User:GravitonsAndGraviolis/Mathematical formulation of quantum mechanics

Postulates of quantum mechanics
The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms. The postulates are canonically presented in six statements, though there are many important points to each.

Description of the State of a System
Each physical system is associated with a (topologically) separable complex Hilbert space $H$ with inner product ⟨φ|ψ⟩. Rays (that is, subspaces of complex dimension 1) in $H$ are associated with quantum states of the system.

In other words, quantum states can be identified with equivalence classes of vectors of length 1 in $H$, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state. "A quantum mechanical state is a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrödinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid."

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J. M. Jauch, Foundations of quantum mechanics, section 11.7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

Description of Physical Quantities
Physical observables are represented by Hermitian matrices on $H$. Since these operators are Hermitian, the measurement is always a real value. If the spectrum of the observable is discrete, then the possible results are quantized.

Measurement of Physical Quantities
By spectral theory, we can associate a probability measure to the values of $A$ in any state $ψ$. We can also show that the possible values of the observable $A$ in any state must belong to the spectrum of $A$. The expectation value (in the sense of probability theory) of the observable $A$ for the system in state represented by the unit vector $ψ$ ∈ H is $$\langle\psi\mid A\mid\psi\rangle$$.

In the special case $A$ has only discrete spectrum, the possible outcomes of measuring $A$ are its eigenvalues. More precisely, if we represent the state $ψ$ in the basis formed by the eigenvectors of $A$, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator $ρ$ normalized to be of trace 1. The expected value of $A$ in the state $ρ$ is $$ \operatorname{tr}(A\rho)$$.

Reduction of the Wavefunction
When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured. If $ρ_{ψ}$ is the orthogonal projector onto the one-dimensional subspace of $H$ spanned by $⟨$, then $$ \operatorname{tr}(A\rho_\psi)=\left\langle\psi\mid A\mid\psi\right\rangle$$.

Time Evolution of a System
Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the DeBroglie hypothesis or path integrals.

Other Implications of Postulates



 * Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (supersymmetry is another matter entirely).


 * Physical observables are represented by Hermitian matrices on $H$.









One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.
 * Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.