Question

In quantum descriptions of atoms why are observables (which we derive from the wave function) attributed to electrons?

Answer 1

hey your ans is
The time-independent Schroedinger equation for the hydrogen atom is

H Ψ(r⃗ N,r⃗ e)=E Ψ(r⃗ N,r⃗ e)
H Ψ(r→N,r→e)=E Ψ(r→N,r→e)
where r⃗ Nr→N is the position of the nucleus, r⃗ er→e the position of the electron and

H=p2N2mN+p2e2me−e24πϵ0∣r⃗ N−r⃗ e∣.
H=pN22mN+pe22me−e24πϵ0∣r→N−r→e∣.
We usually use center of mass coordinates:

r⃗ =r⃗ e−r⃗ NR⃗ =mN r⃗ N+mer⃗ emN+me
r→=r→e−r→NR→=mN r→N+mer→emN+me
so that the Schroedinger equation becomes

H Ψ(R⃗ ,r⃗ )=E Ψ(R⃗ ,r⃗ )
H Ψ(R→,r→)=E Ψ(R→,r→)
where the hamiltonian is now

H=P22M+p22μ+e24πϵ0∣r⃗ ∣
H=P22M+p22μ+e24πϵ0∣r→∣
where M=mN+meM=mN+me is the total mass, P⃗ =p⃗ N+p⃗ eP→=p→N+p→e is the total momentum, μ≡(mNme)/(mN+me)μ≡(mNme)/(mN+me) is the reduced mass and p⃗ =(mNp⃗ e−mep⃗ N)/(mN+me)p→=(mNp→e−mep→N)/(mN+me) is the relative momentum.

The total wave function is the product of the nuclear and the electronic wave function (ΦΦ and ψψ respectively):

Ψ(R⃗ ,r⃗ )=Φ(R⃗ ) ψ(r⃗ )
Ψ(R→,r→)=Φ(R→) ψ(r→)
where

P22M Φ(R⃗ )=ECM Φ(R⃗ )
P22M Φ(R→)=ECM Φ(R→)
and

(p22μ+e24πϵ0∣r⃗ ∣) ψ(r⃗ )=E ψ(r⃗ ).
(p22μ+e24πϵ0∣r→∣) ψ(r→)=E ψ(r→).
The orbitals we always talk about and that you have seen depicted (see following picture for example) are the ψ(r⃗ )ψ(r→) and are usually expressed in polar coordinates: ψ(r,θ,ϕ)ψ(r,θ,ϕ).



So what we usually refer to as "orbital" is not the wave function of the whole atom (which would be ΨΨ) nor the electronic wave function, but the wave function of a fictitous particle with mass μμ , position r⃗ r→ and momentum p⃗ p→.

However, since mN≃1800 memN≃1800 me, you can see that R⃗ ≃r⃗ NR→≃r→N, M≃mNM≃mN and μ≃meμ≃me, so that ΦΦ is in a certain way close to the nuclear wave function and ψψ to the electronic wave function. Also notice that the Schroedinger equation for ΦΦ is a free particle equation, which is trivial to solve and will held a plane wave solution.

I hope its help you plzz mark me brainliest ans and follow me

Answer 2

Wave-function is a complex number because of two properties it should meet. On the one hand it's modulus square is observable and thus should be real (it gives probability density). ... This is possible only in case such a boost is simply a phase of the complex number, which does not affect its modulus.