Quantum Action Principle

• Julian Schwinger
Chapter

Abstract

Consider infinitesimal displacements of the time origin,
$$\bar t = t - \delta t,$$
(5.1.1)
and the implied unitary transformation,
$$U = 1 + \frac{i}{\hbar }Gt\;with\;Gt = - \delta tH$$
(5.1.2)
where H, the Hamiltonian operator, depends upon a set of variables for the system, v α(t), and possibly on t itself. When we shift the origin, the variables are redefined,
$${\upsilon _\alpha }(t) = {\bar \upsilon _\alpha }(\bar t)$$
(5.1.3)
or
$$\begin{array}{*{20}{c}} {{{{\bar{\upsilon }}}_{\alpha }}(t) = {{\upsilon }_{\alpha }}(t + \delta t) = {{\upsilon }_{\alpha }}(t) + \delta t\frac{d}{{dt}}{{\upsilon }_{\alpha }}(t)} \\ { = {{\upsilon }_{\alpha }}(t) - \delta {{\upsilon }_{\alpha }}(t),} \\ \end{array}$$
(5.1.4)
where
$$\delta {\upsilon _\alpha }(t) = \frac{1}{{i\hbar }}[{\upsilon _\alpha }(t),Gt]$$
(5.1.5)
so that
$$- \delta t\frac{d}{{dt}}{\upsilon _\alpha }(t) = \frac{1}{{i\hbar }}[{\upsilon _\alpha }(t), - \delta tH]$$
(5.1.6)

Keywords

Wave Function Action Principle Hamilton Operator Schrodinger Equation Explicit Time Dependence
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