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

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

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

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

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

References

↑ Levinson's Theorem
↑ 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.

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