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Formulation of quantum mechanics on a Hilbert Space
In
mathematical physics
, the
Dirac?von Neumann axioms
give a
mathematical formulation of quantum mechanics
in terms of
operators
on a
Hilbert space
. They were introduced by
Paul Dirac
in 1930 and
John von Neumann
in 1932.
Hilbert space formulation
[
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]
The space
is a fixed complex
Hilbert space
of
countably infinite
dimension
.
- The
observables
of a
quantum system
are defined to be the (possibly
unbounded
)
self-adjoint operators
on
.
- A
state
of the quantum system is a
unit vector
of
, up to scalar multiples; or equivalently, a
ray
of the Hilbert space
.
- The
expectation value
of an observable
A
for a system in a state
is given by the
inner product
.
Operator algebra formulation
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]
The Dirac?von Neumann axioms can be formulated in terms of a
C*-algebra
as follows.
- The bounded
observables
of the quantum mechanical system are defined to be the self-adjoint elements of the C*-algebra.
- The states of the quantum mechanical system are defined to be the
states
of the C*-algebra (in other words the normalized positive linear functionals
).
- The value
of a state
on an element
is the
expectation value
of the observable
if the quantum system is in the state
.
Example
[
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]
If the C*-algebra is the algebra of all bounded operators on a Hilbert space
, then the bounded observables are just the bounded self-adjoint operators on
. If
is a unit vector of
then
is a state on the C*-algebra, meaning the unit vectors (up to scalar multiplication) give the states of the system. This is similar to Dirac's formulation of quantum mechanics, though Dirac also allowed unbounded operators, and did not distinguish clearly between self-adjoint and Hermitian operators.
See also
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]
References
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]
- Dirac, Paul (1930),
The Principles of Quantum Mechanics
- Strocchi, F. (2008), "An introduction to the mathematical structure of quantum mechanics. A short course for mathematicians",
An Introduction to the Mathematical Structure of Quantum Mechanics. Series: Advanced Series in Mathematical Physics
, Advanced Series in Mathematical Physics,
28
(2 ed.), World Scientific Publishing Co.,
Bibcode
:
2008ASMP...28.....S
,
doi
:
10.1142/7038
,
ISBN
9789812835222
,
MR
2484367
- Takhtajan, Leon A. (2008),
Quantum mechanics for mathematicians
,
Graduate Studies in Mathematics
, vol. 95, Providence, RI: American Mathematical Society,
doi
:
10.1090/gsm/095
,
ISBN
978-0-8218-4630-8
,
MR
2433906
- von Neumann, John (1932),
Mathematical Foundations of Quantum Mechanics
, Berlin: Springer,
MR
0066944