The Bohr atom
a0 = 0.053 nm. rn = n2a0 = 0.5 mm.
All the electron wave functions with non-zero angular momentum are zero at the origin. The greater the angular momentum, the farther the electron is kept away from the nucleus.
To find the eigenfunctions and eigenvalues of the
Hamiltonian of a hydrogenic atom we replace in the eigenfunctions of the
Hamiltonian of the hydrogen atom a0 by a0' = ħ2/(μ'Ze2)
= a0(μ/μ')(1/Z), and in the eigenvalues of the Hamiltonian of the
hydrogen atom we replace EI by EI' = μ'Z2e4/(2ħ2)
μ = mμmp/(mμ + mp) ≈ mμ.
Addition of angular momenta
L = 2, S = 1/2 , J = 3/2, 5/2 are possible. From the given choices, only (D) is possible.
The energy of the proton in the field is -∫0zpqe|E|dz
The energy of the electron in the field is ∫0zeqe|E|dz = qe|E|ze.
The potential energy of both particles is -qe|E|(zp - ze) = qe|E|z = -p∙E.
After the measurement ψ(1,2) = α(1)β(2).
Entangled particles, spin
After the measurement of particle 1 ψ(1,2) = α(1)β(2) = (1/21/2)α(1)[αx(2) - βx(2)]
One-dimensional potential wells
The bottom of the well is shifted up. In one dimension, there is no degeneracy.
Spherically symmetric potentials, postulates of QM
For a spherically symmetric potential H, L2, and Lz
commute, their values can be known simultaneously, separation of variables is
Eigenfunctions of any observable with different eigenvalues are orthogonal.
For optically allowed transitions we need Δl = ±1.