By Max von Laue

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Extra resources for Die Relativitatstheorie, Volume 2 Volume 38; Volume 68 of Wissenschaft (Braunschweig, Germany) Die Relativitatstheorie

Example text

Thus, if the field B grows (or decreases respectively) with the distance, the eigenstates feel a stronger (weaker) field and therefore the energies should increase (decrease) as a function of m within a Landau level. , Hm is factorizable and there are ladder operators acting between the Hms. 103) we use the supersymmetry of the operators H% — B(r) and a comparison theorem for the purely angular excitations of ¥L A. 102) First of all, we present a suitable factorization of the reduced Hamiltonians Hm in the case of a constant magnetic field.

As is seen for 0 < n < 7 and 0 < t < 5, we find E(n-1, /+V6) > E(n, t). Furthermore, (E(n-J + >/6) - E(n,f))(E(n9f) - E{n - I,/))" 1 does not exceed 15% for n < 6. A more favourable case is V = r5. 85) has the property that, at V\r) = 0, one has also V"'(r) = 0. This makes it possible to squeeze V between convenient upper and lower bounds. Specifically, after a rescaling to +25 we can prove the following chain of inequalities: 35(p-l))22+21(p-l) 44 < 4 + 2 p 5 - 7 < 35 (- — ^ 1 . s. is a harmonic oscillator perturbed by a quartic term, for which, according to Loeffel, Martin, Simon and Wightman [81] diagonal Pade approximants give a lower bound.

48) where n denotes the number of nodes of the corresponding eigenfunction. Proof: With a change of variable from r to p = r a , the radial Schrodinger equation with potential V^g, angular momentum { and energy zero is transformed into the Schrodinger equation for the Coulomb potential —Z/p, angular momentum X and energy —Z 2/4N2. 47). 49) with Dirichlet boundary conditions at r = 0. One calculates easily, on the one hand, the product of A~ and A+: with and, on the other hand, that the commutator is / ] a ; i V ,z/ + a - Ha>N>z/).

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Die Relativitatstheorie, Volume 2 Volume 38; Volume 68 of Wissenschaft (Braunschweig, Germany) Die Relativitatstheorie by Max von Laue


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