Shankar- Simultaneous Diagonalisation of Hermitian Matrices

In summary, Shankar's simultaneous diagonalisation method is a mathematical technique used to find a common set of eigenvectors for two or more Hermitian matrices. It is important in quantum mechanics for calculating observables and finding eigenstates. This method differs from others by being able to apply to multiple matrices simultaneously. However, it has limitations such as only being applicable to Hermitian matrices and may not be feasible for large matrices. It is also used in other fields of science, such as statistical mechanics and computer science.
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bugatti79
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Folks,

What is the idea or physical significance of simultaneous diagonalisation? I cannot think of anything other than playing a role in efficient computation algorithms?

Thanks
 
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  • #2
bugatti79 said:
Folks,

What is the idea or physical significance of simultaneous diagonalisation? I cannot think of anything other than playing a role in efficient computation algorithms?

Thanks

If operators are simultaneously diagonalizable, then they commute -> no uncertainty relation.
 
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Related to Shankar- Simultaneous Diagonalisation of Hermitian Matrices

1. What is Shankar's simultaneous diagonalisation method?

Shankar's simultaneous diagonalisation method is a mathematical technique used to find a common set of eigenvectors for two or more Hermitian matrices. This allows for the matrices to be simultaneously diagonalised, making it easier to solve certain problems in quantum mechanics and other fields of science.

2. What is the importance of simultaneous diagonalisation in quantum mechanics?

In quantum mechanics, simultaneous diagonalisation is important because it allows for the calculation of observables such as energy and momentum, which are represented by Hermitian matrices. It also simplifies the process of finding the eigenstates of a system, which are crucial in understanding the behavior of quantum particles.

3. How does Shankar's method differ from other diagonalisation techniques?

Shankar's method differs from other diagonalisation techniques in that it can be applied to multiple Hermitian matrices simultaneously, as long as they share a common set of eigenvectors. This makes it more efficient for solving problems involving multiple matrices, compared to methods that only work with one matrix at a time.

4. What are the limitations of Shankar's simultaneous diagonalisation method?

One limitation of Shankar's method is that it can only be applied to Hermitian matrices. This means that it cannot be used for non-Hermitian matrices, which are commonly found in certain applications. Additionally, the method may not always be feasible for matrices with a large number of dimensions.

5. How is Shankar's simultaneous diagonalisation method used in other fields of science?

Shankar's method has applications in various fields of science, such as statistical mechanics, signal processing, and image processing. It is also used in computer science for data compression and encryption algorithms. The method's ability to simplify complex matrices makes it a valuable tool in various scientific and technological applications.

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