78
2 The Discrete Spectrum and the Continuum
Both direct integration and current formula can be used when Γ > 10 −6 keV, as
the current formula is very precise for widths smaller than typically 1 keV and
Γ is still sufficiently large for direct integration to be numerically stable.
Values provided by the current formula when Γ < 10 −6 keV are virtually exact,
as the real part of the wave function can always be calculated precisely with
direct integration and the current formula has already reached its domain of
validity when Γ ∼ 10 −6 keV. Conversely, the imaginary part of the energy
quickly becomes too imprecise to provide with Γ when Γ 10 −6 .
Exercise XVII.
A. The different behavior of width as a function of energy is clearly visible and
changes according to the proton or neutron case, and to the considered partial
wave.
Width decreases quickly with decreasing energy for large orbital angular
momentum in the proton or neutron case, and for nuclei bearing an important
proton charge in the proton case.
B. The polynomial relation between energy and width for neutrons and exponential
relation depending on η for protons is obtained by fitting plots of widths as a
function of (k).
C. The considered expression is valid as long as the width provided by the current
formula is sufficiently small, that is, typically smaller than 10 keV.
References
1. P.M. Morse, H. Feshbach, Methods of Theoretical Physics (Mc Graw-Hill, New York, 1953)
2. A. Messiah, Quantum Mechanics, vol. 1 and 2 (North Holland, Amsterdam, 1961)
3. A.D. Alhaidari, Phys. Rev. A 66, 042116 (2002)
4. I. Talmi, Helv. Phys. Acta 25, 185 (1952)
5. M. Moshinsky, Nucl. Phys. 13, 104 (1959)
6. I.J. Thompson, A.R. Barnett, Comput. Phys. Comm. 36, 363 (1985)
7. N. Michel, Comput. Phys. Comm. 176, 232 (2007)
8. M. Abramowitz, Handbook of Mathematical Functions, National Bureau of Standards, Applied
Mathematics, vol. 55, ed. by M. Abramowitz, I.A. Stegun (National Bureau of Standards,
Gaithersburg, 1972)
9. J. Humblet, L. Rosenfeld, Nucl. Phys. 26, 529 (1961)
10. K.S. Kölbig, Comput. Phys. Comm. 4, 221 (1972)
11. I.J. Thompson, A.R. Barnett, J. Comput. Phys. 64, 490 (1986)
12. A. Dzieciol, S. Yngve, P.O. Fröman, J. Math. Phys. 40, 6145 (1999)
13. N. Mukunda, Am. J. Phys. 49, 910 (1978)
14. C. Eckart, Phys. Rev. 35, 1303 (1930)
15. N. Rosen, P.M. Morse, Phys. Rev. 42, 210 (1932)
16. L. Hulthén, Ark. Mat. Astron. Fys. 28, 5 (1942)
17. M.F. Manning, N. Rosen, Phys. Rev. 44, 953 (1933)
18. S.H. Dong, J. Garcia-Ravelo, Phys. Scr. 75, 307 (2007)
19. G.A. Natanzon, Vestn. Leningr. Univ. Fiz. 10, 22 (1971)
20. G.A. Natanzon, Theor. Math. Phys. 38, 146 (1979)
2 The Discrete Spectrum and the Continuum
Both direct integration and current formula can be used when Γ > 10 −6 keV, as
the current formula is very precise for widths smaller than typically 1 keV and
Γ is still sufficiently large for direct integration to be numerically stable.
Values provided by the current formula when Γ < 10 −6 keV are virtually exact,
as the real part of the wave function can always be calculated precisely with
direct integration and the current formula has already reached its domain of
validity when Γ ∼ 10 −6 keV. Conversely, the imaginary part of the energy
quickly becomes too imprecise to provide with Γ when Γ 10 −6 .
Exercise XVII.
A. The different behavior of width as a function of energy is clearly visible and
changes according to the proton or neutron case, and to the considered partial
wave.
Width decreases quickly with decreasing energy for large orbital angular
momentum in the proton or neutron case, and for nuclei bearing an important
proton charge in the proton case.
B. The polynomial relation between energy and width for neutrons and exponential
relation depending on η for protons is obtained by fitting plots of widths as a
function of (k).
C. The considered expression is valid as long as the width provided by the current
formula is sufficiently small, that is, typically smaller than 10 keV.
References
1. P.M. Morse, H. Feshbach, Methods of Theoretical Physics (Mc Graw-Hill, New York, 1953)
2. A. Messiah, Quantum Mechanics, vol. 1 and 2 (North Holland, Amsterdam, 1961)
3. A.D. Alhaidari, Phys. Rev. A 66, 042116 (2002)
4. I. Talmi, Helv. Phys. Acta 25, 185 (1952)
5. M. Moshinsky, Nucl. Phys. 13, 104 (1959)
6. I.J. Thompson, A.R. Barnett, Comput. Phys. Comm. 36, 363 (1985)
7. N. Michel, Comput. Phys. Comm. 176, 232 (2007)
8. M. Abramowitz, Handbook of Mathematical Functions, National Bureau of Standards, Applied
Mathematics, vol. 55, ed. by M. Abramowitz, I.A. Stegun (National Bureau of Standards,
Gaithersburg, 1972)
9. J. Humblet, L. Rosenfeld, Nucl. Phys. 26, 529 (1961)
10. K.S. Kölbig, Comput. Phys. Comm. 4, 221 (1972)
11. I.J. Thompson, A.R. Barnett, J. Comput. Phys. 64, 490 (1986)
12. A. Dzieciol, S. Yngve, P.O. Fröman, J. Math. Phys. 40, 6145 (1999)
13. N. Mukunda, Am. J. Phys. 49, 910 (1978)
14. C. Eckart, Phys. Rev. 35, 1303 (1930)
15. N. Rosen, P.M. Morse, Phys. Rev. 42, 210 (1932)
16. L. Hulthén, Ark. Mat. Astron. Fys. 28, 5 (1942)
17. M.F. Manning, N. Rosen, Phys. Rev. 44, 953 (1933)
18. S.H. Dong, J. Garcia-Ravelo, Phys. Scr. 75, 307 (2007)
19. G.A. Natanzon, Vestn. Leningr. Univ. Fiz. 10, 22 (1971)
20. G.A. Natanzon, Theor. Math. Phys. 38, 146 (1979)