By D. Iagolnitzer
Overseas Colloquium on complicated research, Microlocal Calculus and Relativistic Quantum idea
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Extra info for Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory
We get En ∼ ω(n − 1/4) n = 1, 2, 3, . . 2 Order-of-magnitude estimate Make an order-of-magnitude estimate of the spectrum: use the BohrSommerfeld quantization rule and approximate the phase space trajectory by a rectangle. Keep the δ correction. 1) can be approximately replaced by the “circumscribed rectangle” of the trajectory, see Fig. 2. The phase space integral then reads: dx 2m(En − mω 2 x2 /2) ∼ 2En /(mω 2 ) xmax ∼ En /ω. 1) gives En ∼ ω(n − 1/4) n = 1, 2, 3, . . August 8, 2013 15:6 World Scientific Book - 9in x 6in B1662-ch03 “Halved” Harmonic Oscillator: A Case Study 45 p p max 0 x x max Fig.
128, 2425 (1962). 2 33 Bohr-Sommerfeld quantization for periodic boundary conditions Derive a Bohr-Sommerfeld quantization rule for a one-dimensional bounded motion with periodic boundary conditions. 3) can be generalized to any number of spatial dimensions. 4 Ground state energy of a logarithmic potential Consider a one-dimensional particle moving in a logarithmic potential: 2 − ∂2 ψ(x) + 2ϵ ln(r/a)ψ(x) = Eψ(x). 13) (a) Estimate the ground state energy using the Heisenberg uncertainty principle.
S. s. (x) is a time-independent ground state of the Hamiltonian H. e. 1. 1 Bohr-Sommerfeld quantization All useful cases of Bohr-Sommerfeld quantization can be concisely expressed in a single formula that reads n + # soft points × ∆soft + # hard points × ∆hard ] dxp(x; En˜ ) = 2π [˜ with n ˜ = 0, 1, 2, 3, . . 15) where π 2 = π, ∆soft = ∆hard # soft points(# hard points) counts the number of the soft(hard) turning points encountered by the classical trajectory during the cycle of motion (see Fig.
Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory by D. Iagolnitzer