By D. Iagolnitzer

ISBN-10: 3540099964

ISBN-13: 9783540099963

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**

**Sample text**

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

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