8/18/2019 summary chapter 5 - Townsend Quantum Mechanics

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Bruna Shinohara de Mendonça

Quantum Mechanics – Summary of Chapter 5 – A system of 2 spin ½ particles

The chapter starts introducing a ne operation !eteen "ectors# the direct product# inhich multiplications are done term !y term# alloing products of to $ets or to !ras%

&sing this feature# e o!tain the 'amiltonian !ased on the spin(spin interaction of a

system of 2 spin ½ particles ( a proton and an electron% )e find that# for the ground

state# there is a single to(particle spin state# hile for the e*cited state# it ill !e three

times degenerated% )e also find the fre+uency for a photon transition !eteen states ,

and 2 to !e ,-2.'/# hich has applications in astrophysics of the interstellar medium%

0art 5%1 defines the total spin angular momentum# S# as the sum of indi"idual momenta%

)e also conclude that S, and S2 ill commute ith S2%

5%- presents the insten(0odols$y(3osen parado* hich shos that# if a spin(. particledecays in to spin ½ particles# hen a measurement is made on the first one# the output

for the measurement of the second one is already defined% instein himself didn4t

appreciate the idea of this !rea$ in the locality principle# so he imagine that the particles

ere 5. in one state and 5. in other# !ut e could not access those states due some

other "aria!le% Another physicist# 6ohn Bell# pro"ed later that a non(+uantum theory#

ith hidden "aria!le# ould !e inconsistent ith some of the e*perimental results%

Chapter 5%7 discuss ho a particle could !e teleported% 'a"ing a particle !y itself# ,#

and an 03 pair of particles# 2 and 1% 8f a Bell State measurement is carried in particles

, and 2# their states cannot !e factored out as to single states# hich is# !y definition#

+uantum entanglement% Therefore# particle 1 state ill no !e defined !y particle ,#

e"en though they hadn4t ha"e a!solutely no contact ith each other%

Chapter 5%9 introduces the density operator# hich allo us to or$ ith non(pure

states% The trace of this operator for a mi*ed state is e+ual to ,% 8f e ha"e to particles

and e ant to $no the e*pectation "alue for the spin of the first# e can use the

reduced density operator# hich is the trace o"er the matri* elements for particle 2%