Levinson's theorem explained

Levinson's theorem is an important theorem in non-relativistic quantum scattering theory. It relates the number of bound states of a potential to the difference in phase of a scattered wave at zero and infinite energies. It was published by Norman Levinson in 1949.^[1]

Statement of theorem

The difference in the

\ell -wave phase shift of a scattered wave at zero energy, \varphi _{\ell(0)} , and infinite energy, \varphi _{\ell(infty)} , for a spherically symmetric potential V(r) is related to the number of bound states n _{\ell} by:\varphi _{\ell(0)} - \varphi _{\ell(infty)} = ( n _{\ell} + (1)/(2) N )\pi

where

N = 0 or 1 . The case N = 1 is exceptional and it can only happen in s -wave scattering. The following conditions are sufficient to guarantee the theorem:^[2] V(r) continuous in ( 0 , inf ty) except for a finite number of finite discontinuitiesV(r) = O(r ) ~ as ~r → 0 ~~\varepsilon> 0 V(r) = O(r ) ~ as ~r → inf ty ~~\varepsilon> 0

External links

• M. Wellner, "Levinson's Theorem (an Elementary Derivation," Atomic Energy Research Establishment, Harwell, England. March 1964.

Notes and References

1. http://physics.nyu.edu/LarrySpruch/LevinsonsTheorem.PDF#Levinson_theorem Levinson's Theorem
2. A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990).