Abstract
The purpose of the chapter is to introduce Hilbert spaces, and more precisely the Hilbert spaces on the field of complex numbers, which represent the abstract environment in which Quantum Mechanics is developed. To arrive at Hilbert spaces, we proceed gradually, beginning with vector spaces, then considering inner-product vector spaces, and finally Hilbert spaces. The Dirac notation will be introduced and used. A particular emphasis is given to Hermitian and unitary operators and to the class of projectors. The eigendecomposition of such operators is seen in great detail. The final part deals with the tensor product of Hilbert spaces, which is the mathematical environment of composite quantum systems.
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Notes
- 1.
If \(\mathcal {S}_0\) is constituted by some points, for example \(\mathcal {S}_0=\{x_1, x_2, x_3\}\), the notation \(\text{ span }(\mathcal {S}_0)=\text{ span }(\{x_1, x_2, x_3\})\) is simplified to \(\text{ span }(x_1, x_2, x_3)\).
- 2.
To proceed in unified form, valid for all the classes \(\mathcal {L}_2(I)\), we use the Haar integral (see [2]).
- 3.
These names are obtained by splitting up the word “bracket”; in the specific case, the brackets are \(\langle \; \rangle \).
- 4.
This is not the ordinary definition of adjoint operator, but it is an equivalent definition, deriving from the relation \(|x\rangle ^*=\langle x|\) (see Sect. 2.8).
- 5.
We take for granted the definition of sum \(A+B\) of two operators, and of multiplication of an operator by a scalar \(kA\).
- 6.
The eigenvector corresponding to the eigenvalue \(\lambda \) is often indicated by the symbol \(|\lambda \rangle \).
- 7.
As we assume \(|x_0\rangle \ne {0}\), to complete \(\mathcal {E}_{\lambda }\) as a subspace, the vector \({0}\) of \(\mathcal {H}\) must be added.
- 8.
Above, the outer product was defined in the Hilbert space \(\mathbb {C}^n\). For the definition in a generic Hilbert space one can use the subsequent (2.48), which defines \({C}=|c_1\rangle \langle c_2|\) as a linear operator.
- 9.
In most textbooks the adjoint operator is indicated by the symbol \(A^\dagger \) and sometimes by \(A^+\).
- 10.
This name comes from the order given to words in the dictionary: a word of \(k\) letters, \({a}=(a_1,\ldots ,a_k)\) appears in the dictionary before the word \(b=(b_1,\ldots ,b_k)\), symbolized \({a}<b\), if and only if the first \(a_i\) which is different from \(b_i\) comes before \(b_i\) in the alphabet. In our context the alphabet is given by the set of integers. Then we find, e.g., that \((1,3)<(2,1)\), \((0,3,2)<(1,0,1)\) and \((1,1,3)<(1,2,0)\).
References
S. Roman, Advanced Linear Algebra (Springer, New York, 1995)
G. Cariolaro, Unified Signal Theory (Springer, London, 2011)
P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958)
R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1998)
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
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Cariolaro, G. (2015). Vector and Hilbert Spaces. In: Quantum Communications. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-15600-2_2
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DOI: https://doi.org/10.1007/978-3-319-15600-2_2
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