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Date Created: 09/23/15
PHYS 113 Quantum Mechanics Waves and Interference In much of our class we have taken the View that everything in the world behaves more or less like a particle Even photons which we describe by their wavelength note the wave are said to have a particular position and velocity We ve spent very little time talking about the wavelike properties of either light or say electrons What do we mean that light has a particular wavelength Consider the classic two slit77 experiment in which a beam of monochromatic all 1 wavelength light is shined at two slits in an otherwise opaque boundary What happens The light behaving like a wave emanates out from each of the slits independently What does this mean from a mathematical point of view Well remember that light is the carrier of the electrical eld Thus the electrical eld and similarly the magnetic eld might at a distance 951 from the rst slit has a value E1lt1 t E0 coskx17wt 1 where k for now is an unknown constant and w is the angular frequency of oscillations We might ask how bright would the light appear on the wall if there were only 1 slit77 Well the intensity of the light is simply the square of the electrical eld Ham 5 E3 cos2kx1 7 wt However if we average this over time we nd that the cos2 averages to 12 and thus the average intensity is just E3 am 7 7 In other words except for the fact that the light is more or less pointed forward and thus the brightness will fade off near the edges the brightness would form a constant patter Now what happens when we add a second slit We get Em E0 coskx1 7 wt coskr2 7 14 Quantum Mechanics7 1 PHYS 113 Quantum Mechanics tot E5 cos2k11 7 wt cos2lcx2 7 wt 2 cosk11 7 wt coskxg 7 0 E5 cos2k117 wt cos2lcx2 7 wt cosk11 7 12 coskx1 x2 7 20 ltItotgt E5 1 coskx1 7 l7ve used a bunch of trig identities here but nothing you haven7t seen before Now what does this mean It means that if 1011712 27m 71 M72 71 0 1 2 then the waves add constructively and you see a bright spot If on other hand 1011 7 x2 27rn 12 then the cos function gives 71 and the average intensity is zero You see a dark spot Remember our basic method We had several waves We added them together and we squared the result Thus the square of the wave7function tells us something important It works with electrons too Now what happens if instead of ring a beam of light we re a beam of electrons through the two slits lmagine further that we have a scintillator on the far wall so that the brightness of the wall is proportional to the number of electrons striking it What pattern would we expect Surely you might think electrons behave like particles and thus you might expect that electrons might simply pass through one slit or the other and form a double peaked pattern like that seen on the left lwl lwiwzl 13 W212 a 4 ludh39rrlunl Accnmul lcd roumsmln umuhmin But in fact electrons behave exactly like light and thus have the multiply peaked behavior as seen on the right What does this mean It means that electrons and all other particles as well have a wave7 like pattern not unlike that of light We call this wave7function 11 and it has the property that P000105 WINde 2 For example consider a particle trapped in a 1d box It might have a particular wave7 Quantum Mechanics7 2