Can anyone please explain me in details about what are the assumptions made for the derivation of Schrödinger Wave Equation?

You can read further about this in the postulates of quantum mechanics.

The assumptions are as follows about #hatH# and #psi#:

• The Hamiltonian operator #hatH# is a linear operator (it distributes through addition, like all other quantum mechanics operators):

#hatH(A(r) + B(r)) = hatHA(r) + hatHB(r)#

It must also be Hermitian, in order to correspond to a physical quantity known as an observable.

This allows us to write things like:

#int_"allspace" psi^"*"hatHpsid tau = int_"allspace" (hatHpsi)^"*"psid tau#,
sometimes called the turnover rule

#hatH^† = (hatH^"*")^T = (hatH^T)^"*" = hatH#
(the adjoint of #hatH# is itself.)

We are used to using scalar Hamiltonians #hatH#, so typically we just say #hatH^"*" = hatH# and don't worry about the transposition part of the adjoint operator #†#.

We also typically use time-independent Hamiltonians, though time-dependent Hamiltonians do exist.

• The wave function, or state function #psi#, belongs to a complete set of eigenfunctions #{psi_i}# such that:

1. #int_"allspace" psi_n^"*"psi_nd tau = 1#,
   for the #n#th state of the particle having a probability of #100%# for the particle being found in the entire universe.

2. #int_"allspace" psi_m^"*"psi_nd tau = overbrace(delta_(mn))^"Kroenecker Delta" -= {(0, m ne n),(1, m = n):}#,
   for the #m#th and #n#th states overlapping in the relevant boundaries. The condition of #m = n# is called normality, and the condition of #m ne n# is called orthogonality. Together they form the orthonormality conditions of #psi#.

3. #hatHpsi= Epsi# follows, i.e. #psi# must be prepared in a state that is an eigenstate of #hatH# that gives the energy #E# (operating with #hatH# must give #E#). Otherwise, either #hatH# or the wave function is not well-chosen.

4. Each #psi_i# in this complete set must be linearly independent, i.e. they can be written as linear combinations #psi_(i) = sum_k c_(ik) phi_k# that do not equal another #psi_i# in the set.

• The wave function #psi# also must be "well-behaved", so it must follow these conditions:

1. #psi# is finite in the relevant boundaries.

2. #psi# goes to zero at #pmoo#

3. #psi# is single-valued and both #psi# and #(dpsi)/(dr)# continuous. (#r# is #sqrt(x^2+y^2+z^2)#.)

4. #psi# is square-integrable (so that the differential equation can be solved in the first place), i.e. #int_"allspace" psi^"*"psid tau# has a solution.