Lecture 12 - Addition of spins A two-particle system Two...

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Addition of spins A two-particle system
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Two spin ½ particles Consider a hydrogen atom in its ground state in a magnetic field. It consists of two particles – electron and proton, each having spin ½ , and interacting therefore with the magnetic field. What are possible eigen states of the Pauli Hamiltonian describing this system? ( 29 (1) (2) (1) (2) z z z z e e e e e e H BS BS B S S m m m = + = + The problem with this notation is that the spin operators 1 and 2 act on spin states of different particles. In order for that expression to make sense we need to consider a new space composed of states of both particles.
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Spin states of two particles We consider spin states of each particle as states , in which each of them have definite value of z-component of the spin. There are four possible linearly independent combinations These states can symbolically be written down as 1 ,2 , 1 ,2 , 1 ,2 , 1 ,2 These states are mutually exclusive, i.e. orthonormal so that we can use them to present any other states by linear superposition.
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Individual S z operators in the two- particle space (1) (2) 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 ; ; 0 0 1 0 0 0 1 0 0 0 0 0 2 2 0 0 0 1 0 0 0 1 0 0 0 1 z z z S S S - = = = - - - - h h h In order to present spin operators in a matrix form we have to introduce a enumeration for the basis vectors: ( 29 ( 29 ( 29 ( 29 1 2 3 4 ↑↑ ↑↓ ↓↑ ↓↓ { { { { { { { { (1) (1) (1) (1) 1 2 3 4 (2) (2) (2) (2) 1 2 3 4 ; ; ; 2 2 2 2 ; ; ; 2 2 2 2 S S S S S S S S ↑↑ = ↑↑ ↑↓ = ↑↓ ↓↑ = ↓↑ ↓↓ = - ↓↓ ↑↑ = ↑↑ ↑↓ = - ↑↓ ↓↑ = ↓↑ ↓↓ = - ↓↓ h h h h h h h h This way we can find the matrices representing individual operators and define the z-component of the total sum as a regular matrix summation
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Z-component operators in the new space These definite values are the diagonal elements of the matrix ( 29 ( 29 ( 29 ( 29 1 0 0 1 m - = ↑↑ ↑↓ ↓↑ ↓↓ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 z S = - h Operator of the z-component of the total spin is diagonal, which means that the states in which each particle has a definite value of their spin component are also
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Lecture 12 - Addition of spins A two-particle system Two...

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