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 Schrödinger 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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