Lecture 12

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

This preview shows pages 1–6. Sign up to view the full content.

Addition of spins A two-particle system

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 16

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online