# Quantum Mechanics

From *M

## Quantum Mechanical Math

The following is a break down of the definitions of the definitions one would need to understand in order to understand this brief account of Quantum mechanical mathematics. In other words, if You wanted to learn QM via the math you would need to set your curriculm to all these topics as you work your way up to the QM itself.

The possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. The time evolution of a quantum state is described by the SchrÃ¶dinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

[Quantiki]

### Quantum Computation Name List

• John Preskill
• Peter Zoller in Innsbruck
• David Deutsch

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