130_notes.dvi

= ̄h^2 ( χ 00 − 1 2 √ 2 ( χ(1)+χ(2)− −χ(1)−χ(2)+ +χ(1)+χ(2)− −χ(1)−χ(2)+ )) = ̄h^2 (1−1)χ 00 = 0 ̄h^2 χ 00 21.8.4 Adding anyℓplu ...

TheYℓ(m+1)χ−term gives βj(j+ 1) =β ( ℓ(ℓ+ 1) + 3 4 −(m+ 1) ) +α √ ℓ(ℓ+ 1)−m(m+ 1). Collectingαterms on the LHS andβterms on the ...

So we have completed the calculation of the coefficients. We will make use of these in the hydrogen atom, particularly for the a ...

Consider a system of two spinless identical particles. Show that the orbital angular momentum of their relative motion can only ...

A hydrogen atom is in an eigenstate (ψ) ofJ^2 ,L^2 , and ofJz such thatJ^2 ψ =^154 ̄h^2 ψ, L^2 ψ= 6 ̄h^2 ψ,Jzψ=−^12 ̄hψ, and of ...

22 Time Independent Perturbation Theory Perturbation Theory is developed to deal withsmall corrections to problems which we have ...

c(1)nk = 〈φk|H 1 |φn〉 E(0)n −E(0)k En(2) = ∑ k 6 =n |〈φk|H 1 |φn〉|^2 E (0) n −E (0) k audio So thefirst order correction to the ...

degenerate state perturbation theory ∑ i∈N 〈φ(j)|H 0 +H 1 |φ(i)〉αi=Eαj, This is an eigenvalue equation with as many solutions as ...

22.3.2 Hydrogen Atom Ground State in a E-field, the Stark Effect. We have solved the Hydrogen problem with the following Hamilto ...

= a^30 E^2 ∑∞ n=2 2 f(n) −1 +n^12 = − 2 a^30 E^2 ∑∞ n=2 n^2 f(n) n^2 − 1 This is all a little dissatisfying because we had to in ...

This is just the eigenvalue equation forH 1 which we can write out in (pseudo)matrix form  H 1      α 1 α 2 α 3 α 4    ...

E=E 0 ± √ γ^2 + ∆^2 This is OK in both limits, ∆≫γ, andγ≫∆. It is also correct when the two corrections are of the same order. 2 ...

and dotted intoφkyields 〈φk|λH 1 |φn〉+E (0) k λc (1) nk=E (0) n λc (1) nk. From these it is simple to derivethe first order corr ...

The full Schr ̈odinger equation is. (H 0 +λH 1 )   ∑ i∈N αiφ(i)+ ∑ k6∈N cnk(λ)φk  = (E(0)n +λE(1)+...)   ∑ i∈N αiφ(i)+ ∑ k ...

22.6 Sample Test Problems Assume an electron is bound to a heavy positive particle with a harmonic potentialV(x) = 1 2 mω (^2) ...

23 Fine Structure in Hydrogen In this section, we will calculate the fine structure corrections to the Hydrogen spectrum. Some o ...

Like all the fine structure corrections, this is down by a factor of orderα^2 from the Hydrogen binding energy. The second term, ...

Enjmjℓs=En(0)+ En(0) 2 2 mc^2 [ 3 −n 4 ∓j+^21 2 ℓ+^12 ] We can write (ℓ+^12 ) as (j+^12 ∓^12 ), so that 4 ∓j+^21 2 ℓ+^12 = 4 j+ ...

23.2 Hydrogen Atom in a Weak Magnetic Field One way to break the spherical symmetry is to apply an external B field. Lets assume ...

add B-field n=2 S1/2 P3/2 P1/2 S P1/2 1/2 P3/2 add relativity add Darwin spin-orbit The factor ( 1 ± 2 ℓ^1 +1 ) is known as th ...