Starting Toward a Unified Theory
Most recent answer: 12/15/2014
- david (age 26)
Edmonds, WA, USA
First, we've passed your concern about the text box problems on to our web expert. We hope it will be fixed soon. Unfortunately I'm as bad at that sort of programming as at the deep physics which you ask about. I'll just give a first stab at some of your starter questions.
For an isolated system, the expectation value of the Hamiltonian doesn't change with time. In fact, the set of eigenvectors and eigenvalues and the squared amplitude of the state component along each eigenvector don't change with time. Thus the mean energy, its standard deviation, and all higher statistical moments don't change in time.
Various other quantities (e.g. position) do change because the interference between state components with different energies changes with time. That is often obscured in first courses on quantum mechanics, which focus too much on pure eigenstates of the Hamiltonian. Those don't do anything as a function of time, except rotate in a non-physical complex plane.
The business of adding small perturbing Hamiltonians (within the isolated system) is just a mathematical method for approximating solutions to equations that are too hard if handled in their full complexity. It doesn't change the physics described above.
Considering a small interaction Hamiltonian of the system with the outside environment does allow actual fluctuations in energy and other conserved quantities, through exchange with that outside environment.
I think the Wikipedia introduction to Lagrangians is good: . The Lagrangian is, like the Hamiltonian, just what you surmised- a way to look at the time dependence of the system. One key physical point that doesn't seem clear in the Wikipedia description is why objects should follow trajectories that minimize the time integral of the Lagrangian, i.e. minimize the classical action. That integral is proportional to the change in the phase of the quantum wave from start to end of the trajectory. Having it minimized (or maximized) means that the components of the wave on a range of nearby trajectories are adding in phase, giving constructive interference. So that's where the wave mostly shows up.
The Lagrangian is not itself a symmetry, but it can have symmetries. In that sense it's similar to Newton's dynamical laws. They have certain symmetries, such as remaining true if all the objects they describe are displaced and/or rotated by the same amounts. Those symmetries lead to corresponding conservation laws.
That didn't tell you much but maybe it'll help a little.
Mike W.
(published on 12/15/2014)