Eigenvalues of Hermitian Operator

Prove that the eigenvalues of any Hermitian operator must be real numbers.

Hermitian operators, also known as self-adjoint operators, play a crucial role in quantum mechanics because they correspond to observable quantities. A key property of these operators is that their eigenvalues, which represent possible measurement outcomes, are always real numbers. This is fundamentally tied to the requirement that measurements yield real and physically meaningful results, as imaginary numbers do not correspond to anything observable in reality. Understanding why Hermitian operators have real eigenvalues involves considering the mathematical properties of these operators, which are defined such that their transpose is equal to their complex conjugate transpose.

To prove that Hermitian operators have real eigenvalues, one often starts by considering the inner product space in which quantum states exist. The concept of inner product holds particular significance as it allows one to define angles and lengths in complex vector spaces. For a Hermitian operator, when applied to an eigenvector, the inner product of the resulting vector with the eigenvector itself yields a value that satisfies specific symmetry properties, guaranteeing that the output is a real number. This property is critically linked to the definition of Hermitian operators and is deeply rooted in the linear algebra and functional analysis principles, which underpin much of quantum theory. Moreover, this proof often illustrates the broader feature of quantum operators in which symmetries and conservation laws are expressed through mathematical constructs such as Hermitian operators, making them a cornerstone of not just theory but practical quantum mechanics.