Derivations Flashcards
Derive the reduced mass if
-ℏ^2/2me ∇^2e Ψ(re,rn) - ℏ^2/2mn ∇^2n Ψ(re,rn) + V(re-rn) Ψ(re,rn) = Etotal Ψ(re,rn)
R + (X,Y,Z) = 1/M (me xe + mn xn…)
r = (x,y,z) = re - rn = (xe - xn…)
M = me + mn
chain rule for d/dxe and d/dxn
square both sides
substitute into Schrodinger equation
mixed derivatives cancel
giving as required with definition of reduced mass defined
use separation of variables for Ψ(re,rn) = Ψ(r)Φ(R)
to solve
ℏ^2/2M ∇^2 Ψ(re,rn) - ℏ^2/2μ ∇^2 Ψ(re,rn) + V(re-rn) Ψ(re,rn) = Etotal Ψ(re,rn)
first term corresponds to R and second r. So they attach to derivative.
Divide by Ψ(r)Φ(R) and identify the first term as ECoM and second term as E
such that EcoM + E = Etotal
use separation of variables for Ψ(r,θ,Φ) = R(r)Y(θ,Φ) to solve
eq.1.30
substitute into eq.1.30 where r corresponds to first term and θ,Φ the second term.
multiply by r^2/RY and rearrange to find the terms independent of r and terms independent of
θ,Φ.
Term independent of r is l(l+1)
use separation of variables for Y(θ,Φ) = Θ(θ)Φ(ϕ) to solve
eq. 1.36
substitute into eq. 1.36 and multiply by sin^2θ/ΘΦ
find the term independent of θ and the term independent of ϕ
where the term independent of θ is the constant m^2
Solve the Legendre differential equation (eq. 2.7) by using the taylor series
Θ(x) = (∞∑k=0) ck x^k
set m = 0 and divide by sin^2 = 1-x^2
substitute the taylor series in and take the derivative.
Expand the brackets and take the derivative
shift k such that each x term is x^k
collect like terms
set equal to zero and rearrange for ck+2
solve the associated legendre differential equation eq.2.7
using the substitution
Θ = (1 − x^2)^m/2 y
here m does not equal 0
use the product rule
and substitute into the legendre differential equation
divide through by (1 − x^2)^m/2 y gives the transformed equation
Solve the Laguerre differential equation (eq.2.50) by using the taylor series
L(p) = (∞∑k=0) ck p^k
substitute the series into eq. 2.50
take the derivatives and collect like terms
shift ck by 1 so that they are all to the power of p^k
which gives the recursion relations for ck
Solve the centrifugal barrier eq.3.10 through change of variables
change the dependent variable R(r) = U(r)/r and divide through by r
we can identify Veff(r) = V(r) + ℏ^2/2µ l(l+1)/r^2
Find the constants of proportionality Clm and Dlm
L+ |l,ml > = Clm|l,ml+1>
L- |l,ml > = Dlm|l,ml-1>
unit normalise (mod square)
replace with definitions of L+ and L-
expand brackets and replace with with operator eigenvalues
giving as required
derive L.S
J = L + S
J^2 = (L+S)^2
J^2 = L^2 + S^2 + 2L.S
L . S = 1/2(J^2-L^2-S^2)
or L.S = L+S- + L-S+ + 2LzSz
Calculate the corrections of pertubation
H(λ) = Hˆ(0) + λHˆ(1)
H(λ)|n(λ)⟩ = En(λ)|n(λ)⟩
where n(λ) and En(λ) are a taylor series of |n^(0)> + λ|n^(1)> …
substitute into H(λ)|n(λ)⟩ = En(λ)|n(λ)⟩
multiply out each term and collect like factors of λ
Derive the relativistic correction based on its Hamiltonian eq.5.10
H^(0) = p^2/2me - Ze^2/r4πϵ0r
rearrange for p^2/2me and substitute
calculate the first order pertubation
replace H^(0) with En^(0) and expand brackets
divide through by (En^(0))^2 and substitute in 1/r and 1/r^2
Derive the spin orbit correction based on its Hamiltonian eq. 5.15
substitute L.S based on J^2 = (L+S)^2
calculate first order pertubation
put in 1/r^3 and L^2, S^2 and J^2
which gives two solutions for j
Derive the total fine structure corrections
Add together Erel + Eso if l ≠ 0
then Erel + Edar if l = 0
Adding all three gives one whatever the l.
Derive the electric dipole selection rules
if [Lz,z] = 0, {P,z} = 0 and [L^2,[L^2,z]] = 2h^2(L^2z + zL^2)
0 =! <n,l,ml | … |n’,l’,ml’>
start with first relation giving ml = ml’
second relation states parity must change
and third that ∆l = |l −l’| =0
