Electron Spin and Quantum Entanglement

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Electron spin and quantum entanglement stand as two of the most fundamental and conceptually challenging pillars of quantum mechanics. These phenomena, lacking any direct classical analogues, not only highlight the profound departure of the quantum world from our everyday experiences but also form the bedrock of emerging quantum technologies. The intricate relationship between electron spin, an intrinsic form of angular momentum, and quantum entanglement, a peculiar correlation between quantum systems, has been instrumental in reshaping our understanding of physical reality at the quantum scale.
The concept of electron spin arose in the early 20th century from the need to explain experimental observations that contradicted classical atomic theory. The Stern-Gerlach experiment in 1922, which showed the splitting of a beam of silver atoms passing through a magnetic field, suggested an intrinsic magnetic moment beyond that attributable to orbital motion. Wolfgang Pauli proposed a two-valued quantum property in 1924, which was later formalized as electron spin by George Uhlenbeck and Samuel Goudsmit in 1925. Paul Dirac subsequently incorporated spin into his relativistic quantum theory of the electron in 1928.
In quantum mechanics, electron spin is described by a quantum number s = 1/2, classifying the electron as a spin-1/2 particle or fermion. When measured along any axis, spin can only take discrete values: +ħ/2 ("spin-up", $|\uparrow\rangle$) or -ħ/2 ("spin-down", $|\downarrow\rangle$), illustrating quantum discreteness. The spin state of an electron can be represented as a superposition of these two states: $|\psi\rangle = \alpha|\uparrow\rangle + \beta|\downarrow\rangle$. Mathematically, spin is associated with the Pauli matrices, which are non-commuting operators, leading to the uncertainty principle for spin components along different axes. In systems with multiple electrons, the fermionic nature of electrons gives rise to the Pauli exclusion principle, stating that no two electrons can occupy the same quantum state simultaneously, necessitating opposite spins for electrons in the same orbital. The interaction between electron spins in many-electron systems also underlies phenomena like ferromagnetism and antiferromagnetism, arising from the quantum mechanical exchange interaction.
Quantum entanglement represents a unique form of correlation between multiple particles where their quantum states become linked in a way that cannot be explained by classical physics. For two spin-1/2 particles, an entangled state cannot be factored into the tensor product of individual particle states. Prominent examples of entangled states are the Bell states, which are maximally entangled. The singlet state $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$ is particularly notable for exhibiting perfect anti-correlation: a measurement of spin-up for one particle along any axis invariably yields spin-down for the other along the same axis.

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