consider a system having only
two eigenstates X and Y of some observable . Now I perform a
"measurement"
of this observable using an apparatus that is in some initial state K.
Case 1
initially system is in state X and the apparatus in state K. After
measurement the entire thing(system+ apparatus) is in some state a.
Case 2
initially system is in state Y and the apparatus in state K(same as in
case1). After measurement the entire thing(system+ apparatus) is in
some
state b.
Problematic case
initially system is in state X+Y and the apparatus in state K(same as
in
case1). After measurement the entire thing(system+ apparatus) should
be in
state a+b from the schrodinger equation. This is where my problem lies.
The
"collapse" to a particular eigenstate of the system violates
linearity.

Hence, from this I conclude that Schrodinger equation cannot explain
measurement. Also, there is a problem with reversibility. The
schrodinger
equation allows only unique time evolution. i.e. given a state at t=0
there
is only one state at t=T completely determined by the schrodinger
equation.
I can also integrate the equation and determine the state at t= 0 if i
know
the state at t=T.
Now, if you consider measurement, i cannot do so. if the final state as
a
result of a single measurement leaves the system in state X, then the
inital
state could be anything of the form uX+vY.
This is another problem.

lastly, quantum mechanics requires classical mechanics in its
formulation to explain measurement. But since we know classical
mechanics predicts bizzare things like instability of atoms, should we
really accept a theory based on it!

Please suggest some reference where such questions have been tackled