CC BY
33
UDC 519.62; 530.145; 519.614
Model of Hydrogen Atom Quantum Measurements on Rigged
Hilbert Spaces
A. V. Zorin
Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198
The measurement procedure makes the isolated (closed) quantum system to be the open one. The operators of observables of rather simple explicit form are converted into pseudodifferential operators of more complex form. The author has proposed the method of establishing consistency between the theoretical data of conventional quantum mechanics of (isolated) quantum objects and experimental data on the measured values of the observables of corresponding open quantum objects. In this paper, the proposed correspondence is used for the construction of rigged Hilbert spaces, in which the operators of measured the observables of hydrogen-like atom admit spectral decomposition.
Key words and phrases: operator of measured quantum observable, rigged Hilbert space, spectral decomposition of unbounded self-adjoint operator.
1. Introduction
Let us recall the basic structures of quantum measurement model in Kuryshkin-Wodkiewicz and Kuryshkin-Weil representations.
The configuration space of an isolated physical object (for example, the Kepler problem - a hydrogen atom) Q = R3, the phase space T*Q = R3 © R3. Classical observables A (q,p) are distributions on the phase space [1-3] of the object (system). According to the Weyl rule of quantization [4-7] quantum observables Ow (A) are self-adjoint (unbounded) operators in rigged Hilbert space $ C H = L2 (Q) C $*.
The results of Shewell-Kuryshkin [8-10] on a one-to-one correspondence of quantization rules and quantum distribution functions (QDFs) puts the Weil rule into correspondence with Wigner QDF, so that
(A)^ = (Ow (A) = J A (q,p)W^ (q,p) dqdp (1)
or, more generally,
(A)p = Tr (Ow (A) p) = j A (q,p)Wp (q,p) dqdp. (2)
In [11-13] on the basis of statistical correspondence, formulated by Blokhintsev and Terletsky, is developed the model of quantum mechanics with nonnegative QDF. Kuryshkin quantization rule to each A (q,p) and some function 0 e H = L2 (Q) or density matrix p assigns the operator an explicit form of which is convenient to write with the help of auxiliary functions. Namely, to each function 0 (q) e L2 (Q), we assign its Fourier image 0 (p) e L2 (Q) and an auxiliary function
Received 6th October, 2015.
on the phase space. To each mixed state p = Y1 °k \$k) | we assign an auxiliary
k
function
= £ fa (?) fa (P).
k
(2nK)
Then the rule for constructing operators of quantum mechanics with nonnegative QDF can be summarized as follows: the classical function A (q,p) corresponds to a linear operator O (A), whose action on an arbitrary function ^ (^admitting the Fourier transform is defined by the equation:
O^ (A) ^ (q) = (2nK)-3 (Ç - q,V - p) A (£, r,) e * (q') d£dr)dpdq.
The mean values of the operators (3) represented as
(K = (°4> (A) 4>,4>) = J A (q,p)F+ (q,p) dqdp,
where
(q,p) = (2nK)
-3
fa (Q - 0 i> (0 e-* (i'p)dt
or, more generally,
(A)pp1 = Tr (Op (A) pi) = J A (q,p)Fp1 (q,p) dqdp,
where
and
Fppi (q,p) = (2nh)-3J2 ckJ2 c
(q - 0 ^ (0 e~ *
Pi = £ °3 ) № 1 .
(3)
(4)
(5)
In [14-16] shown that the relation (5) may be written in an equivalent manner in the form proposed by Wodkiewicz [17,18]. Namely
(A)ppl = A (q,p)Pppi (q,p) dqdp,
(6)
where QDF of Wodkiewicz Pppi (q,p) is given by convolution of two Wigner functions
Pppi (q,p) = (Wp * Wp!)(q,p).
(7)
One of them is QDF of a mixed state p\ of a quantum object, the other is QDF of a mixed state p of the quantum filter instrument.
For pure state relations (6) and (7) retain the same form. These formulations allowed [19] to prove that the operator of the measured observable Op (A) is given by the Weyl quantization rule for the "measured" classical observable Ow (Ap), where Ap (q,p) = (A * Wp) (q,p) and Wp (q,p) are Wigner's QDFs.
Theorem 1. Quantization rule of Kuryshkin-Weil (3) to each of slowly increasing generalized functions A (q,p) associates Weyl operator Ow (A * Wp), where Wp is QDF Wigner of density matrix p = ck |0k) (fa I of the quantum filter.
2
2
2. Kuryshkin-Wodkiewicz construction in Kuryshkin-Weil
representation
This theorem makes it possible to build a rigged Hilbert space (RHS), provides spectral decomposition of the operator having mixed, discrete and continuous spectrum. In [20—22] and [23-25] conducted theoretical and mathematical studies of the structure of the RHS needed to describe the quantum system with the Hamiltonian H = Ow (H) and the Schrodinger equation ih(t) = (t). Further, in [26] stated that the operator Ow (Ap) satisfies the Schrodinger-Heisenberg equation,
dOp (A)
dt
that coincides with the Dirac equation
{(^-) 2 itl)" i-^Op (A)
(8)
^ = -h [Op (A) ,Ow (H)] (9)
on those operators for whom the latter relation is uniquely determined.
Let us recall briefly the arguments of [24] on the need to use the RHS to model quantum systems in order to justify in a similar manner the need for the RHS to model quantum measurements.
To find the eigenfunctions (classical and generalized) of the operator Op (H) first by von Neumann for bounded operators (with a discrete spectrum) in a Hilbert space, then by Gelfand and Vilenkin for unbounded operators (with mixed spectrum) in a rigged Hilbert space was justified spectral decomposition of essentially self-adjoint operators.
In addition to eigenfunctions of the operator's Op (H) the interpretation of the theory of quantum measurements needed in "observed average measured observable H (using quantum filter of measuring apparatus in the state p) in the state
(^,Op (H)$) or Tr (Op (H)pl) (10)
in the state pi = Y1 cj Vh) (ipjI, and the dispersion of the measured values H in the j
state pi = £ Cj \ipj) (ipj \. j
From (10), (11) we see that we need such vectors ^ G L2 (Q) and their combination in mixed states which belong to the domains of Op (H) and of Op (H2). It is possible by formal reasoning of the solution of the equation (8) or (9) in the form of the operator exponential, power series expansion of Taylor, to show that the state ^ G L2 (Q) on which these solutions are well defined, belongs to the domain of degrees of the measured observable ^ G D (Op (Hn)).
Thus, the quantum mechanics of the measured values Op (H) is not functioning either in H = L2 (Q)and in D (Op (H)), but is functioning in the dense subspace $ of infinitely differentiable functions on the configuration space Q, decreasing at infinity faster than any polynomial. By analogy with [20-25] let us postulate on the subspace $ C H = L2 (Q) a countable system of norms.
3. Rigged Hilbert space with the system of norms, generated by operators of the measured observable of hydrogen-like atom
Let us consider the operators of the measured observables Op (q), Op (p) Op (H) and their degrees, which depend on the original quantum object (a hydrogen atom), on the state of the quantum filter and maybe some other parameters of original
Hilbert space $ C H = L2 (Q). Let us recall the marginal probability densities given by the integrals [27]
«o (q) = J q,p)dp
and
^ (q) = [$(q,p)dq.
With their help, in [27] are built:
- operators Op (A (q)) = J a0 ^ A^q + ^ in particular, the operators
3 \ 3 I nj
( n«71 = m E«k0)0^-kJ [,
<3=1 ) 3=1 Uj=0
where {A (q))0 = / a0 (q)A (q) dq,
operators Op (A (p)) = f ß0 (rj)A {^p — iïïV^ dtf, in particular, the operators
3 \ 3 I nj / q ^ nj-kj
3 \ 3 nJ / pi \ n
oP ( n p? I = n|Ç cki№) X>, (A2)
where {A (p))0 = / ¡30 (p)A (p) dp.
For building an explicit form of the operator Op (H) one need more specific information on the construction, in particular in [28] is proposed a method of constructing the operators Op (H) of a hydrogen atom with the Hamiltonian function H (q,p) =
— via the basis of Sturmian functions of the hydrogen atom. The results of calculation with a mixed states p = ^ ck ) | and 0k (q) = Stni (%nl ) Ytm (V,4>),
k ^ '
k = (n, I, m) give
OA £) = ow(£) +^^
(!)=-(C)+!çbi ^
and 2 2
op(—w) = °w (—w)+Ç ck yk (q, cos6] bk ). (AA)
A notion of an abstract rigged Hilbert space and its classic implementation are given in [29,30]. It's Hilbert space H = L2 (Q) and a subset S of infinitely differentiable functions ^ G H that decrease at infinity faster than any polynomial, so that the quantities \\^\\n t are limited for each function ^ G S:
n,l
= max
qeQ
( 2\n dl 1+12+13é i1 + k1) ^
(11)
The values ||^||ni define a countable system of norms in the space S (Schwartz space). Three continuously embedded spaces
5 C H = L2 (Q) C S', S C H = L2 (Q) C 5*,
(12)
where S' (S*) is a space conjugate (anti conjugate) to S, i.e. the space of linear (antilinear) functionals continuous in the topology defined by the system (11) of norms))^)^ t define a rigged Hilbert space.
In [23-25] is built a system of norms |^|n|ra, generated by observable operators of an isolated quantum system
= f\ Ow (pn)Ow (ql)Ow (Hm(q)\dq. (13)
In this case, the space $ of infinitely differentiable rapidly decreasing functions continuously (with respect to all norms|^|n t m) embedded in H = L2 (Q), invariant
with respect to the Schrodinger equation in the Heisenberg representation i h d0^it(A) =
[ Ow (A) ,Ow (H)].
For the model of quantum measurements, i.e., for quantum mechanics with nonnegative QDF, the system of norms (13) takes the form
pn,i,m = j \ OP (p2n)Op (q21 )Op (Hm) ^ (q)\dq. (14)
The space $p of infinitely differentiable rapidly decreasing functions continuously (with respect to all norms l^m i m) embedded in H = L2 (Q), is invariant with respect to the Schrodinger equation in the Heisenberg representation of the form (8) and (9).
The system of norms (14) by virtue of (A1)-(A4) is equivalent to the system of norms
p _ n,Lm
J \(1 + |q21)p (1 + A) Op (Hm)^(q)
dq.
Thus, the construction of nuclear rigged Hilbert spaces is modified for the model of quantum measurements. It should be noted that the explicit form of the operators Op (Hm), the moments of measured energy, is properly described in [31], but wrongly described in [32].
4. Conclusion
One of the important problems in the description of quantum-mechanical systems is to describe the characteristics of the measurement results, for example, spectral data of systems. After all the measured characteristics of quantum objects tell us the properties of these objects The author has proposed the method of establishing consistency between the theoretical data of conventional quantum mechanics of (isolated) quantum objects and experimental data on the measured values of the observables of corresponding open quantum objects. The measurement procedure makes the isolated (closed) quantum system to be the open one The operators of observables of rather simple explicit form are converted into pseudo-differential operators of more complex form. In this paper, the proposed correspondence is used for the construction of rigged Hilbert spaces, in which the operators of measured the observables of hydrogen-like atom admit spectral decomposition. Thus, the applicability of stable numerical method of investigation of discrete spectra of the measured quantum observables is proved.
References
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20. M. Gadella, F. Gomez, A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics, Found. of Phys. 32 (6) (2002) 815-869.
21. M. Gadella, F. Gomez, A Measure-Theoretical Approach to the Nuclear and Inductive Spectral Theorems, Bull. Sci. Math. 129 (2005) 567-590.
22. M. Gadella, F. Gomez, Eigenfunction Expansions and Transformation Theory, Acta Appl. Math. 109 (2009) 721-742.
23. R. de la Madrid, Rigged Hilbert Space Approach to the Schrodinger Equation, J. Phys. A: Math. Gen. 35 (2002) 319-342.
24. R. de la Madrid, A. Bohm, M. Gadella, Rigged Hilbert Space Treatment of Continuous Spectrum, Fortsch. Phys. 50 (2002) 185-216.
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УДК 519.62; 530.145; 519.614 Модель квантовых измерений водородоподобного атома в оснащенном гильбертовом пространстве
А. В. Зорин
Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
Процедура измерения превращает изолированную (замкнутую) квантовую систему в открытую. При этом операторы наблюдаемых достаточно простого явного вида преобразуются в псевдо-дифференциальные операторы более сложного вида. Ранее автором был предложен метод установления соответствия между теоретическими данными общепринятой квантовой механики (изолированных) квантовых объектов и экспериментальными данными об измеренных значениях наблюдаемых соответствующих открытых квантовых объектов. В настоящей работе предложенное соответствие использовано для построения оснащенного гильбертова пространства, в котором операторы измеренных наблюдаемых водородоподобного атома допускают спектральное разложение.
Ключевые слова: оператор измеренной квантовой наблюдаемой, оснащенное гильбертово пространство, спектральное разложение неограниченных самосопряженных операторов.
Литература
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2. Березин Ф. А., Шубин М. А. Лекции по квантовой механике. — М.: Изд-во МГУ, 1972.
3. Березин Ф. А. Квантование // Известия АН СССР. Сер. матем. — 1967. — Т. 38. — С. 1116-1175.
4. Тарасов В. Е. Quantum dissipative systems I.Canonical quantization and quantum Liouville equation // ТМФ. — 1994. — Т. 100. — С. 402-417.
5. Тарасов В. Е. Quantum dissipative systems. III. Definition and algebraic structure // ТМФ. — 1997. — Т. 110. — С. 73-85.
6. Березин Ф. А., Шубин М. А. Уравнение Шредингера. — М.: Изд-во МГУ, 1983.
7. Тарасов В. Е. Квантовая механика. — М.: Вузовская книга, 2000.
8. Mehta C. L. Phase-Space Formulation of the Dynamics of Canonical Variables // J. Math. Phys. — 1964. — Vol. 5. — Pp. 677-686.
9. Cohen L. Generalized Phase-Space Distribution Functions // J. Math. Phys. — 1966. — Vol. 7, issue 5. — Pp. 781-948.
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13. Kuryshkin V. V. Some Problems of Quantum Mechanics Possessing a NonNegative Phase-Space Distribution Function // J. Theoret. Phys. — 1973. — Vol. 7. — P. 451.
14. Зорин А. В., Севастьянов Л. А. Математическое моделирование квантовой механики с неотрицательной КФР // Вестник РУДН. Серия «Физика». — 2004. — Т. 11, № 12. — С. 64-80.
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16. Зорин А. В., Севастьянов Л. А. Модель квантовых измерений Курышкина-Вудкевича // Вестник РУДН. Серия «Математика. Информатика. Физика». — 2010. — № 3, вып. 1. — С. 98-103.
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18. Kochanski P., Wodkiewicz K. Operational Measurements in Quantum Mechanics // Rep. Math. Phys. — 1997. — Vol. 40. — Pp. 245-253.
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20. Gadella M., Gomez F. A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics // Found. of Phys. — 2002. — Vol. 32, No 6. — Pp. 815-869.
21. Gadella M., Gomez F. A Measure-Theoretical Approach to the Nuclear and Inductive Spectral Theorems // Bull. Sci. Math. — 2005. — Vol. 129. — Pp. 567-590.
22. Gadella M., Gomez F. Eigenfunction Expansions and Transformation Theory // Acta Appl. Math. — 2009. — Vol. 109. — Pp. 721-742.
23. de la Madrid R. Rigged Hilbert Space Approach to the Schrodinger Equation // J. Phys. A: Math. Gen. — 2002. — Vol. 35. — Pp. 319-342.
24. de la Madrid R., Bohm A., Gadella M. Rigged Hilbert Space Treatment of Continuous Spectrum // Fortsch. Phys. — 2002. — Vol. 50. — Pp. 185-216.
25. de la Madrid R. The Role of the Rigged Hilbert Space in Quantum Mechanics // Eur. J. Phys. — 2005. — Vol. 26. — Pp. 287-312.
26. Зорин А. В. Операционная модель квантовых измерений Курышкина-Вудкевича // Вестник РУДН, серия «Математика. Информатика. Физика». — 2012. — № 2. — С. 42-54.
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32. Englert B.-G., Wodkiewicz K. Intrinsic and Operational Observables in Quantum Mechanics // Phys. Rev. A. — 1995. — Vol. 51. — P. 2661.
