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13.6: Problems

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13.1. Generalized momentum is defined as

M=L(dVdr).

(a) Find the generalized momentum for the system described by the Lagrangian of Equation 13.3.51.

(b) The generalized momentum does not have the units of momentum. Identify the units of this generalized momentum.

(c) Write the Hamiltonian of Equation 13.3.50 as a function of r, V, and M but not as a function of dVdr.

(d) Write the Lagrangian of Equation 13.3.51 as a function of r, V, and M but not as a function of dV dr .

(e) Show that the Hamiltonian and Lagrangian found above satisfy the equation H=MdVdrL.

13.2. In the analysis of this chapter, the generalized path was chosen as V and the generalized potential was chosen as ρch. The opposite choice is also possible where the generalized path is ρch and the generalized potential is V.

(a) Write the Hamiltonian of Equation 13.3.50 as functions of ρch instead of V, so it has the form H(r,ρch,dρchdr).

(b) Repeat the above for the Lagrangian of Equation 13.3.51.

(c) Find the Euler-Lagrange equation using ρch as the generalized path.

13.3. Verify that y=144t3 is a solution of the Thomas Fermi equation [46].

(While this solution satisfies the Thomas Fermi equation, it is not useful in describing the energy of an atom. In the t0 limit, this solution approaches infinity, y(0). However, in the t0 limit, the solution should approach a constant, y(0)1, to correctly describe the physical behavior of an atom [180].)

13.4. The previous problem discussed that y=144t3 is a solution of the Thomas Fermi equation. Show that y=72t3 is not a solution.

13.5. Prove that the Thomas Fermi equation is nonlinear.


This page titled 13.6: Problems is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform.

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