INTRODUCTORY MODERN PHYSICS
INTRODUCTORY MODERN PHYSICS PH 314
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This 14 page Class Notes was uploaded by Mrs. Demarcus Breitenberg on Monday October 19, 2015. The Class Notes belongs to PH 314 at Oregon State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/224540/ph-314-oregon-state-university in Physics 2 at Oregon State University.
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Date Created: 10/19/15
Simpleminded Wave Packet tutorial We will be summing superposing simple waves of the form fx sinkx with kvalues from a narrow range Let the k range width be always 10 of the average k value For simplicity let the average value always be kaverage Let s begin with just two waves The two k values should be then k1 095 and k2 105 As you can see in the plot below the resultant wave exhibits characteristic groups or packets Superposition of two waves 1 05x 4 105x I s sin sin sinO95x 10 100 50 50 100 Spatial coordinate x Now let s take three waves The kvalues should be now k1 095 k2 10 and k3 105 As you can see in the plot below new the packets shift apart a bit A smaller packet forms between each pair of large packets Superposi ion of 3 waves sin95x11 sin10x8 sin105x5 InXsin105x 95Xs39 sin 15 50 100 150 200 Spatial coordinate x 50 Now let s take five waves with k1 095 kg 0975 k3 10 k4 1025 and k5 105 Now the main packets shift apart even more and more baby packets form in between 35 3O 25 20 15 Superposition of 5 Waves sin95xsin975xsinxsin1025xsin105x n n n n n J J I39 i I39 I39 I I II sin975x17 sin1025x11 IA quot quot A Iquot quot I I I II H I vVIv 39I3939 nn 1 n I M 1 sin95x2o39 sinx14 sin105x3 I A I HH HI u I v A I IV 39439 39 I i quot 393939 I J 4 391 391 i j j quotj quotj quotj quotLI 39 quot Iquot I39 I39 I39 I l39 l I quotquot39v quoti J I I I 100 150 Spatial coordinate x 200 250 Next take as many as eleven waves with k1 095 k2 096 km 104 and k11105t s difficult to show everything in a single plot so let s make two one displaying the 11 waves and the other the sum only The distance between the main packets is now even larger and there is a whole bunch of baby packets in between 095 096104 105 11 waves sinkx with k I I I I l I I I I I I I ax J R I r I Ix Hv UV An H an V I l rxl I R am Wu H Hy AM My an Hy on Iy nl I R I III I I II II II II II II II II In 100 50 50 100 Spatial coordinate x WAVEFUNCTION RESULTANT WAVEFUNCTION 15 n A39 H39 Iii 1 nquot 39 lll HP 39 vnguill wr5139iI5EI 39l5nm L 14Hquot quotthequot NIH l quot I 39 39 ll 200 300 400 SPATIAL COORDINATE X 500 We may continue and keep adding more and more waves and the distance between the main packets will further increase Note that the central packet the one that forms around x O does not move So if the number of waves we take goes to in nity only the central packet will remain The next one to its right side will shift to plus in nity and the next one to its left side will shift to minus in nity But in nity means nowhere So by superposing an infinite number of constituent waves we obtain a single wave packet However there are still those annoying baby packets Is there a way of eliminating them The answer is yes Note that until now the constituent waves we were taking were all of the same amplitude Let s now take 11 waves with a Gaussian distribution of amplitudes They are shown in the next plot The wave with the middle k value has the largest amplitude and for larger and smaller k values the amplitude gradually decreases And as you can see in the plot showing the sum of those 11 waves the baby packets are no longer there 11 waves with Gauusian distrib of amplitudes 15 pquot quotgquotzquot I quot quot 39quot 39 quotquotquotr quot gquotzf y 39 39 xx H r r 39x quot quot 39 lquot m quot M r 10 V V w w 39 39 J V I 239 I u o u n r u I I X I 15 r 39 100 50 0 50 100 Spatial coordinate x Superposition of 11 waves with Gaussian distribut of amplitudes ZOOZDumgtltgtgt O 100 200 300 400 500 600 700 SPATIAL COORDINATE X 100 Conclusion by adding up an infinite number of de Broglie waves from a narrow range of wavenumbers k and with a Gaussian distribution of amplitudes we can obtain a single wave packet representing a localized particle This solves the localization problem The sum of an infinite number of waves can be replaced by an integral In fact we usually write functions representing wave packets in an integral form The wave packet formalism solves the localization problem It also solves the particle velocity problem In contrast to a single de Broglie wave a packet of de Broglie waves propagates with a velocity that is consistent with the particle velocity as for now please accept this without a proof
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