Ch3. Wave Properties of Particles 1
Introduction Quantum theory of light particle properties of waves particles might show wave behavior. For a photon hf E = pc hf = pc p= = c h h or λ = = ; where k = h (1) λ 1 p km 2 0v v 1 c 2 De Broglie suggested that (1) is a completely general one that applies to material particles as well as to photon. The wave and particle aspects of moving bodies never be observed at the same time. 2
Which set of properties is most conspicuous depends on how its de Broglie wavelength compares with its dimensions and the dimensions of whatever it interacts with. Example: Find the de Broglie wavelengths of (a) a 46-g golf ball with a velocity of 30 m/s, and (b) an electron with a velocity of 10 7 m/s. Sol: 3
The dimensions of atoms are comparable with this figure the radius of hydrogen atom, for instant, is 5.3x10-11 m. It is not surprising that the wave character of moving electron is the key to understanding atomic structure and behavior. Example Find the kinetic energy of a proton whose de Broglie wavelength is 1.000fm = 1.000x10-15 m, which is roughly the proton diameter. Sol: 4
de Broglie wave was verified by experiments involving the diffraction of electron by crystal.(we will talk about it latter) Wave of what? wave of probability 5
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1. Ψ=Ψ( xyzt,,, ) matter wave We call Ψ the wave function. The value of the wave function associated with a moving body at a particular point x,y,z in space at the time t is related to the likelihood of finding the body there at the time. 2. The wave function Ψ itself, however has no direct physical significance. The probability of experimentally finding the body described by the wave function Ψ at the point x,y,z at time t is proportional to the value of first made by Max Born in 1926. 2 Ψ there at t. This interpretation was 3. Bragg wrote: Everything in the future is a wave, everything in the past is a particle. 2 Ψ 7
Wave formula 8
v p : phase velocity k 2π ω =, ω = 2π f, = λ f λ k 9
2 2 h E h mc c vp = λ f = c p = = > h mv g h vg v g group velocity. Phase and Group velocity The wave representation of a moving body corresponds to a wave packet or wave group. 10
Wave packet? sound beat f 1 =440Hz f 2 =442Hz we will hear a fluctuating sound of frequency 441Hz with two loudness peaks, call beats, per second. 11
1. above represents a wave of angular frequency w and wave number k 12
that superimpose upon it a modulation of angular frequency 1 2 w and of wave number 1 2 k. 2. cosine wave with indefinitely small k and w difference, group wave (or localized particle). 13
v = v g The de Broglie wave group associated with a moving body travels with the same velocity as the body. Example An electron has a de Broglie wavelength of 2.00x10-12 m. Find its kinetic energy and the phase and group velocities of its de Broglie waves. Sol: 14
Particle Diffraction In 1927 Davison and Germer confirmed de Broglie s hypothesis by demonstrating that electron beams are diffracted when they are scattered by the regular atomic arrays of crystals. 15
( ) smoothly 16
De Broglie s hypothesis suggested that electron waves were being diffracted by the target much as x-rays are diffracted by planes in a crystal. 17
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: target work function. 19
The Davisson-Germer experiment thus directly verifies de Brogli hypothesis of the wave nature of moving bodies. neutron, atom diffraction Electron Microscopes 20
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Particle in a Box: Why the energy of a trapped particle is quantized (We will discuss this topic in detail in Ch.5) particle E( E = KE) Ignore relativistic consideration. From a wave point of view, a particle trapped in a box is like a standing wave in a string stretched between the box s wall. 24
box standing wave : Each permitted energy is called an energy level, n is called quantum number. 25
: Any particle confined to a certain region of space ( even if the region does not have a well-defined boundary) : 1. A trapped particle cannot have an arbitrary energy, as a free particle can. 2. A trapped particle cannot have zero energy. standing wave 3. Quantization of energy is conspicuous only when m and L are also small. E E = E n n 1 ( particle ) Example An electron is in a box 0.10nm, which is the order of magnitude of atomic dimensions. Find its permitted energies. Sol: 26
Indeed, energy quantization is prominent in the case of an atomic electron. 27
Example A 10-g marble is in a box 10cm across. Find it permitted energies. Sol: The permissible energy levels are so close together, then, there is no way to determined wherever the marble can take on only those energies or any energy whatever. Uncertainty Principle: We cannot know the future because we cannot know the present In 1927 Heisenberg Uncertainty principle: It is impossible to know both the exact position and exact momentum of an object at the same time. 28
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Fourier series or Fourier transform theory Fourier Series: Fourier Transform f( x) = a cos( nkx) n= 0 n f( x) = g( k)coskxdk 0 1 g( k) = f ( x)coskxdx 2π 0 gx ( ) f( x) Inverse Foureie Transform or spectrum of f(x) 31
A narrow de Broglie wave group thus means a well-defined position ( x smaller) but a poorly defined wavelength and a large uncertainty p in 32
the momentum of the particle the group represents. A wide wave group means a more precise momentum but a less precise position. Gaussian Function When a set of measurement is made of some quantity x in which the experiment errors are random, the result is often Gaussian distribution whose form is a bell-shaped curve. 33
+x x 0 f(x) normalized 34
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x k We cannot know exactly both where a particle is right now and what its momentum is. We cannot know the future for sure because we cannot know the present for sure. de Broglie matter The narrower the original wave packet that is, the more precisely we know its position at that time the more it spreads out because it is made up of a greater span of waves with different phase velocities ( 1 x k ) 2 37
Example A measurement establishes the position of a proton with an accuracy of ±1.00x10 11 m. Find the uncertainty in the proton s 1.00s later. Assume v c. Sol: 38
x ( p ) match box house : 39
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xt ( = 0) x ( xt ( = 0) pt ( = 0) Uncertainty Principle : 1. particle 2. particle 3. If we are to see the electron λ x p 41
Applying the Uncertainty Principle: A useful tool, not just a negative statement. Example A typical atomic nucleus is about 5.0x10-15 m in radius. Use the uncertainty principle to place a lower limit on the energy an electron must have if it is to be part of a nucleus. Sol: 42
Example A hydrogen atom is 5.3x10-11 m in radius. Use the uncertainty principle to estimate the minimum energy an electron can have in this atom. Sol: Actually, the kinetic energy of an electron in the lowest energy level of a hydrogen atom is 13.6eV. Energy & Time Uncertainty principle : E t 2 43
particle ( ) Example An excited atom gives up its energy by emitting a photon of characteristic frequency. The average period that elapse between the excitation of an atom and the time it radiates is 1.0x10-8 s. Find the inherent uncertainty in the frequency of the photon. Sol: 1.0x10-8 s : For a photon whose frequency is, say, 5.0x10 14 Hz, f / f =1.6x10-8 44