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2b. 11 Now suppose the harmonic oscillators were capped at energy 0', Le. they are finite systems of dimension 451' corresponding to the states l0),
2b. 11 Now suppose the harmonic oscillators were capped at energy 0', Le. they are finite systems of dimension 451' corresponding to the states l0), ll), , Id 1). Thus the total energy is capped at 0 S E S N(d 1), or 0 5 u g d 1.You can think of them as spin-% particles with an external magnetic field if you prefer. For 0' = 2, repeat the above derivation. What happens to T as u > %? What happens for u > %? Answer: Ford = 2, Z = l + 6"} u__6an_ 1 =>_lnlu _ as e+1 _ u . 1 s = ln(2e_E' +'u = 1n(1u)+uln \" u 2d. Let's optimize f under the constraints. Show that in the d-dimensional unconstrained space of distributions {n; ), f is convex-up, (f" is negative-definite, i.e. all its eigenvalues are negative, everywhere) in the physically allowed region, so that there is a unique global maximum. Show that f" must therefore be convex-up in the (d - 2)-dimensional constrained space of distributions, so that there is a unique global maximum in the constrained space. Show that { n* }, where the constrained maximum of f occurs, is in the domain of integration. Use Lagrange optimisation to find this constrained maximum. Define S to be the Lagrange multiplier of the constraint fixing _; in; = u, and show that - BK d-1 n* = Z Z = >e-BK K= 0 Find u as a function of (at fixed d). Since it is difficult to invert u(B) to find B(u), find s and T as functions of B, u(B) and Z(B) Answer: Use two Lagrange multipliers a and B. Define T = N In ( N) - E N. In ( N.) + 0 N - EN. + B E - EN.E. =-N E n; In ( 1 ) + a N - N E n +B E - NE ME . Now, the maximisation requires - = 0. This gives - [In(n;) + 1] - [a] - [BE;] = 0, where we have used dx [x In x] = Inx + 1 This yields the solution2i. Suppose you had a really large finite-dimensional system with /
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