Undergrad What Are the Key Properties and Measurements of Pauli Matrices?

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SUMMARY

The discussion focuses on the properties and measurements of Pauli matrices, specifically their relationship to the identity matrix and their Hermitian nature. Participants clarify that Pauli matrices satisfy the equation σ² = I, indicating their role in quantum mechanics. They also highlight that these matrices represent a homomorphism from Clifford algebra, which simplifies the representation of operators while allowing for numerical approximations. Understanding these concepts is crucial for grasping the implications of Pauli matrices in quantum physics.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with linear algebra and matrix operations
  • Knowledge of Hermitian operators
  • Basic concepts of Clifford algebra
NEXT STEPS
  • Study the properties of Hermitian matrices in quantum mechanics
  • Explore the applications of Pauli matrices in quantum computing
  • Learn about the Stern-Gerlach experiment and its significance in quantum measurement
  • Investigate the role of Clifford algebra in advanced physics
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Students and researchers in quantum mechanics, physicists exploring quantum computing, and anyone interested in the mathematical foundations of quantum theory.

QuasarBoy543298
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TL;DR
questions about Pauli matrices -

why do they need to be Hermitian, what are they trying to measure and why do they need to

satisfy that those matrices squared equals the identity matrices.
Hi :)
I have several questions about the Pauli matrices,
I have seen them when the lecturer showed us Stern-Gerlach experiment
, and we did some really weird assumptions on what we think they should be.

1- why did we assume that all of those matrices should satisfy
σ2 = I (the identity matrices)

2- why do they have to be Hermitian?

3- what they are trying to measure? (when we insert <-| for example, we get -<-|, why is that? )
thanks for helping !
 
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They are a representation of Clifford algebra.

Clifford algebra has operators, abstract objects that have certain properties. Representation is a homomorphism from the operator algebra into matrices. The properties of the matrices reflect the properties of the operators.

https://en.wikipedia.org/wiki/Clifford_algebra#Physics
By using matrices instead of operators you are losing certain general properties, but at the same time you gain certain simplification and also possibility to do numerical approximations.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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