) x min 2 x  p 1.055 10 24(kg.m / s) x min 2 x p 1.055 10 24(kg.m / s) p2 (1.055 10 24)2 KE 6.1 10 19 J 3.82eV 31 2me 2 9.1 10
21.   E. t E 3.3 10 27 J 2.1 10 8eV 2 2 t hc E l ldE El const; ldE Edl 0; dl E l E 589nm 2.1 10 8eV l 589 10 8 nm E 2.105eV