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Mathematics of Complexity and Dynamical Systems pp 1351–1375Cite as

Perturbation Theory in Quantum Mechanics

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Article Outline

Glossary

Definition of the Subject

Introduction

Presentation of the Problem and an Example

Perturbation of Point Spectra: Nondegenerate Case

Perturbation of Point Spectra: Degenerate Case

The Brillouin–Wigner Method

Symmetry and Degeneracy

Problems with the Perturbation Series

Perturbation of the Continuous Spectrum

Time Dependent Perturbations

Future Directions

Bibliography

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Abbreviations

Hilbert space:

A Hilbert space \({\mathcal{H}}\) is a normed complex vector space with a Hermitian scalar product. If \({\varphi,\,\psi\in\mathcal H}\) the scalar product between φ and ψ is written as \({(\varphi,\psi)\equiv (\psi,\varphi)^*}\) and is taken to be linear in ψ and antilinear in φ: if \({a,\,b\in \mathbb{C}}\), the scalar product between \({a\, \varphi}\) and \({b\, \psi}\) is \({a^*b(\varphi,\psi)}\). The norm of ψ is defined as \({\Vert \psi\Vert\equiv \sqrt{(\psi,\psi)}}\). With respect to the norm \({\Vert\cdot\Vert}\), \({\mathcal H}\) is a complete metric space. In the following \({\mathcal{H}}\) will be assumed to be separable, that is any complete orthonormal set of vectors is countable.

States and observables:

In quantum mechanics the states of a system are represented as vectors in a Hilbert space \({\mathcal H}\), with the convention that proportional vectors represent the same state. Physicists mostly use Dirac's notation: the elements of \({\mathcal H}\) are represented by \( \,\vert\, \cdot\,\rangle \) (“ket”) and the scalar product between \( \,\vert\, \varphi\,\rangle \) and \( \,\vert\, \psi\,\rangle \) is written as \( \langle\, \varphi\mid \psi\,\rangle \) (“braket”). The observables, i. e. the physical quantities that can be measured, are represented by linear Hermitian (more precisely: self‐adjoint) operators on \({\mathcal H}\). The eigenvalues of an observable are the only possible results of the measurement of the observable. The observables of a system are generally the same of the corresponding classical system: energy, angular momentum, etc., i. e. they are of the form \({f(q,p)}\), with \( q\equiv (q_1,\dots,q_n), p\equiv (p_1,\dots,p_n) \) the position and momentum canonical variables of the system: q i and p i are observables, i. e. operators, which satisfy the commutation relations \( [q_i,q_j]\equiv q_iq_j-q_jq_i=0 \), \( [p_i,p_j]=0 \), \( [q_i,p_j]=i\hbar\,\delta_{ij}\), with \({\hbar}\) the Planck's constant h divided by \({2\pi}\).

Representations:

Since separable Hilbert spaces are isomorphic, it is always possible to represent the elements of \({\mathcal H}\) as elements of l 2, the space of the sequences \( \{u_i\},\; {u_i\in\mathbb{C}}\), with the scalar product \( (v,u)\equiv \sum_i v_i^*u_i \). This can be done by choosing an orthonormal basis of vectors e i in \( \mathcal H : (e_i,e_j)=\delta_{ij}\) and defining \( u_i=(e_i,u) \); with Dirac's notations \( \,\vert\, A\,\rangle \to \{a_i\},\ a_i=\langle\, e_i\mid A\,\rangle \). Linear operators ξ are then represented by \( \{\xi_{ij}\}, \xi_{ij}= (e_i,\xi\, e_j)\equiv \langle\, e_i\mid \xi\mid e_j\,\rangle \). The ξ ij are called “matrix elements” of ξ in the representation e i . If \({\xi^\dagger}\) is the Hermitian‐conjugate of ξ, then \( (\xi^\dagger)_{ij}=\xi_{ji}^* \). If the e i are eigenvectors of ξ then the (infinite) matrix ξ ij is diagonal, the diagonal elements being the eigenvalues of ξ.

Schrödinger representation:

A different possibility is to represent the elements of \({\mathcal H}\) as elements of \({L^2[\mathbb{R}^n]}\), the space of the square‐integrable functions on \({\mathbb{R}^n}\), where n is the number of degrees of freedom of the system. This can be done by assigning how the operators q i and p i act on the functions of \({L^2[\mathbb{R}^n]}\): in the Schrödinger representation the q i are taken to act as multiplication by x i and the p i as \({-i\hbar\partial/\partial x_i}\): if \({\,\vert\, A\,\rangle\to \psi_A(x_1,\dots,x_n)}\), then

$$ \begin{aligned} q_i\vert A\rangle&\to x_i\psi_A(x_1,\dots,x_n)\:, \\ p_i\vert A\rangle &\to -i\hbar\partial\psi_A(x_1,\dots,x_n)/\partial x_i\:. \end{aligned}$$
Schrödinger equation:

Among the observables, the Hamiltonian H plays a special role. It determines the time evolution of the system through the time dependent Schrödinger equation

$$ i\hbar\frac{\partial \psi}{\partial t} =H\psi\:, $$

and its eigenvalues are the energy levels of the system. The eigenvalue equation \({H\psi=E\psi}\) is called the Schrödinger equation.

Bibliography

Primary Literature

  1. Baker GA, Graves‐Morris P (1996) Padé approximants. Cambridge Univ. Press, Cambridge

    Google Scholar 

  2. Bargmann V (1964) Note on Wigner's theorem on symmetry operations. J Math Phys 5:862–868

    Article  MathSciNet  MATH  Google Scholar 

  3. Bender CM, Wu TT (1969) Anharmonic oscillator. Phys Rev 184:1231–1260

    Article  MathSciNet  Google Scholar 

  4. Bloch C (1958) Sur la théorie des perturbations des états liées. Nucl Phys 6:329–347

    Article  Google Scholar 

  5. Böhm A (1993) Quantum mechanics, foundations and applications. Springer, New York, pp 208–215

    Google Scholar 

  6. Borel E (1899) Mémoires sur le séries divergentes. Ann Sci École Norm Sup 16:9–136

    MathSciNet  MATH  Google Scholar 

  7. Born M, Heisenberg W, Jordan P (1926) Zur Quantenmechanik, II. Z Phys 35:557–615

    Article  MATH  Google Scholar 

  8. Born M (1926) Quantenmechanik der Stossvorgänge. Z Phys 38:803–827

    Article  Google Scholar 

  9. Brillouin L (1932) Perturbation problem and self consistent field. J Phys Radium 3:373–389

    Article  Google Scholar 

  10. Courant R, Hilbert D (1989) Methods of mathematical physics, vol I. Wiley, New York, pp 343–350

    Book  Google Scholar 

  11. Dirac PAM (1926) On the theory of quantum mechanics. Proc Roy Soc A112:661–677

    Article  MATH  Google Scholar 

  12. Dyson FJ (1952) Divergence of perturbation theory in quantum electrodynamics. Phys Rev 85:631–632

    Article  MathSciNet  MATH  Google Scholar 

  13. Dyson FJ (1949) The radiation theories of Tomonaga, Schwinger and Feynman. Phys Rev 75:486–502

    Article  MathSciNet  MATH  Google Scholar 

  14. Dyson FJ (1949) The S‑matrix in quantum electrodynamics. Phys Rev 75:1736–1755

    Article  MathSciNet  MATH  Google Scholar 

  15. Epstein ST (1954) Note on perturbation theory. Amer J Phys 22:613–614

    Article  MATH  Google Scholar 

  16. Epstein ST (1968) Uniqueness of the energy in perturbation theory. Amer J Phys 36:165–166

    Article  Google Scholar 

  17. Feynman RP (1939) Forces in molecules. Phys Rev 56:340–343

    Article  MATH  Google Scholar 

  18. Fock VA (1935) Zur Theorie des Wasserstoffatoms. Z Phys 98:145–154

    Article  Google Scholar 

  19. Graffi S, Grecchi V, Simon B (1970) Borel summability: Application to the anharmonic oscillator. Phys Lett B32:631–634

    Article  MathSciNet  Google Scholar 

  20. Grossman A (1961) Schrödinger scattering amplitude I. J Math Phys 3:710–713

    Article  Google Scholar 

  21. Hamermesh M (1989) Group theory and its application to physical problems. Dover, New York, pp 32–114

    Google Scholar 

  22. Hannabuss K (1997) Introduction to quantum theory. Clarendon, Oxford, pp 131–136

    MATH  Google Scholar 

  23. Hellmann H (1937) Einführung in die Quantenchemie. Deuticke, Leipzig

    Google Scholar 

  24. Joachain J (1983) Quantum collision theory. North–Holland, Amsterdam

    Google Scholar 

  25. Kato T (1966) Perturbation theory for linear operators. Springer, New York

    MATH  Google Scholar 

  26. Kato T (1949) On the convergence of the perturbation method I. Progr Theor Phys 4:514–523

    Article  Google Scholar 

  27. Kramers HA (1957) Quantum mechanics. North-Holland, Amsterdam, pp 198–202

    MATH  Google Scholar 

  28. Krieger JB (1968) Asymptotic properties of perturbation theory. J Math Phys 9:432–435

    Article  MathSciNet  MATH  Google Scholar 

  29. Landau LD, Lifshitz EM (1960) Mechanics. Pergamon Press, Oxford, p 129

    MATH  Google Scholar 

  30. Lippmann BA, Schwinger J (1950) Variational principles for scattering processes I. Phys Rev 79:469–480

    Article  MathSciNet  MATH  Google Scholar 

  31. Loeffel J, Martin A, Wightman A, Simon B (1969) Padé approximants and the anharmonic oscillator. Phys Lett B 30:656–658

    Article  Google Scholar 

  32. Padé H (1899) Sur la représentation approchée d'une fonction pour des fonctions rationelles. Ann Sci Éco Norm Sup Suppl 9(3):1–93

    Google Scholar 

  33. Rayleigh JW (1894–1896) The theory of sound, vol I. Macmillan, London, pp 115–118

    Google Scholar 

  34. Reed M, Simon B (1975) Methods of modern mathematical physics, vol II. Academic Press, New York, pp 282–283

    Google Scholar 

  35. Reed M, Simon B (1978) Methods of modern mathematical physics, vol IV. Academic Press, New York, pp 10–44

    Google Scholar 

  36. Rellich F (1937) Störungstheorie der Spektralzerlegung I. Math Ann 113:600–619

    Article  MathSciNet  Google Scholar 

  37. Rellich F (1937) Störungstheorie der Spektralzerlegung II. Math Ann 113:677–685

    Article  MathSciNet  MATH  Google Scholar 

  38. Rellich F (1939) Störungstheorie der Spektralzerlegung III. Math Ann 116:555–570

    Article  MathSciNet  Google Scholar 

  39. Rellich F (1940) Störungstheorie der Spektralzerlegung IV. Math Ann 117:356–382

    Article  MathSciNet  Google Scholar 

  40. Riesz F, Sz.-Nagy B (1968) Leçons d'analyse fonctionelle. Gauthier-Villars, Paris, pp 143–188

    Google Scholar 

  41. Rollnik H (1956) Streumaxima und gebundene Zustände. Z Phys 145:639–653

    Article  MATH  Google Scholar 

  42. Sakurai JJ (1967) Advanced quantum mechanics. Addison-Wesley, Reading, pp 39–40

    Google Scholar 

  43. Scadron M, Weinberg S, Wright J (1964) Functional analysis and scattering theory. Phys Rev 135:B202-B207

    Article  MathSciNet  Google Scholar 

  44. Schrödinger E (1926) Quantisierung als Eigenwertproblem. Ann Phys 80:437–490

    Google Scholar 

  45. Schwartz J (1960) Some non-self‐adjoint operators. Comm Pure Appl Math 13:609–639

    Article  MathSciNet  MATH  Google Scholar 

  46. Simon B (1970) Coupling constant analyticity for the anharmonic oscillator. Ann Phys 58:76–136

    Article  Google Scholar 

  47. Simon B (1991) Fifty years of eigenvalue perturbation theory. Bull Am Math Soc 24:303–319

    Article  MATH  Google Scholar 

  48. Thirring W (2002) Quantum mathematical physics. Springer, Berlin, p 177

    Google Scholar 

  49. von Neumann J, Wigner E (1929) Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys Z 30:467–470

    MATH  Google Scholar 

  50. Watson G (1912) A theory of asymptotic series. Philos Trans Roy Soc Lon Ser A 211:279–313

    Article  Google Scholar 

  51. Weyl H (1931) The theory of groups and quantum mechanics. Dover, New York

    MATH  Google Scholar 

  52. Wigner EP (1935) On a modification of the Rayleigh–Schrödinger perturbation theory. Math Natur Anz (Budapest) 53:477–482

    Google Scholar 

  53. Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York

    MATH  Google Scholar 

  54. Yosida K (1991) Lectures on differential and integral equations. Dover, New York, pp 115–131

    MATH  Google Scholar 

  55. Yukalov VI, Yukalova EP (2007) Methods of self similar factor approximants. Phys Lett A 368:341–347

    Article  Google Scholar 

  56. Zemach C, Klein A (1958) The Born expansion in non‐relativistic quantum theory I. Nuovo Cimento 10:1078–1087

    Article  MathSciNet  MATH  Google Scholar 

Books and Reviews

  1. Hirschfelder JO, Byers Brown W, Epstein ST (1964) Recent developments in perturbation theory. In: Advances in quantum chemistry, vol 1. Academic Press, New York, pp 255–374

    Google Scholar 

  2. Killingbeck J (1977) Quantum‐mechanical perturbation theory. Rep Progr Phys 40:963–1031

    Article  Google Scholar 

  3. Mayer I (2003) Simple theorems, proofs and derivations in quantum chemistry. Kluwer Academic/Plenum Publishers, New York, pp 69–120

    Google Scholar 

  4. Morse PM, Feshbach H (1953) Methods of theoretical physics, part 2. McGraw-Hill, New York, pp 1001–1106

    Google Scholar 

  5. Wilcox CH (1966) Perturbation theory and its applications in quantum mechanics. Wiley, New York

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università di Pisa, Pisa, Italy

    Luigi E. Picasso, Luciano Bracci & Emilio d'Emilio

  2. Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy

    Luigi E. Picasso, Luciano Bracci & Emilio d'Emilio

Authors
  1. Luigi E. Picasso
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  2. Luciano Bracci
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  3. Emilio d'Emilio
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Picasso, L.E., Bracci, L., d'Emilio, E. (2012). Perturbation Theory in Quantum Mechanics. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_85

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