Question

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

Answer 1

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

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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.