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American Journal of Physics

On a concavity property of the ground state in quantum mechanics

P. Gärtner, Norman Baer, Sandra Ritter

Published May 12, 2010• 0 citations
Publisher

Abstract

The concavity of the ground-state energy as a function of the coupling constant of the perturbation is shown using Ritz’s variational principle. The derivation is simple and general, and the result is a useful tool for checking the accuracy of standard approximations such as perturbation theory and variational methods.
DOI
10.1119/1.3308567

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A variational functional has been derived for the expectation value of the energy of a quantum system governed by a time-dependent Hamiltonian. Application to a stationary system perturbed by a time-dependent force shows that a perturbational wave function correct to first order in the perturbation, when inserted into the functional, delivers an expression for the energy that is correct up to third order in the perturbation. The use of the variational functional by means of a trial function is illustrated on the forced harmonic oscillator where well-known results are reproduced easily.
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S u m m a r y . A rigorous theory of variational principles in quantum mechanics is formulated. This result is obtained by generalizing the standard theory of the inverse problem of the variational cahmlus, according to new concepts of differentiability on complex Banach spaces. Our theory provides sound mathematical foundations to all known variational principles of quantum mechanics and a powerful tool to find new ones. As a simple application, we derive rigorously two variational principles for the time-dependent SchrSdinger equation: the conventional one, requiring both an initial and a final condition, and a new one, requiring an initial condition only.
Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
A recently proposed quantum mechanical criterion `concavity of energy' for the second law of thermodynamics is studied also for classical particle systems confined in a bounded region by a potential with a time-dependent coupling constant. It is shown that the time average of work done by particles in a quench process cannot exceed that in the corresponding quasi-static process, if the energy is a concave function of the coupling constant. It is proven that the energy is indeed concave for a general confining potential with certain properties. This result implies that the system satisfies the principle of maximum work in the adiabatic environment as an expression of the second law of thermodynamics.