A Pedagogical Matrix Derivation of the Many-Worlds Interpretation and Its Comparison with Copenhagen and Classical Probability

This paper provides a self-contained mathematical derivation of the Many-Worlds Interpretation of quantum mechanics, relying exclusively on unitary Schrödinger evolution and quantum entanglement without a wave-function collapse postulate. The emergence of distinct physical branches is demonstrated through system, apparatus, and environment entanglement followed by decoherence. Using explicit matrix representations for spin-$\frac{1}{2}$ states, Pauli operators, and measurement interactions, we analyze a two-branch model where a quantum superposition produces world branches weighted by their corresponding squared amplitudes. This construction is extended to a sequence of $N$ measurements, generating a branching tree of $2^N$ worlds. We directly compare this unitary framework with the von Neumann–Lüders collapse mechanism of the Copenhagen interpretation and the epistemic probabilities of classical statistical mechanics. By evaluating these three approaches at the level of the same $2 \times 2$ density matrices, we show why they are locally equivalent while differing significantly regarding global state evolution, reversibility, and the status of the Born rule. We further use the Wigner’s-friend thought experiment and its modern generalization by Frauchiger and Renner to show how the question of what counts as an observer sharpens this comparison. Additionally, we address the preferred-basis problem and examine the decision-theoretic derivation of the Born rule by Deutsch and Wallace. Designed for graduate and advanced undergraduate courses in quantum foundations, this work bridges the gap between qualitative overviews and abstract Hilbert-space treatments by equipping students with concrete algebraic tools to evaluate measurement interpretations.