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 phase of a scattered wave at zero energy, ${\displaystyle \phi _{l}(0)}$, and infinite energy, ${\displaystyle \phi _{l}(\infty )}$, for a spherically symmetric potential $V(r)$ is related to the number of bound states ${\displaystyle {\bar {n}}_{l}}$ by:

${\displaystyle \phi _{l}(0)-\phi _{l}(\infty )=(n+{\frac {1}{2}}N)\pi \ }$

where $N=1$ for $s$-wave scattering, $N=2$ for ${\displaystyle l\geq 1}$ and $N=0$ otherwise. Furthermore, the potential must satisfy the following asymptotic conditions:^[2]

${\displaystyle r^{2}|V(r)|\rightarrow 0{\text{ at }}r\rightarrow 0}$

${\displaystyle r^{3}|V(r)|\rightarrow 0{\text{ at }}r\rightarrow \infty }$

References

1. Levinson's Theorem
2. Shi-Hai Dong, Zhong-Qi Ma, #One_Dimension Levinson's Theorem for the Schrodinger Equation in One Dimension, International Journal of Theoretical Physics, Vol 39, No 2, 2000.

External links

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