Question

In Hilbert space, what property must a set of basis vectors satisfy to uniquely represent any quantum state as a linear combination?

Accepted Answer

To uniquely represent any quantum state in a Hilbert space, a set of basis vectors must form an orthonormal basis. This requirement consists of two specific properties. First, the set must be orthogonal, meaning that the inner product of any two distinct vectors in the set is zero. The inner product acts like a geometric measure of the angle between two vectors; orthogonality ensures that the vectors are pointing in entirely independent directions, which prevents redundancy in the representation. Second, the set must be normalized, meaning that each individual vector has a length of one, calculated by the inner product of a vector with itself. Normalization ensures that the probability amplitudes of the quantum state remain consistent. Finally, the set must be complete, which means the vectors span the entire Hilbert space. A span is the collection of all possible linear combinations of the basis vectors. If the set is complete, any vector within the space can be constructed as a unique sum of these basis vectors multiplied by specific scalar coefficients. If these three conditions—orthogonality, normalization, and completeness—are met, every unique quantum state corresponds to a single, distinct set of coefficients, ensuring a mathematically precise and unambiguous representation.

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In Hilbert space, what property must a set of basis vectors satisfy to uniquely represent any quantum state as a linear combination?