Near a stationary configuration of a two-atom system

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

Near a stationary configuration

(x,y) = (0,0)

of a two-atom system, the potential energy is well approximated by

V(x,y) = \tfrac12\mathbf r^{\mathrm T} H \mathbf r

, where

\mathbf r = (x,y)

are small displacements in metres from the stationary point and

H = \begin{pmatrix}375&-125\\-125&375\end{pmatrix}\,\mathrm{N\,m^{-1}}

is the Hessian (second-derivative) matrix of

V

there. Take the Boltzmann constant

k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}

. ((a)) Find the eigenvalues of

H

and classify the stationary point (minimum, maximum or saddle). ((b)) Give the singular value decomposition of

H

, treated as a matrix in its own right, and its condition number

\kappa(H)

in the Euclidean norm. ((c)) Using the principal axes found in part (a) and the two-dimensional Gaussian integral, evaluate

Z = \iint_{-\infty}^{\infty} \exp\!\left(-\frac{V(x,y)}{k_BT}\right)dx\,dy

T = 300\,\mathrm K

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