# Heisenberg's reasoning concerning matrices

• jackmann
In summary, "Quantum" by Manjit Kumar explores the history of quantum mechanics and how Heisenberg's work with matrices led to the development of the Uncertainty Principle. He multiplied two matrices together to study the behavior of electrons and their energy transitions, but realized that the non-commutativity of the matrices made it impossible to know both the exact position and momentum of a particle simultaneously. This discovery led to the concept of indeterminacy and the Uncertainty Principle.
jackmann
I'm reading "Quantum" by Manjit Kumar, a history of quantum mechanics. It tells how Heisenberg designed an array to track the frequencies of all possible spectral lines being emitted by hydrogen electrons as they "jumped" between energy levels. Heisenberg was troubled because when he multiplied two of these matrices together, the answer was non-commutative, so that AB-BA did not always equal zero. Further on, this led to his Uncertainty Principle.

I am missing his reasoning.
1. Why did he multiply two of these matrices together?
2. What made the matrix properties non-commutative? Was it because of using complex numbers inside the matrix, was it it because it was a certain type of Matrix, such as Hermitian, or none of these?

3. How did this lead to the development of the Uncertainty Principle? 1. Heisenberg multiplied two of these matrices together to study the behavior of electrons as they transition between energy levels. This allowed him to analyze the relationship between the energies of different states and their corresponding frequencies. 2. The matrix properties were non-commutative because of Heisenberg's use of complex numbers inside the matrix. This caused the matrix elements to be dependent on both their position in the matrix and their values, which led to non-commutativity. 3. Heisenberg's discovery of the non-commutativity of the matrices led him to develop the Uncertainty Principle. He realized that it was impossible to know both the exact position and momentum of a particle at the same time due to the inherent uncertainty in the matrices. This led to the concept of indeterminacy and the Heisenberg Uncertainty Principle.

## 1. What is Heisenberg's reasoning concerning matrices?

Heisenberg's reasoning concerning matrices is a mathematical concept that he used in his uncertainty principle. He argued that the position and momentum of a particle cannot be known simultaneously because the act of measuring one affects the other.

## 2. How did Heisenberg use matrices in his uncertainty principle?

Heisenberg used matrices to represent the quantum mechanical operators for position and momentum. These operators do not commute, and their non-commutativity leads to the uncertainty principle.

## 3. Can you explain the mathematical basis for Heisenberg's reasoning?

Heisenberg's reasoning is based on the mathematical properties of matrices, particularly their non-commutativity. When two operators do not commute, their order of application affects the result, leading to uncertainty in the measurement of certain physical quantities.

## 4. How does Heisenberg's reasoning apply to other physical quantities?

Heisenberg's reasoning can be applied to other physical quantities besides position and momentum. Any two quantities that do not commute will have a degree of uncertainty in their measurements, as their order of application affects the result.

## 5. Can Heisenberg's reasoning be applied to classical mechanics?

No, Heisenberg's reasoning is a fundamental principle of quantum mechanics and does not apply to classical mechanics. In classical mechanics, the position and momentum of a particle can be known precisely, without affecting each other.

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