Physics:Quantum Mathematical Foundations of Quantum Theory

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A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product.[1] It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.[2] The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension.

Quantum Mathematics: Foundations of Quantum Theory Visualized

Definition

A Hilbert space ℋ is a vector space over ℝ or ℂ together with an inner product

⟨ϕ∣ψ⟩

such that the induced norm

‖ψ‖=⟨ψ∣ψ⟩

makes ℋ a complete metric space.[3]

In quantum theory, Hilbert spaces are usually taken to be complex. A normalized state vector satisfies

⟨ψ∣ψ⟩=1.

Geometric interpretation

Hilbert space extends the geometry of ordinary three-dimensional space to possibly infinitely many dimensions. The inner product determines:

  • the length of a vector through its norm;
  • the angle between vectors through their overlap;
  • the notion of orthogonality, when ⟨ϕ∣ψ⟩=0;
  • the projection of one vector onto another or onto a subspace.

These ideas are central in quantum mechanics.[4]

Examples

Euclidean space

The finite-dimensional space ℝn with the standard dot product is a simple example of a Hilbert space. Likewise, ℂn with

⟨ϕ∣ψ⟩=∑i=1nϕi*ψi

is the standard Hilbert space of finite-dimensional quantum systems.[5]

Function spaces

An important infinite-dimensional example is the space L2 of square-integrable functions. The inner product is

⟨ϕ∣ψ⟩=∫ϕ*(x)ψ(x)dx.

Wavefunctions in nonrelativistic quantum mechanics are elements of such spaces.[6]

Sequence spaces

Another standard example is the space ℓ2 of square-summable sequences.

Basis and expansion

A Hilbert space may have an orthonormal basis {en}, meaning

⟨em∣en⟩=δmn.

Any vector can be expanded as

ψ=∑ncnen.

These expansions generalize Fourier series.[7]

Orthonormal expansions in Hilbert space.

Hilbert space in quantum mechanics

Hilbert space provides the formal setting for quantum states.[8]

The probability amplitude between two states is

⟨ϕ∣ψ⟩

and probabilities are given by its squared magnitude.

Operators and observables

Physical quantities are represented by operators acting on Hilbert space.[9]

Observables correspond to self-adjoint operators with real eigenvalues.[10]

The expectation value is

⟨A^⟩=⟨ψ∣A^∣ψ⟩.

Commutators

The commutator

[x^,p^]=iℏ

leads to the uncertainty principle.[11]

Spectral theorem

The spectral theorem decomposes self-adjoint operators into projection operators.[12]

A^=∑nanPn

This provides the mathematical basis for quantum measurement.

Density matrices

A general quantum state is described by a density operator ρ.[13]

⟨A⟩=Tr(ρA)

Pure states satisfy ρ2=ρ, while mixed states satisfy Tr(ρ2)<1.

See also

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press. ISBN 9780125850506. 
  2. ↑ Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press. https://archive.org/details/principlesofquan0000dira. 
  3. ↑ Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer. https://link.springer.com/book/10.1007/978-1-4614-7116-5. 
  4. ↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson. 
  5. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. https://doi.org/10.1017/CBO9780511976667. 
  6. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  7. ↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press. 
  8. ↑ von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press. https://archive.org/details/mathematicalfoun0000vonn. 
  9. ↑ Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press. 
  10. ↑ Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer. 
  11. ↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson. 
  12. ↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press. 
  13. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. 
Author: Harold Foppele

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