Physics:Levinson's theorem
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 [math]\displaystyle{ \ell }[/math]-wave phase shift of a scattered wave at zero energy, [math]\displaystyle{ \varphi_\ell(0) }[/math], and infinite energy, [math]\displaystyle{ \varphi_\ell(\infty) }[/math], for a spherically symmetric potential [math]\displaystyle{ V(r) }[/math] is related to the number of bound states [math]\displaystyle{ n_\ell }[/math] by:
- [math]\displaystyle{ \varphi_\ell(0) - \varphi_\ell(\infty) = ( n_\ell + \frac{1}{2}N )\pi \ }[/math]
where [math]\displaystyle{ N = 0 }[/math] or [math]\displaystyle{ 1 }[/math]. The case [math]\displaystyle{ N = 1 }[/math] is exceptional and it can only happen in [math]\displaystyle{ s }[/math]-wave scattering. The following conditions are sufficient to guarantee the theorem:[2]
- [math]\displaystyle{ V(r) }[/math] continuous in [math]\displaystyle{ (0,\infty) }[/math] except for a finite number of finite discontinuities
- [math]\displaystyle{ V(r) = O(r^{ -3/2 + \varepsilon}) ~\text{ as } ~r\rightarrow 0 ~~\varepsilon\gt 0 }[/math]
- [math]\displaystyle{ V(r) = O(r^{ -3 - \varepsilon}) ~\text{ as } ~r \rightarrow \infty ~~\varepsilon\gt 0 }[/math]
References
- ↑ Levinson's Theorem
- ↑ A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990).
External links
- M. Wellner, "Levinson's Theorem (an Elementary Derivation," Atomic Energy Research Establishment, Harwell, England. March 1964.
de:Compton-Effekt#Compton-Wellenlänge
 |  |
