Answered step by step
Verified Expert Solution
Link Copied!

Question

1 Approved Answer

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),

image text in transcribedimage text in transcribedimage text in transcribedimage text in transcribedimage text in transcribed
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 /

Step by Step Solution

There are 3 Steps involved in it

Step: 1

blur-text-image

Get Instant Access to Expert-Tailored Solutions

See step-by-step solutions with expert insights and AI powered tools for academic success

Step: 2

blur-text-image_2

Step: 3

blur-text-image_3

Ace Your Homework with AI

Get the answers you need in no time with our AI-driven, step-by-step assistance

Get Started

Recommended Textbook for

Microoptics Technology Fabrication And Applications Of Lens Arrays And Devices

Authors: Nicholas F Borrelli

2nd Edition

1351836668, 9781351836661

More Books

Students also viewed these Physics questions

Question

Under what conditions are two qualitative variables independent?

Answered: 1 week ago

Question

analyze how research and writing unites with design.

Answered: 1 week ago