Equations of motion, variational principles, and WKB approximations in quantum mechanics and quantum field theory: Bound states

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Abstract
We describe, within the framework of the Heisenberg form of quantum mechanics, a general method for obtaining a quantization condition for bound states at the WKB level of accuracy. The method, applicable to both quantum mechanics and quantum field theory, proceeds as follows: (i) Relevant matrix elements of the equations of motions are studied in the large-quantum-number limit including first quantum corrections. (ii) These equations then are derived from several variational principles which generalize the classical versions of Hamilton's principle or the principle of least action, respectively. (iii) The quantization condition emerges in differential form from consideration of the change in either of the stationary functionals upon unit change of the quantum number of the bound state. (iv) The quantum condition in integral form thus involves an integration constant describing quantum fluctuations which is determined for every example considered by a suitable "connection formula". (v) The energy is computed in several ways, but most powerfully by employing the consequences of the quantization condition in the calculation of the expectation value of the Hamiltonian. The program outlined above is illustrated by application to one-dimensional quantum mechanics, to the nonlinear Schrödinger equation, and to the sine-Gordon model (in one space and one time dimension).