We present a new derivation of Time-Independent Perturbation Theory (PT) that is shorter and more general than the usual ones presented in Quantum Mechanics (QM) texts. Expressing the task of PT in terms of Green's functions, the key to our approach is to write the bound-state equation for the perturbed wave function directly in terms of the unperturbed Green's function. Not only does this greatly simplify the derivation of PT expressions for energies and wave functions, but it does so without using the orthogonality or completeness properties of unperturbed Hamiltonian eigenstates, which form the basis of standard derivations of PT. As a result, our approach does not require the potentials to be energy independent as is required in standard derivations, nor does it require the inverse free Green's operator G0−1(E) to be a linear function of energy, as is the case in QM. Consequently, our approach is applicable not only to QM but also to various extensions of QM, including Quantum Field Theory. For students, our approach provides an easy-to-understand introduction to the use of Green's functions in QM while at the same time presenting a description of PT that is shorter and more fundamental than the standard ones currently advocated.
We propose a new derivation of Time-Independent Perturbation Theory (PT) that has a fundamental advantage over the usual derivations presented in textbooks on Quantum Mechanics (QM): it is simpler and much shorter. As such, it can provide an easier and quicker way for students to learn PT, than afforded by current methods. In spite of that, our approach does not require the potentials to be energy independent or the inverse free Green function
G0−1(E) to be a linear function of energy
E, as is the case in QM, and can be applied directly to various extensions of QM including Relativistic QM, the Bethe-Salpeter equation, and all kinds of quasipotential approaches in Quantum Field Theory.
A general nonperturbation problem, described by the Hamiltonian
H, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's
T4 correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for
〈Sz〉, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.