Non-Hermitian Hamiltonian Treatment of Stark Effect in Quantum Mechanics

Randal Hallford, Preet Sharma


The Non-Hermitian aspect of Quantum Mechanics has been of great interest recently. There have been numerous studies on non-Hermitian Hamiltonians written for natural processes. Some studies have even expressed the hydrogen atom in a non-Hermitian basis. In this paper the principles of non-Hermitian quantum mechanics is applied to both the time independent perturbation theory and to the time dependant theory to calculate the Stark effect. The principles of spherical harmonics has also been used to describe the development in the non-Hermitian case. Finally, the non-Hermitian aspect has been introduced to the well known Stark effect in quantum mechanics to find a condition in which the Stark effect will still be true even if a non-Hermitian Hamiltonian is used. This study completes the understanding at a fundamental level to understand the well known Stark effect.


Doi: 10.28991/esj-2020-01242

Full Text: PDF


Quantum Mechanics; Non-Hermitian Hamiltonians; Hydrogen Atom; Stark Effect; Spherical Harmonics.


Bender, Carl M. “Introduction to 𝒫𝒯-Symmetric Quantum Theory.” Contemporary Physics 46, no. 4 (July 2005): 277–292. doi:10.1080/00107500072632.

Ashida, Yuto, Zongping Gong, and Masahito Ueda. "Non-Hermitian Physics." arXiv preprint arXiv:2006.01837 (2020).

Hamazaki, Ryusuke, Kohei Kawabata, Naoto Kura, and Masahito Ueda. “Universality Classes of Non-Hermitian Random Matrices.” Physical Review Research 2, no. 2 (June 4, 2020). doi:10.1103/physrevresearch.2.023286.

Bender, Carl M., and Stefan Boettcher. “Real Spectra in Non-Hermitian Hamiltonians HavingPTSymmetry.” Physical Review Letters 80, no. 24 (June 15, 1998): 5243–5246. doi:10.1103/physrevlett.80.5243.

Ernzerhof, Matthias, Alexandre Giguère, and Didier Mayou. “Non-Hermitian Quantum Mechanics and Exceptional Points in Molecular Electronics.” The Journal of Chemical Physics 152, no. 24 (June 28, 2020): 244119. doi:10.1063/5.0006365.

Causey, Robert L. “Computing Eigenvalues of Non-Hermitian Matrices by Methods of Jacobi Type.” Journal of the Society for Industrial and Applied Mathematics 6, no. 2 (June 1958): 172–181. doi:10.1137/0106010.

Bender, Carl M., and Philip D. Mannheim. “No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model.” Physical Review Letters 100, no. 11 (March 18, 2008). doi:10.1103/physrevlett.100.110402.

Chen, T., J. Demmel, M. Gu, Y. Saad, R. Lehoucq, D. Sorensen, K. Maschhoff, et al. “7. Non-Hermitian Eigenvalue Problems.” Templates for the Solution of Algebraic Eigenvalue Problems (January 2000): 149–231. doi:10.1137/1.9780898719581.ch7.

Frith, Thomas. "Time-dependence in non-Hermitian quantum systems." arXiv preprint arXiv:2002.01977 (2020).

Romero, Juan M., O. González-Gaxiola, and R. Bernal-Jaquez. “Non-Hermitian Hydrogen Atom.” Journal of Mathematical Chemistry 51, no. 8 (June 4, 2013): 1990–2000. doi:10.1007/s10910-013-0194-z.

Bagarello, Fabio, and Francesco Gargano. "Eigenvalues of non‐Hermitian Matrices: A Dynamical and an Iterative approach—Application to a Truncated Swanson Model." Mathematical Methods in the Applied Sciences 43, no. 9 (March 15, 2020): 5758–5775. doi:10.1002/mma.6317.

Zettili, N. "Quantum Mechanics Concepts and Applications: John Wiley & Sons Ltd UK." (2009).

Feinberg, Joshua, and A. Zee. “Non-Hermitian Random Matrix Theory: Method of Hermitian Reduction.” Nuclear Physics B 504, no. 3 (November 1997): 579–608. doi:10.1016/s0550-3213(97)00502-6.

Sakurai, J. J., and Jim Napolitano. “Modern Quantum Mechanics” Cambridge University Press (September 21, 2017): 550. doi:10.1017/9781108499996.

Ipsen, I. C. F., and B. Nadler. “Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices.” SIAM Journal on Matrix Analysis and Applications 31, no. 1 (January 2009): 40–53. doi:10.1137/070682745.

Sussman, Benjamin J. “Five Ways to the Nonresonant Dynamic Stark Effect.” American Journal of Physics 79, no. 5 (May 2011): 477–484. doi:10.1119/1.3553018.

Hezel, T. P., C. E. Burkhardt, M. Ciocca, and J. J. Leventhal. “Classical View of the Stark Effect in Hydrogen Atoms.” American Journal of Physics 60, no. 4 (April 1992): 324–328. doi:10.1119/1.16875.

Barratt, C. “The Stark Effect on an Excited Hydrogen Atom.” American Journal of Physics 51, no. 7 (July 1983): 610–612. doi:10.1119/1.13185.

Fernández, Francisco M., and Eduardo A. Castro. "Stark Effect in a One‐dimensional Model Atom." American Journal of Physics 53, no. 8 (August 1985): 757–760. doi:10.1119/1.14307.

Full Text: PDF

DOI: 10.28991/esj-2020-01242


  • There are currently no refbacks.

Copyright (c) 2020 Preet Sharma, Randal Hallford