Trying to interpret matrix representations of operators

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SUMMARY

The discussion centers on the interpretation of matrix representations of operators, specifically a 3x3 operator Q, and its eigenvectors and eigenvalues. The participant confirms that the eigenvectors correspond to eigenfunctions, and when measuring the state represented by a 3x1 matrix [b], it can be expressed as a linear combination of these eigenvectors. The analogy to Schrödinger's cat is drawn, illustrating that prior to measurement, the state is a superposition, and upon measurement, the system collapses into one of the eigenstates of Q, confirming the probabilistic nature of quantum mechanics.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly eigenvectors and eigenvalues.
  • Familiarity with quantum mechanics principles, especially wavefunction collapse.
  • Knowledge of matrix representations of operators in quantum systems.
  • Basic comprehension of superposition in quantum states.
NEXT STEPS
  • Study the mathematical formulation of quantum mechanics, focusing on operators and their eigenstates.
  • Learn about the implications of measurement in quantum mechanics, particularly the collapse of the wavefunction.
  • Explore the relationship between linear combinations of eigenvectors and physical states in quantum systems.
  • Investigate further into the concept of superposition and its applications in quantum computing.
USEFUL FOR

Quantum physicists, students of quantum mechanics, and anyone interested in the mathematical foundations of quantum theory and operator theory.

nabeel17
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Say I have a 3x3 operator Q and I find its eigenvectors and eigenvalues. Now i know that those eigenvectors are the same as eigenfunctions so if i act on them with Q i will get the corresponding eigenvalue.

What the question I am trying to solve asks is, Measure the quantity Q in state where b is given as a 3x1 matrix. I know how to do it mathematically, I just express as a linear combination of my eigenvectors. But I'm trying to interpret what this means. Is the state some state my wavefunction is in and when I measure it, it will be in a state of one of the eigenvectors of Q?

I am comparing this to Schrodingers cat where before I measure it is in a linear combination of alive and dead and there is a probability that can be alive or dead and when I measure (ie look) the wavefunction collapses to either alive or dead. Am i correct in thinking this way about Q and ?
 
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yeah, that's right. When they say a measurement of the physical quantity associated with Q, that means the state suddenly jumps into one of the eigenstates of Q.
 
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