Bounds On Scattering Phase Shifts Static Central Potentials
Bounds On Scattering Phase Shifts Static Central Potentials
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E, ... E. , ... . The bound states are charac- 1 n' J' terlzed by the requirement that the function vanish at r = R. Now consider that function which Is Identical with the J'th bound state solution for r S R and which Is a solution of the free wave equation for the energy E for r> R, with contlnuoiis value and slope at r = R. From \inlqueness, this amst be Identical with the scattering solution, which Is a multiple of BlnTk r + Tj(k. )3 In the external region. Since this must vanish at r = R, I...t follows that k R + Ti(k ) must be a multiple of n. J J To prove the equivalence of the two different definitions of k. , Eqs. (5. 2) and (5. 4), it must still be shown that the multiple of « is in fact J itself. To see this, we note firstly that by Levlnson's theorem.
13a Figure 1 {N> Z)ir N-fl N+2 A schematic plot of ^(k) a kR + T](k) versus k; ^(k) need be defined only for non-negative energies. Since there are N negative energy states, Le-vlnson's theorem gives ^(O) = Nrt. By Wigner's causal Inequality, d5(k)/dk > for any interval of k vhlch satisfies Jx ^ C(l^) - (J + 2)jt> where J is an integer, which, from the above dis- cussion, must be greater than N.
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To quickly assess the difficulty of the text, read a short excerpt:
E, ... E. , ... . The bound states are charac- 1 n' J' terlzed by the requirement that the function vanish at r = R. Now consider that function which Is Identical with the J'th bound state solution for r S R and which Is a solution of the free wave equation for the energy E for r> R, with contlnuoiis value and slope at r = R. From \inlqueness, this amst be Identical with the scattering solution, which Is a multiple of BlnTk r + Tj(k. )3 In the external region. Since this must vanish at r = R, I...t follows that k R + Ti(k ) must be a multiple of n. J J To prove the equivalence of the two different definitions of k. , Eqs. (5. 2) and (5. 4), it must still be shown that the multiple of « is in fact J itself. To see this, we note firstly that by Levlnson's theorem.
13a Figure 1 {N> Z)ir N-fl N+2 A schematic plot of ^(k) a kR + T](k) versus k; ^(k) need be defined only for non-negative energies. Since there are N negative energy states, Le-vlnson's theorem gives ^(O) = Nrt. By Wigner's causal Inequality, d5(k)/dk > for any interval of k vhlch satisfies Jx ^ C(l^) - (J + 2)jt> where J is an integer, which, from the above dis- cussion, must be greater than N.
What to read after Bounds On Scattering Phase Shifts Static Central Potentials?
You can find similar books in the "Read Also" column, or choose other free books by Tony Randall to read onlineMoreLess
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