1.2 Quantum Mechanics
7
1.2 Quantum Mechanics
1.2.1 In formal schemes of all theories considered in this work, the basic concepts
are ‘states’, ‘observables’ and their transformations ascribed to a considered physical
system. We shall not discuss here details of the empirical meaning of these concepts.
Roughly, states are prepared by some standard empirical procedures and represent
the situation, what has to be measured, observables describe (equivalence classes
of) measuring apparatuses (i.e. the role of their function in the theory) giving certain
empirically obtained responses if applied to states, and transformations include time
evolution of the system in given conditions as well as various changes of equivalent
descriptions of the system (symmetries).
In this section, we shall outline a simple standard scheme of the formalism of nonrelativistic (resp. Galilean-relativistic) quantum mechanics of finite systems (QM),
i.e. the nonrelativistic view on physical systems containing only finite number of
their further indecomposable elementary constituents (particles, spins,…).
7
1.2.2 Observables: A separable complex Hilbert space H corresponds to any
physical system in QM. Let L(H) denote the set of all bounded linear operators
from H to H, where the boundedness (equiv. continuity) is defined with respect to
the norm of H coming from the scalar product (x, y), (x, y ∈ H), which is linear in the second factor y. Observables in QM (i.e. physical quantities empirically
identifiable by some realizable(?) measuring devices) are represented by selfadjoint
operators on H (in general unbounded). It is useful to consider along with any selfadjoint operator A (corresponding to an equally denoted observable A) its spectral
measure E A defined on Borel subsets of the real line R with values in projectors
E A (•) in L(H), E A (R) = I := id H (:= the identity of the algebra L(H)), cf. [37,
Appendices B & C].
It is important to stress here, that in the conventional QM of finite systems (atoms,
molecules, and finite collections of them) the set of observables contains the whole
set L(H) of operators representing these observables. Hence, the algebra L(H) acts
on H by the irreducible manner (i.e. no nontrivial subspace of H is by the actions of
the whole L(H) left invariant). This also implies the impossibility, resp. inadequacy,
of interpretation of the “mixed states” as representing some statistical mixture of
systems occurring in the states decomposing the corresponding “mixture” (cf. 1.2.3)
in this QM of finite systems.
1.2.3 States in QM are conventionally represented by density matrices, i.e. positive trace class operators on H with unit trace (=the trace norm): T r() = 1.
Density matrices form a convex subset in the linear space T(H) of all trace class
operators which is closed in the trace norm A 1 := T r
√
A ∗ A. Denote this set of
states S ∗ . The extreme points of S ∗ are represented by the one-dimensional orthogonal projectors P x ∈ L(H) (projecting H onto one-dimensional subspaces x containing x, 0 = x ∈ H). Any ∈ S ∗ can be expressed as a weak limit of finite convex
7 The concepts of “system”, and “physical system” are taken here to be as intuitively clear.
7
1.2 Quantum Mechanics
1.2.1 In formal schemes of all theories considered in this work, the basic concepts
are ‘states’, ‘observables’ and their transformations ascribed to a considered physical
system. We shall not discuss here details of the empirical meaning of these concepts.
Roughly, states are prepared by some standard empirical procedures and represent
the situation, what has to be measured, observables describe (equivalence classes
of) measuring apparatuses (i.e. the role of their function in the theory) giving certain
empirically obtained responses if applied to states, and transformations include time
evolution of the system in given conditions as well as various changes of equivalent
descriptions of the system (symmetries).
In this section, we shall outline a simple standard scheme of the formalism of nonrelativistic (resp. Galilean-relativistic) quantum mechanics of finite systems (QM),
i.e. the nonrelativistic view on physical systems containing only finite number of
their further indecomposable elementary constituents (particles, spins,…).
7
1.2.2 Observables: A separable complex Hilbert space H corresponds to any
physical system in QM. Let L(H) denote the set of all bounded linear operators
from H to H, where the boundedness (equiv. continuity) is defined with respect to
the norm of H coming from the scalar product (x, y), (x, y ∈ H), which is linear in the second factor y. Observables in QM (i.e. physical quantities empirically
identifiable by some realizable(?) measuring devices) are represented by selfadjoint
operators on H (in general unbounded). It is useful to consider along with any selfadjoint operator A (corresponding to an equally denoted observable A) its spectral
measure E A defined on Borel subsets of the real line R with values in projectors
E A (•) in L(H), E A (R) = I := id H (:= the identity of the algebra L(H)), cf. [37,
Appendices B & C].
It is important to stress here, that in the conventional QM of finite systems (atoms,
molecules, and finite collections of them) the set of observables contains the whole
set L(H) of operators representing these observables. Hence, the algebra L(H) acts
on H by the irreducible manner (i.e. no nontrivial subspace of H is by the actions of
the whole L(H) left invariant). This also implies the impossibility, resp. inadequacy,
of interpretation of the “mixed states” as representing some statistical mixture of
systems occurring in the states decomposing the corresponding “mixture” (cf. 1.2.3)
in this QM of finite systems.
1.2.3 States in QM are conventionally represented by density matrices, i.e. positive trace class operators on H with unit trace (=the trace norm): T r() = 1.
Density matrices form a convex subset in the linear space T(H) of all trace class
operators which is closed in the trace norm A 1 := T r
√
A ∗ A. Denote this set of
states S ∗ . The extreme points of S ∗ are represented by the one-dimensional orthogonal projectors P x ∈ L(H) (projecting H onto one-dimensional subspaces x containing x, 0 = x ∈ H). Any ∈ S ∗ can be expressed as a weak limit of finite convex
7 The concepts of “system”, and “physical system” are taken here to be as intuitively clear.