Manifest Gauge Invariance in Nonrelativistic Quantum Mechanics with Classical Electromagnetic Fields

  • Donald H. Kobe
Part of the NATO ASI Series book series (volume 110)


Since gauge invariance is such a fundamental symmetry principle, it is surprising that a manifestly gauge-invariant formulation of nonrelativistic quantum mechanics did not emerge until a few years ago. In 1976 Yang1 gave such a formulation in which he took as physical observables only Hermitian, gauge-invariant operators.2,3 For a charged particle in an external classical electromagnetic field, the Hamiltonian is not in general the energy operator. An energy operator is defined as the sum of the kinetic and potential energies, which is the Hamiltonian minus the scalar potential of the external time-dependent electromagnetic field.4–6 The eigenvalue equation for the energy operator gives the energy eigenvalues and energy eigenstates at any time. The probability amplitude for finding the particle in an energy eigenstate is the inner product of the eigenstate of the energy operator and the wave function. This amplitude is gauge invariant, and therefore can be used to obtain a valid probability. The energy eigenstates are coupled by the matrix elements of the quantum-mechanical power operator.7


Gauge Transformation Unitary Transformation Energy Operator Probability Amplitude Schrodinger Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K.-H. Yang, Ann. Phys. (N.Y.) 101: 62 (1976).ADSCrossRefGoogle Scholar
  2. 2.
    D. H. Kobe, and K.-H. Yang, J. Phys. A: Math. Gen. 13: 3171 (1980).ADSCrossRefGoogle Scholar
  3. 3.
    C. Cohen-Tannoudji, B. Diu, F. Laloë, “Quantum Mechanics,” Vol. 1, Wiley, New York (1977), pp. 315–328.Google Scholar
  4. 4.
    D. H. Kobe, and A. L. Smirl, Am. J. Phys. 46: 624 (1978).ADSCrossRefGoogle Scholar
  5. 5.
    D. H. Kobe, Int. J. Quant. Chem. S 12: 73 (1978).Google Scholar
  6. 6.
    D. H. Kobe, Phys. Rev. A 19:205 (1979); 19: 1876. (1979).ADSCrossRefGoogle Scholar
  7. 7.
    D. H. Kobe, E. C.-T. Wen, and K.-H. Yang, Phys. Rev. D 26: 1927 (1982).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    L. I. Schiff, “Quantum Mechanics,” 3rd. edn., McGraw-Hill, New York (1968), pp. 398–403.Google Scholar
  9. 9.
    C. Leubner, and P. Zoller, J. Phys. B: At. Mol. Phys. 13: 3613 (1980).MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    D. H. Kobe, and K.-H. Yang, Am. J. Phys. 51: 163 (1983).ADSCrossRefGoogle Scholar
  11. 11.
    D. H. Kobe, Am. J. Phys. 50: 128 (1982).ADSCrossRefGoogle Scholar
  12. 12.
    D. H. Kobe, and E. C.-T. Wen, J. Phys. A: Math. Gen. 15: 787 (1982).ADSCrossRefGoogle Scholar
  13. 13.
    D. H. Kobe, J. Phys. A: Math. Gen. 16: 737 (1983)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    P. K. Kennedy, and D. H. Kobe, J. Phys. A: Math. Gen. 16: 521 (1983).ADSCrossRefGoogle Scholar
  15. 15.
    D. H. Kobe, J. Phys. B: At. Mol. Phys. 16: 1159 (1983).MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    K.-H. Yang, Phys. Lett. 92A: 71 (1982).CrossRefGoogle Scholar
  17. 17.
    K.-H. Yang, J. Phys. A: Math. Gen. 15: 437 (1982).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • Donald H. Kobe
    • 1
  1. 1.Department of PhysicsNorth Texas State UniversityDentonUSA

Personalised recommendations