Heisenberg's Matrix Mechanics
Introduction to Matrix Mechanics
Matrix mechanics, developed by Werner Heisenberg in 1925, represents a fundamental shift in the way quantum mechanics was understood. Unlike classical mechanics, which relies on deterministic equations of motion, matrix mechanics describes quantum systems in terms of matrices that encapsulate the probabilities of various outcomes.Fundamental Concepts
Matrices and Observables
In matrix mechanics, physical quantities such as position, momentum, and energy are represented as matrices. The state of a quantum system is described by a state vector (or wavefunction), which is an eigenvector of the observable’s matrix. The eigenvalues of this matrix correspond to the possible measurement outcomes of that observable.The Heisenberg Uncertainty Principle
Matrix mechanics naturally incorporates the Heisenberg Uncertainty Principle, which states that certain pairs of observables, like position (x) and momentum (p), cannot be simultaneously measured with arbitrary precision. This is represented mathematically as:$$ { ext{Δ}x ext{Δ}p} ext{ ≥ } rac{ ext{ℏ}}{2} $$
where ℏ (h-bar) is the reduced Planck constant.
Commutation Relations
A crucial aspect of matrix mechanics is the commutation relations. For two matrix observables, A and B, the commutator is defined as:$$ [A, B] = AB - BA $$
If the commutator of two observables is non-zero, this implies that the observables cannot be simultaneously diagonalized, reinforcing the uncertainty principle. For instance, the position and momentum operators satisfy:
$$ [x, p] = i ext{ℏ} $$
Example: Position and Momentum Matrices
To illustrate how matrix mechanics works, consider a simple quantum system with discrete energy levels. If we define the position matrix x and momentum matrix p in a 2D space, they can be represented as:$$ x = egin{pmatrix} 0 & 1 \\ 0 & 0 \\ \\ 1 & 0 \\ 0 & 0 \\ \\ 0 & 1 \\ 0 & 0 \\ \\ 0 & 0 \\ 1 \\ \\ 0 & 0 \\ 0 & 1 \\ \\ 0 & 0 \\ 1 & 0 \\end{pmatrix}, $$
$$ p = egin{pmatrix} 0 & -i \\ 0 & 0 \\ \\ i & 0 \\ 0 & 0 \\ \\ 0 & 0 \\ i & 0 \\ \\ 0 & -i \\ 0 & 0 \\ \\ 0 & 0 \\ i & 0 \\ \\ 0 & 0 \\ 0 & -i \\end{pmatrix} $$
The product of these matrices gives insight into their behavior when measured in sequence. The non-commutativity signifies that measuring the position followed by momentum (or vice versa) results in different outcomes and is indicative of the uncertainty principle in action.