8/19/2019 summary chapter 3 townsend quantum mechanics modern approach

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Bruna Shinohara de Mendonça

Quantum Mechanics – Summary of Chapter 3 -Angular Momentum

This chapter starts with a gedanken eperiment to familiari!e the reader with the ideathat the order of rotations changes the final state" #or a mathematical eample$ it is

considered two rotation% one &y an angle '( a&out y$ followed &y a rotation of the same

angle$ &ut a&out the -ais and another rotation of '( a&out $ followed &y a rotation of 

'( a&out the y" )et '( &e a small angle and using Taylor series$ we conclude that the S

operator does not commute"

*owe+er$ as we ha+e seen in chapter ,$ the rotation operator can &e epressed as an

eponential function of the generator of functions and the angle" #ollowing the steps

a&o+e$ we reach the commutator of the generator of the rotations$ which is the product

of this operator su&tracted &y the same product$ &ut in in+erse order" The &oo also

mentions that the commutator for the or&ital momentum has the same format"

.art 3", shows the importance of the non-commutati+e property &y showing the special

features of commutati+e operators" /f two operators commute$ they will share

eigenstates" /f an operator has more than one eigenstate with the same eigenvalue$ the

state is called degenerated"

Chapter 3", pro+es that the operator 0, commutes with the other generator operators"

Therefore$ they will ha+e eigenstates in common 1m and 2" 4sing a numerical eample$

it is shown that although 0 and 0y together are not a diagonal matri$ there are linear 

com&inations 10 5 i0y that generates a pattern" 06 is called a raising operator$ &ecause it

creates and eigenstate  of 1m67 times h-&ar" 0-  is called a lowering operator$

analogously"

8nowing that the pro9ection of the angular momentum should not eceed it magnitude$

we find a constraint to the eigenvalues" The restriction leads to a maimum 2 of 91967

and a minimum 2 of :"

Chapter 3"; shows how to find the elements of the matri 06 and 0-" Chapter 3"< shows

that operators that do not commute and ha+e a factor of i in the right hand side leads to

an uncertainty relation using Schwar! ine=uality"

.art 3"> uses the concepts from the whole chapter to find the intrinsic spin states of fermions" To represent the $ y and ! components of the matri$ the &oo o&tained first

the coefficients for raising and lower matrices and use the relations with the cartesian

components" The ,, matrices o&tained are referred as .auli Spin Matrices" Then$ linear 

alge&ra is used to find the eigenvalues and eigenstates.

The last chapter presents a Stern-?erlach eperiments for &osons and demonstrates

alge&raically that the pro&a&ilities will not &e e=ual &y finding the eigen+alues and

normali!ing it" /f the particles pass &y a S?y and then a S?! de+ice$ <:@ will not &e

defected$ ,<@ will &e up and ,<@ will &e down"