FI 3103 Quantum Physics Spin - Multisite ITB

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FI 3103 Quantum Physics

Alexander A. Iskandar

Physics of Magnetism and Photonics Research Group Institut Teknologi Bandung

Spin

Spin Operator

Spin ½Eigenstate Representation

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Evidence for the need of new quant. #

▪ Recall the Zeeman Effecton a orbitingelectron

Alexander A. Iskandar Spin 3

m_BB m_BB

B L μB 



 ˆ ˆ

ˆ



H mB

Evidence for the need of new quant. #

▪ Instead of the following Hydrogen spectrum

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Evidence for the need of new quant. #

▪ It was observed a Fine Structurein the Hydrogen spectrum

Evidence for the need of new quant. #

▪ For some other elements, it was found that there is an even number of splitting (anomalous Zeeman Effect)

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Stern – Gerlach Experiment

▪ The prediction, already made by the ‘old’ quantum theory that developed out of Bohr’s work, was that the spatial components of angular momentum could only take discrete values, so that the direction of angular momentum vector was restricted to only number of limited possibilities.

▪ This could be tested by making use of the fact that orbiting electron will give rise to a magnetic moment proportional to the orbital angular momentum of the electron.

▪ So, by measuring the magnetic moment of an atom, it should be possible to determine whether or not space quantization existed.

Stern – Gerlach Experiment

▪ The expectation based on classical physics is that due to random thermal effects in the oven, the magnetic dipole moment vectors of the atoms will be randomly oriented in space, so there should be a continuous spread in

the direction of the magnetic moments of the silver atoms as they emerge from the oven, ranging from −|μ| to |μ|. A line should then appear on the observation screen along the z direction.

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Stern – Gerlach Experiment

▪ To explain the result of Stern-Gerlach experiment (1922, Nobel prize in Physics 1943), Pauli (1924, Nobel in Physics 1945), Uhlenbeck and

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Spin Operator

▪ Introduce the spin operator (in analogy with the angular momentum operator) as

▪ With eigen-equation

▪ From which to obtain the two-splitting, we have to choose s=

½.

▪ Further, as was done for the angular momentum operator, we define the raising and lowering operator for the spin as

 

Sˆx,Sˆy iSˆz cyclic

s s m m

s m m

s S

m s s s m

s S

s s

z

s s

, , with

, ˆ ,

, ) 1 (

ˆ² , ²













y x iS S Sˆ_  ˆ  ˆ

Matrix Representation

▪ From the eigen-equation, we can find the matrix representation of the spin operator

▪ And from

the matrix representation of the raising and lowering operators are obtained as

z m

m s s

z s

z s m S s m m _s _s

S





2 1 0

0 1 2 2 1 0

0 2 1

ˆ ,

, _,



 







 





 



 





 

 

 _

1

) ,

1 ( ) 1 ( ˆ ,

, _ _ _

      

s

sm

m s s s

s S s m s s m m

m s

S  



 



 



 



  _

 1 0

0 and 0

0 0

1

0 

 S

S

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Matrix Representation

▪ From the raising and lowering operators, we can find the matrix representation of the other spin operators

▪ The matrices _x, _yand _zare called the Pauli matrices that satisfy the following commutation and anti-commutation relation

▪ Further, these Pauli matrices have the following property

y y

x

x i

S i

S  

2 0 0 and 2

2 0 1

1 0 2



 

 



 







 



 



x^,y



²iz ^and^cyclic^permutatioⁿ



 







 0 1

0

2 1

2 2

z y

x  





x,y



xyyx 0 andalso



y,z



0



x,z



Representation of Spin ½ Eigenstate

▪ Using the matrix representation of the spin operators, we can find the representation of its eigen-state.

▪ For spin ½state, the eigen-state of S_zwill be represented as a two-component column vector called spinor.

▪ Which gives



 



 





 







 





 



 





v u v

u v

S_z u

1 2 0

0 1 2







 

 

 _



and 0

1 



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Representation of Spin ½ Eigenstate

▪ Any arbitrary spinor can be expanded in terms of this complete set of eigen-spinor

▪ For a normalized spinor state, the following condition of the expansion coefficients has to be satisfied

▪ And from the expansion postulate, the probability of the S_z measurement on the state ayielding the eigen-spinor₊ (spin up) is and the probability of finding it in the eigen- spinor_- (spin down) is .











  



 



 



 



 



 



 a a a  a 

a a a

1 0 0

1

2 1

2 _ 

 a

a

2

a

2

a

A complete set of quantum numbers

▪ Hence the full wavefunction of an electron in the H atom is

▪ Where the eigenspinnor is given as

▪ Note that the spin functions do not depend on the electron spatial coordinates r, q, j ; they represent a purely internal degree of freedom.

▪ The complete set of quantum numbers is:

n,l,m,s,m_swith s = ½ and m_s= +/- ½.

( ) ( ) ( ) ,

s s

nlmsm R r Ynl lm s m

 r  q  ,

1/ 2,1/ 2 1/ 2, 1/ 2

1 0

0 , 1

 ^{ }   _ ^{ } 

   

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More on Matrix Representation

▪ It is not necessary to have a diagonal matrix representation of S_x.

▪ Consider the eigensystem of the following operator (the parameter lis given to make the equation more flexibility)

or,

▪ Yielding,

▪ Hence,

 

_



 



 



 



 

v u v

S u

S_x  _y  l

sin 2

cos 



 



 



 







 







 







 







 ^

v u v

u e

e v

u i

i

i l



0 0 0

sin cos

0

v u e

u v e

i i

l l





 

1 0

) 1

( l²   l uv

More on Matrix Representation

▪ Hence, for l= 1, then and, the eigenvector is

▪ Since we can multiply a state vector by an arbitrary constant phase, then we can write the above eigenvector as

▪ And for l= –1, the eigenvector has to be orthogonal to the previous eigenvector, then we have



 







ei

1 2 1

u v e^ⁱ^ 



 



  ^

 2

2

2 1



 i

i

e u e

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More on Matrix Representation

▪ Example 1