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
-wave phase shift of a scattered wave at zero energy,
, and infinite energy,
, for a spherically symmetric potential
is related to the number of bound states
by:
\varphi\ell(0)-\varphi\ell(infty)=(n\ell+
N)\pi
where
or
. The case
is exceptional and it can only happen in
-wave scattering. The following conditions are sufficient to guarantee the theorem:
[2]
continuous in
except for a finite number of finite discontinuities
V(r)=O(r)~as~r → 0~~\varepsilon>0
V(r)=O(r)~as~r → infty~~\varepsilon>0
External links
Notes and References
- http://physics.nyu.edu/LarrySpruch/LevinsonsTheorem.PDF#Levinson_theorem Levinson's Theorem
- A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990).