The time-dependent Schrodinger equation for one electron

For a single electron, the time-independent Schrodinger eigenvalue problem is determined by the variational condition 

The simplified notation (xt) is used here to denote (x, t). If Ti is independent of time, these two equations are equivalent. The wave function fix) is modified by a time-dependent phase factor exp(ef/ifi), which has no physical consequences. 

Following Hamilton s principle in classical mechanics, the required time dependence can be derived from a variational principle based on a seemingly artificial Lagrangian density, integrated over both space and time to define the functional 

Treating variations of f and f as independent, because they lead to equivalent Euler-Lagrange equations, and integrating the time integral by parts as in Euler s theory, the resulting variational expression is 

The action integral A is not changed if the trial function f is multiplied by a phase factor exp(/f y t )dt /ifi), while Ti is increased by a time-dependent but spatially uniform potential y(t). This is an example of gauge invariance, taken out of the usual context of electromagnetic theory. Indicating the modified wave function by fy, the modified action integral is 

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