Unbounded Operators and the Incompleteness of Quantum Mechanics

1990 ◽  
Vol 57 (3) ◽  
pp. 523-534 ◽  
Author(s):  
Adrian Heathcote
Keyword(s):  
Quantum Mechanics ◽  
Unbounded Operators

scholarly journals ALGEBRAS OF UNBOUNDED OPERATORS AND PHYSICAL APPLICATIONS: A SURVEY

2007 ◽  
Vol 19 (03) ◽  
pp. 231-271 ◽  
Author(s):  
F. BAGARELLO
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Approach ◽  
Unbounded Operators ◽  
Large Systems

After a historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance in physical applications.

Spherical Harmonics, the Weyl Transform and the Fourier Transform on the Heisenberg Group

1984 ◽  
Vol 36 (4) ◽  
pp. 615-684 ◽  
Author(s):  
Daryl Geller
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Quantum Mechanics ◽  
Spherical Harmonics ◽  
Separable Hilbert Space ◽  
Dense Subspace ◽  
Unbounded Operators ◽  
Weyl Transform ◽  
The Fourier Transform ◽  
Image Position

In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋa separable Hilbert space. Given certain (unbounded) operators W1,…,Wn,W1+, …, Wn+ on ℋ satisfying(on a dense subspace D of ℋ) with all other commutators vanishing. Given also a function where ζ ∈ Cn. Let W = (W1 …, Wn) W+ = (W1+ …, Wn+). How does one associate to f an operator f(W, W+)? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W, Q = Im W+ which represent momentum and position. In this paper, however, we wish to exploit the unitary group on Cn and so prefer complex notation.)

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