Distribution of the quantum mechanical time-delay matrix for a chaotic cavity

We calculate the joint probability distribution of the Wigner-Smith time-delay matrix Q = -i (h) over bar S(-1)partial derivative S/partial derivative epsilon and the scattering matrix S for scattering from a chaotic cavity with ideal point contacts. To this end we prove a conjecture by Wigner about the unitary invariance property of the distribution functional P[S(epsilon)] of energy-dependent scattering matrices S(epsilon). The distribution of the inverse of the eigenvalues tau(1),...,tau(N) of Q is found to be the Laguerre ensemble from random-matrix theory. The eigenvalue density rho(tau) is computed using the method of orthogonal polynomials. This general theory has applications to the thermopower, magnetoconductance, and capacitance of a quantum dot.