**Usage in Quantum Mechanics**

The mathematical structure of quantum mechanics is based in large part on linear algebra:

- Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra-ket notation, for example, an electron might be in "the state ". (Technically, the quantum states are
*rays*of vectors in the Hilbert space, as corresponds to the same state for any nonzero complex number*c*.) - Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state is in a quantum superposition of the states and .
- Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.
- Dynamics is also described by linear operators on the Hilbert space. For example, in the SchrÃ¶dinger picture, there is a linear operator
*U*with the property that if an electron is in state right now, then in one minute it will be in the state, the same*U*for every possible . - Wave function normalization is scaling a wave function so that its norm is 1.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often *does* involve, bra-ket notation. A few examples follow:

Read more about this topic: Bra-ket Notation

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