Quantum Mechanics I
Quantum Mechanics I PHY 851
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This 3 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 851 at Michigan State University taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/207632/phy-851-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
Lecture 23 February 7 2001 Quantum Fields and Second Quantization Many transitions of interest involve not only a particle changing from one state to another but the actual creation of a new particle For instance the radiative decay of an atom or nucleus results in a system with a photon that did not exist before the decay Most decays in particle physics also involve the creation of new particles In fact relativistic quantum eld theory considers all interactions eg the Coulomb force consisting of the creation and absorption of particles which are exchanged over a suf ciently short time so not to contradict the energytime uncertainty principle For this class we will constrain our goals to understanding simple decays Ireation and destruction operators were introduced to describe the creation of energy quanta in a harmonic oscillator The operators obeyed the relations fa 61 midi 0 0 1 where i speci es which oscillator is being affected If one had N harmonic oscillators one would have N creation operators and N destruction operators Note that the operators corresponding to different operators commute with one another they are unrelated The essential feature of creation and destruction operators is that they increase decrease the number of quanta where the number of quanta of the oscillator i is N However instead of counting only energy quanta in the case of the harmonic oscillator the number operator could also refer to a number of particles in that level Let us then consider a creation operator for each momentum eigenstate of a system The momentum state is created by operating on the vacuum k am 2 The state is normalized to unity just one would expect for creation operators k k 5M 3 By operating twice with 1 one creates a state with two particles of momentum k Example Two oscillators with a mixing term Consider two oscillator described by the creation operators and Let the Hamiltonian be H H0 V 4 H0 claim e2a2a2 5 V 8 6 Consider the operators bi and 6 de ned by bi E cos 9a sin 9a b E cos 9a sin 9a A 00 VV Lecture 23 February 7 2001 First show that bl bg bl and b obey the commutation rules for destruction operators Find E1 E2 and 9 such that H E15151 E25352 9 To nd the answer substitute the expressions for and bi and compare to the original expression for H E E E E algal alga cos 29 I 2 2 Lila algal 61 62 H E1E2 T T T alal agag alal agag lt dial aim 13 TM 03 11 By inspection one sees that the Hamiltonians are equivalent when 28 tan 29 12 61 62 E1Eg 6162 E1E2 2 2 61 62 7 z 7 14 lt 2 l 398 Ti 2 One can note the algebraic equivalence of this problem to the twocomponent problem with Hamiltonian 6 6 6 e v I 2 az 60m 1a Hzi 2 Field operators Creation and destruction in coordinate space Field operators are the coordinatespace analogs to They are de ned 8 ikx 1Tx W 16 They obey the commutation rules 1 x H WXL I TQ H V 8 ilk k ygtlak9aLl 1 m39 z 222 ik39ix y 18 k 1 V 2703 ldtte k39xyl 19 53X Y 20 These operators can create the state x EX New 21 which is a state with one particle at the position x and are normalized as KEY 53X Y 22 2 Lecture 23 February 7 2001 One can easily check that e ikx W One should keep in mind the Ilx is an operator not a wave function If 0 referes to a oneparticle state the relation to the wave function is given by 1 X 5gt mm where the wave function is de ned by ltxkgt 23 Charge densities and currents can also be considered as operators 0X PX PX 51x 5wltxgtgtrltxgt re llx0 If the state 0 and are oneparticle states one sees that ltX iltxgt agt we he we we 24 Free particle One can write the Hamiltonian for freely moving particles 72 H0 3 f l3x 1TXV2 llx 25 ifd3xZc2eilk k l39xala 26 2mV MC i k k The integral over x gives zero due to the varying phase unless k k at which point the phase is unity and the integral over x cancels the volume in the denominator 192 k 2m 7202 l aLak 27 Thus even though the Hamiltonian looks like the familiar expression for a wave function it is far more powerfull it correctly express the energy even when many particles are present in the system Interaction with an external potential An interaction with a potential would be written Hm 13r12rx1 rxpr 28 3
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