Re: Schrodinger's Equation

Hi Robert,

Yeah, I agree that the idea of a trajectory in CM is not valid in QM
where only the expectation values are important. But the Schrodinger's
equation is a PDE, 1st order in time and 2nd order in space. And,
usually, for solving PDEs, we use the method of characteristics. So, I
was wondering if this method of characteristics applies to the
Schrodinger's equation also. If it does, then what are the
characteristics? If it does not apply, then why so? Both these cases
would be interesting.

The Schrodinger's Equation is not provable in the strict sense, but
this equation has emerged out of a very smart way of looking at wave
motion in optics and coupling that to Hamiltonian systems. The
Schrodinger's equation has a very deep connection with Classical
Hamiltonian Mechanics. A reading of Schrodinger's original paper can
be very enlightening in this regard.

Thanks,
Kushal.

On Aug 10, 12:08 am, robert bristow-johnson
<r...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Aug 8, 3:39 pm, kushal <atmabo...@xxxxxxxxx> wrote:



What are the characteristics of the Schrodinger's Equation? Are they
the single particle trajectories that we get from Newton's Force laws?

it doesn't describe a trajectory of a particle in the same sense as
classical mechanics does.  it describes the probablity of existence of
the particle at a particular place in space and time.

If I remember correctly, Schrodinger had derived this equation
starting from classical action.

i always thought he hypothesized it as one of several differential
equations that have the DeBroglie wave equation as a solution.

So, there must be a connection between
the single particle trajectories and the solutions and characteristics
of the Schrodinger's equation.

i think the connection between what classical physics predicts (an
equation for a trajectory) and what wave mechanics predicts is in the
Expectation Value.  long ago when i was a student, i tried to apply
this "correspondance principle" in a naive way.  i tried to solve the
time-dependant Schrodingers equation for a simple particle in a box
(or maybe  it was the hamonic oscillator) and then, leaving t
variable, solve for the expection value of x (as a function of t).  i
was hoping to see as the mass of the particle got large, that the
result of the expection would approach what classical mechanics would
say.  but i was unsuccessful and never really understood my prof's
explanation for why.

r b-j

one thing that i tried to do

.

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