On the Formal Theory of Collision And Reaction Processes
On the Formal Theory of Collision And Reaction Processes
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FVom this, formulas for the eigenvalues and the eigenstates will be derived.
We consider an unperturbed Harailtonian H which we assume to have a ptirely discrete spectrum. The eigenvalues, denoted by E, will be assumed to be simple. The total Hamiltonian H = H + V will also have a disci^te spectirura with simple eigenvalues, provided the pertiirbation V satisfies some conditions which we shall not investigate here. We are interested in the matrix elements /e |g(X)|E \ « G (X) of the resolvant G...(X) " (X - H) in the representation in which the unperturbed Hamiltonian is diagonal. These matrix elements, considered as functions of the complex parameter X, have polar singularities in the eigenvalues of H» We consider that eigenvalue X of H which tends to a particular eigenvalue E of E as the perturbation tends to zero. We would likB our formulas for G__(^) *o exhibit clearly the polar singularity for X = X. .
To this purpose we introduce the operator | '(X), which is defined in the Ir- representation by the equation Clearly ("^ (X) depends \:^on the value n of the index which is omitted in the sum - 5 - occurring in the right-hand side, but we shall not indicate this dependence ex- plicitly, since we want to concentrate on a particular value of n« Since the polar singularity for X " E does not appear in the equation, | can be ass'umed to be regular in a neighborhood of E .
What to read after On the Formal Theory of Collision And Reaction Processes?
You can find similar books in the "Read Also" column, or choose other free books by Bruno Zumino to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
FVom this, formulas for the eigenvalues and the eigenstates will be derived.
We consider an unperturbed Harailtonian H which we assume to have a ptirely discrete spectrum. The eigenvalues, denoted by E, will be assumed to be simple. The total Hamiltonian H = H + V will also have a disci^te spectirura with simple eigenvalues, provided the pertiirbation V satisfies some conditions which we shall not investigate here. We are interested in the matrix elements /e |g(X)|E \ « G (X) of the resolvant G...(X) " (X - H) in the representation in which the unperturbed Hamiltonian is diagonal. These matrix elements, considered as functions of the complex parameter X, have polar singularities in the eigenvalues of H» We consider that eigenvalue X of H which tends to a particular eigenvalue E of E as the perturbation tends to zero. We would likB our formulas for G__(^) *o exhibit clearly the polar singularity for X = X. .
To this purpose we introduce the operator | '(X), which is defined in the Ir- representation by the equation Clearly ("^ (X) depends \:^on the value n of the index which is omitted in the sum - 5 - occurring in the right-hand side, but we shall not indicate this dependence ex- plicitly, since we want to concentrate on a particular value of n« Since the polar singularity for X " E does not appear in the equation, | can be ass'umed to be regular in a neighborhood of E .
What to read after On the Formal Theory of Collision And Reaction Processes?
You can find similar books in the "Read Also" column, or choose other free books by Bruno Zumino to read onlineMoreLess
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