Pilot Waves and the wave function

Transcription

1 Announcements: Pilot Waves and the wave function No class next Friday, Oct. 18! Important Lecture in how wave packets work. Material is not in the book. Today we will define the wave function and see how it works. Physics 2170 Fall

2 Friday Facts -- Earth s Core Physics 2170 Fall

3 Seismic Waves Earthquakes generate three types of seismic waves: P (primary) waves, S (secondary) waves and surface waves, which arrive at seismic recording stations one after another. Both P and S waves penetrate the interior of the Earth while surface waves do not. Surface waves arrive last and are the least interesting to seismic tomographers because they don't penetrate deep inside the Earth, hence provide little information. Physics 2170 Fall

4 P and S Waves P WAVES compressional Waves longitudinal waves First to arrive at seismic stations Travel at km/sec in the Earth's crust Shake the ground in the direction they are propagating Travel through the Earth's core S WAVES shear waves Second to arrive at siesmic stations 1.7 times slower than P waves Shake the ground perpendicular to the direction in which they are propagating Do not travel through liquid (ie. water, molten rock, the Earth's outer core) Physics 2170 Fall

5 de Broglie s Pilot Waves For electromagnetic radiation: λ=w/ν, where w=c, ν=e/h, λ=c/ν, and E = pc, λ = h/p De Broglie postulated that the wavelength λ and the frequency ν of the pilot waves associated with a particle of momentum p and total relativistic energy E are given by the equations λ = h/p ν = E/h and that the motion of the particle is governed by the wave propagation properties of pilot waves. Let s compute the propagation velocity w of the pilot waves associated with a particle w=λ ν = h/p E/h = E/p Physics 2170 Fall

6 De Broglie s Pilot waves cont. w = E/p = c 2 p 2 + (m 0 c 2 ) 2 = p p 2 + (m c 0 c) 2 p w = c 1+ (m 0 c / p) 2 Note w is greater than c. This seems, at first, quite disturbing because the velocity, v, must necessarily be less than c, and it would appear that the particle could not keep up with its own pilot waves. Actually, there is no problem as we will show -- Physics 2170 Fall

7 Group of Pilot Waves Imagine a free particle (experiencing no forces) and its associated pilot waves are moving such they can be described by a single spatial coordinate x. It must look qualitatively like: x The amplitude of the wave must be modulated so that their value is non-zero in only a finite region of space in the vicinity of the particle. The pilot waves form a group of waves and the group must move along the x-axis with the velocity of the particle. It is important to distinguish between the velocity g of the group and the velocity w of the individual oscillations of the waves. Need to find a relationship between g and E and p Physics 2170 Fall

8 A Simple Wave Consider the simplest type of wave motion a sinusoidal wave of frequency ν and wavelength λ, which is of constant amplitude from - to +, but moving with uniform velocity in the + x direction: or Ψ(x,t) = sin2π x λ υt Ψ(x,t) = sin2π( kx υt);k =1/λ 1. Holding x fixed, function oscillates in time with frequency ν 2. Holding t fixed, function has sinusoidal dependence on λ 3. Zeros of function correspond to nodes of the wave, at positions x n for which: 2π(kx n νt) = πn n = 0,±1,±2,... or x n = n 2k + ν k t k=wavenumber Physics 2170 Fall

9 Wave Velocity x n = n 2k + ν k t Thus these nodes, and in fact all points on the wave, are moving in the direction of increasing x with velocity w = dx n /dt = ν/k = νλ Let s try two waves where where Now Ψ(x,t) = Ψ 1 (x,t) + Ψ 2 (x,t) Ψ 1 (x,t) = sin2π kx υt ( ) Ψ 2 (x,t) = sin2π (k + dk)x (υ + dυ)t [ ] sin A + sin B = 2 cos½(a-b) sin½(a+b) Ψ(x,t) = 2cos2π dk 2 x dυ 2 t (2k + dk) (2υ + dυ) sin2π x 2 2 Ψ(x,t) 2cos2π dk 2 x dυ 2 t sin2π kx υt [ ] t Physics 2170 Fall

10 Group Velocity Ψ(x,t) 2cos2π dk 2 x dυ 2 t sin2π kx υt [ ] w = υ /k g = dυ /2 dk /2 = dυ dk ν=e/h and k=1/λ=p/h dν=de/h and dk=dp/h g=de/dp finally! Physics 2170 Fall

11 So Group Velocity Equal to Particle Velocity g = de/dp E 2 =c 2 p 2 +(m 0 c 2 ) 2 2E de = c 2 2p dp de/dp = c 2 p/e Therefore g = c 2 p/e ; where E = γmc 2 and p = γmv or g = c 2 γmv/γmc 2 g = v The velocity of the group of pilot waves is equal to the velocity of the particle. Also w = E/p ; so g w = c 2 The wave velocity is always larger than g, the individual waves are constantly moving through the group from rear to front. The same situation occurs in a group of water waves. Physics 2170 Fall

12 Maxwell s Equations describe EM radiation in vacuum: E, B, and k form a right-handed system, with the wave traveling in the direction of k at speed, c. E max =peak amplitude c standard wave equation Physics 2170 Fall

13 Two slit interference Standard two slit interference with light causes waves to interfere The amplitude A(x) gives the electric and/or magnetic field of the wave as a function of position. It can be positive or negative. The intensity is proportional to the square of the amplitude and is 0. This is what we see! The intensity also tells us the probability that a photon will land in a particular x region. Physics 2170 Fall

14 Same thing with electrons Electrons have a wavelength and so two slit interference has the same effect as for light The probability of any given electron hitting near x is determined by is called the wave function. It contains everything we know about the electron but only gives probabilities about where the electron is located Physics 2170 Fall

15 Probability versus probability density is the probability density for finding the electron at point x. To get probability, multiply the probability density by a region δx. The probability of finding the electron in the region δx shown is Similar to the relationship between linear mass density and mass. x Physics 2170 Fall

16 Probability and probability density are similar to mass and linear mass density. Physics 2170 Fall

17 Clicker question 1 This is the wave function for a neutron. Around which point is the neutron most likely to be found? A. x = 0 B. x = x A C. x = x B D. x = x C E. There is no most likely place Set frequency to AD Wave functions exist for any object including electrons and neutrons. Physics 2170 Fall

18 Clicker question 1 This is the wave function for a neutron. Around which point is the neutron most likely to be found? A. x = 0 B. x = x A C. x = x B D. x = x C E. There is no most likely place Set frequency to AD Wave functions exist for any object including electrons and neutrons. The wave functions we have been looking at came from two-slit interference and so were simple sine waves but different functions are also possible. Remember the probability is related to so need to square the wave function. Physics 2170 Fall

19 The wave function The wave function does not tell you the path of an object like a normal traveling wave function. ψ(x) The wave function (squared) gives the probability of finding the object at a given x. Physics 2170 Fall L Some details about the wave function: Wave function is really a function of the 3D position, ψ(x,y,z) but many of the problems are 1D so we will start with that: ψ(x). Wave functions are complex valued (have real and imaginary parts) and are never directly observable. The probability density is observable and is L x ψ*(x) is complex conjugate: replace i by i in ψ(x).

20 Getting probabilities from wave functions ψ(x) Wave function ψ(x): L L x Probability density For an infinitesimal distance δx the probability is To find the probability over a finite distance we need to integrate. L a b L x Probability of finding electron between a and b is the area under the curve. Probability of electron being between a and b is Physics 2170 Fall

21 Normalization Probability density = Probability of electron being between a and b is L a b L x What if this calculation gives a number larger than 1? Doesn t make any sense! A properly normalized wave function obeys the normalization constraint: This is simply a statement that the electron is located somewhere! That is, the probability of finding the electron somewhere is 100%. Physics 2170 Fall