LECTURE 3 BASIC QUANTUM THEORY

Transcription

1 LECTURE 3 BASIC QUANTUM THEORY Matter waves and the wave function In 194 De Broglie roosed that all matter has a wavelength and exhibits wave like behavior. He roosed that the wavelength of a article of momentum is λ π! where P is the momentum of the article The incentive for this exression was the corresonding one for hotons roosed by Einstein. He described an electron as a standing wave around the circumference of an atomic orbit such that, n λ πr This was combined with Niels Bohr s ostulate that angular momentum was quantized (which we will later see imlies a limited number of energies), to give a wavelength mvr n! π! λ Lecture 3: Basic Quantum Theory Setember, 000 1

2 e.g., Calculate the de Broglie wavelength of an electron which has a kinetic energy of 4 ev ( 1/) mv ( 1/ ( m ) E λ h h me 6.13 A The wavelength has the dimensions of interatomic sacing. What s the de Broglie wavelength of a tennis ball having a mass of 50 g and traveling with a velocity of 00 km/h? Similar calculations show that 4 λ A If electrons are just articles, they should behave as such: Behavior of a stream of articles and a wave iminging on a screen with a small oening. The wave is said to be diffracted, diffraction is observed when a wave is distorted by an obstacle which has dimensions comarable to the wavelength of the wave. Lecture 3: Basic Quantum Theory Setember, 000

3 The wave like behavior of electrons was exerimentally confirmed by the Bragg diffraction of electrons by crystals. This henomenon is analogous to crystal diffraction of an EM wave (X-rays). Bragg icture of X- ray Diffraction Reflection is in hase if nλ dsinθ (Bragg condition), and we get constructive interference and a diffraction attern is obtained. It was observed that the same diffraction attern was roduced by X-rays of wavelength λ h/ and that roduced by electrons with the same de Broglie wavelength. nλ dsinθ nh/ This is a result of quantum theory, and holds for hotons for relativistic and nonrelativistic articles (with a velocity small comared with that of light). Other articles include neutral atoms, rotons, ositive ions, etc. Lecture 3: Basic Quantum Theory Setember, 000 3

4 Back to Quantum Mechanics Unified scheme of Mechanics based uon notions of Planck s quantum theory (E hν) and utilizing de Broglie s ideas of the wave like nature of matter. It is a revision of the laws of mechanics design to extend the subject into the realm of atomic and nuclear henomena. Classically an EM wave is interreted as changing electrical and magnetic fields. If a article has a wavelength, what is the field? What is meant by a wave to be associated with a article? Born showed that the wave amlitude is related to the robability of locating the article in a given region of sace. More secifically, in quantum mechanical roblems, we attemt to find a quantity Ψ called the wave function. While Ψ itself has no direct hysical meaning, it is defined in such a way that the robability of finding the article in the region of sace between x and x + dx and y and y + dy, and z and z + dz is given by ψ ψ dx dy dz The interretation of the wavelength of an electron is through the function Ψ. Since the robability that the article will be found somewhere in sace is unity, we must require that the wave function be normalized such the integral being taken over all sace. ψ ψ dx dy dz 1 This makes it clear that in quantum mechanics robability statements are often obtained, whereas in classical mechanics the location of a article can be determined exactly. The Schrodinger Equation Solution to the calculation and interretation of Ψ rovided by Schrodinger in 195: Motion of electron with energy E described by a wavefunction : The roerties of the wave function can be exressed in the form of ostulates: 1.- Associated with the article there is a comlex wave function Ψ(x,y,z,t), where x,y,z are sace coordinates and t is the time. Ψ jωt ( r, t) Ψ( r) e (1) Lecture 3: Basic Quantum Theory Setember, 000 4

5 where, e () r Ψ jωt - osition deendent art - time deendent art.- The classical exression for the total energy (Et), given by E t m + v ( X, Y, Z) where is the momentum of the article, m its mass and V(x,y,z) its otential energy, may be converted into a wave equation by associating certain oerators. Let s define the momentum oerator! i such that when substituted in the exression for Et, we get E t 1 m! m ( i ) + v( X, Y, Z) + v( X, Y, Z)!. When we oerate with this exression on Ψ, we obtain,! m Ψ+ V() r E This is the Time Indeendent Schrodinger Equation and is obeyed by Ψ(x,y,z). Ψ Ψ x where, - Lalacian oerator defined before. Ψ + y Ψ + z What does Ψ mean? What is the interretation of the wave function? Lecture 3: Basic Quantum Theory Setember, 000 5

6 Ψ( r, t) Ψ ( r, t) robability density (er unit volume) of finding electron at osition r at time t where, Ψ ( r, t) - comlex conjugate. Schrodinger equation is a guess; it can t be derived but it has been confirmed exerimentally. Solution to Schrodinger equation for secial roblems Free Electrons! " Free" V() r 0 everywhere Ψ Ψ (5) E m! d Ψ consider 1- dim case : m dx E Ψ (6) jkx Solution : ψ (x) Ce where C is a constant (7) d Ψ dx k Ψ! k need m E or k me! (8) classical hysics : m E identify! k electron momentum (9) Ψ( x, t) Ce jkx e jωt Ce j( kx wt) (10) This is the equation of a travelling wave. It reresents a free article of momentum and energy given by Equation 9. Lecture 3: Basic Quantum Theory Setember, 000 6

7 Phase velocity defined as: k x ω t Consider a harmonic lane wave, the hase velocity is the seed at which the rofile moves: ψ t 0 time t later X x The hase velocity can also be exressed as: λ x ω E! Phase velocity t k! me E m More on Plane waves: Lecture 3: Basic Quantum Theory Setember, 000 7

8 Harmonic functions should reeat themselves in sace after a dislacement of λ in the direction of roagation (K). Then, λk ψ ( r) ψ ( r + ) k where k is the magnitude of K and K/k is a unit vector arallel to it, then Ae ik r Ae Ae ik ( r +λk / k ) ik r e iλk for this to be true e 1 e iλk iπ and finally λ π k We can generalize to 3 dim: m! jk r Ψ EΨ with Ψ Ce ( 11) where, k k - wave vector π λ Ψ j (, ) ( k r ωt r t Ce ) (1) - Travelling lane wave solution - Wave moves in direction of k - Planes of constant hase erendicular to k - Wavelength given by, π λ k Lecture 3: Basic Quantum Theory Setember, 000 8