Formulae Flashcards
the overlap of the same state
<n|n> = 1
commutator
[A,B] = [AB-BA]
something is minimised when
the derivative is set equal to zero i.e. d/dx = 0
reduced mass
µ = (me mN)/(me+mN) = (m1 m2)/(m1+m2)
spherical harmonics are correctly normalised if
(2π ∫ 0) (π ∫ 0) |(Y^m l)|^2 sinθdθdφ
Remember to square everything in the wave function
a wave function is normalised in general if
( ∞ ∫ 0) |Ψ|^2 dr^3 = 1
expectation value of <r^3>
calculate ∫dr^3 |Ψ|^2 r^3 as before with extra expectation value multiplied in
Rydberg equation
E = Ry (1/n2^2 - 1/n1^2)
Planck’s equation
E = hf = hλ/c
Derive the angular momentum operators Lx, Ly and Lz
L = r x p
p = (-iℏ∇)
and r = (x,y,z)
take the cross product
lowering operator for L
L- = Lx - iLy
the conjugate of |x>
<x|
total angular momentum quantum number
j = l ± 1/2
how is mj related to ms and ml
mj = ml + ms
how is J related to L and S
J = L + S
first order pertubation theory
|Ψn(λ)> = |Ψn(0)> + λ|Ψn(1)> + O(λ^2)
to show something is orthonormal
take the conjugate
hydrogen atom wavefunction
Ψnlm on formula sheet
virial theorem
<V> = -2<T>
</T></V>
Write the Hamiltonian in the form of its kinetic and potential energy
<H> = <T> + <V>
</V></T></H>
how are the relativstic corrections and the spin-orbit coupling related
En^(0) = E^(1) rel + E^(1) SO
on the formula sheet = En (Zα)^2/n^2 …
in the stark effect z can be written as
z = rcosθ
energy shift in the Paschen-back regime
strong magnetic field
∆E = ωlℏ(m+2ms) on formula sheet
how is m related to l
-l ≤ m ≤ l
what does >JJ mean
j,mj>
what does >LS mean
ml,ms>
how is >JJ related to >LS
|jmax,jmax>JJ = l,s>LS
how does n relate to l
n = 1,2,3,…
l = 0,1,2,…
the total energy of two electrons to first order pertubation
Etot = E(0) + E(1)
binding energy
En ~ - Ry Z^2/n^2
transition energy
E(gamma) = Ry(Z^2/n1^2 - Z^2/n2^2)
labelling of shells and subshells
K, L, M, N
n = 1,2,3,4
l = 0,1,2,3
s, p, d, f
n = 1,2,3,4
l = 0,1,2,3
[p,x] =
iℏ
quantum harmonic oscillator ladder operators
a-|n> = √n|n-1>
a+|n> = √n+1|n+1>
harmonic oscillator restoring force
F = -kx = - mω^2x
force constant
k = mω^2
what is the first order perturbation
H(1)
what is the energy of the first order perturbation
E(1) = <n|H(1)|n>
rotational energy
E = 1/2ℏω
reduced mass for hydrogen
µ = 1/2 mp
Operator L^2 and S^2 and J^2
J = h(bar)^2 j(j+1) similarly for L and S
Zeroth order perturbation
H(0)|n> = En(0)|n>
<n|m>
δnm
Standard deviation
Δr = (<r^2> -<r>^2)^(1/2)</r>
Stark energy shift
ΔEstark = eEz.z
Hstark = εE.z
eε ∫ z dr^3 |psi|^2
Diagonalisation of the matrix of eigenvalues
det|H(1) - λI|
gives the energy splitting
Applying the lowering operator to the ground state
a-|n0> = 0
K alpha
n = 2 to n= 1
K beta
n = 3 to n = 1
Energy levels for harmonic oscillators.
En = ℏω(n+1/2)
The number of spin states for para spin states
N = (s+1)(2s+1)
The number of spin states for ortho spin states
N = s(2s+1)
In a weak magnetic field
(L . 2S) . B is small
Re write Lz + 2Sz in terms of Jz and Sz
Lz + Sz + Sz
=> Jz + Sz
x(hat)
x(hat) = i (ℏ/2mω)^1/2 (A- - A+)
p(hat)
p(hat) = (mℏω/2)^1/2 (a+ + a-)
Screening factors
Z1 ~ (Z-a) and Z2 ~ (Z-b)
Parity
Pr = r
Pθ = π - θ
PΦ = Φ + π
the orbital magnetic moment
mL = -e/2me L
L x L
iℏL
S x S
iℏS
spin magnetic moment
ms = -eg/2me S
nuclear magneton
μN = eℏ/2mN
{A,B}
AB + BA
energy of a multielectron states
Ex(0) = -Z^2 Ry (1/n1^2 - 1/n2^2)
state labelling the the L-S coupling scheme (diatomic molecules)
2S+1 XJ where X = S,P,D,F
S1 . S2
S(1,+)S(2,-) + S(1,-)S(2,+) + 2S(1,z)S(2,z)
L . S
L+S- + L-S+ + 2LzSz
[Lz,z]
=0
[Lz,L±]
±ħL±
[L^2,L ±]
=0
[L,S]
=0
A probability distribution is maximised
When x tends to infinity
Energy shift to First order perturbation of the stark effect
En < psi | eεz | psi >
= eε ∫ dr^3 z |psi|^2
Probability density
|Rnl|^2 |Yml|^2 r^2 sin theta d theta d phi dr
Property of factorials
n! = n x (n-1)!
|jmax, -jmax> JJ
= | -l, -s>LS
|jmax,jmax>JJ
|l,s>
