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Atomic Systems > Formulae > Flashcards

Formulae Flashcards

1
Q

the overlap of the same state

A

<n|n> = 1

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2
Q

commutator

A

[A,B] = [AB-BA]

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3
Q

something is minimised when

A

the derivative is set equal to zero i.e. d/dx = 0

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4
Q

reduced mass

A

µ = (me mN)/(me+mN) = (m1 m2)/(m1+m2)

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5
Q

spherical harmonics are correctly normalised if

A

(2π ∫ 0) (π ∫ 0) |(Y^m l)|^2 sinθdθdφ

Remember to square everything in the wave function

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6
Q

a wave function is normalised in general if

A

( ∞ ∫ 0) |Ψ|^2 dr^3 = 1

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7
Q

expectation value of <r^3>

A

calculate ∫dr^3 |Ψ|^2 r^3 as before with extra expectation value multiplied in

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8
Q

Rydberg equation

A

E = Ry (1/n2^2 - 1/n1^2)

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9
Q

Planck’s equation

A

E = hf = hλ/c

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10
Q

Derive the angular momentum operators Lx, Ly and Lz

A

L = r x p

p = (-iℏ∇)

and r = (x,y,z)

take the cross product

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11
Q

lowering operator for L

A

L- = Lx - iLy

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12
Q

the conjugate of |x>

A

<x|

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13
Q

total angular momentum quantum number

A

j = l ± 1/2

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14
Q

how is mj related to ms and ml

A

mj = ml + ms

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15
Q

how is J related to L and S

A

J = L + S

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16
Q

first order pertubation theory

A

|Ψn(λ)> = |Ψn(0)> + λ|Ψn(1)> + O(λ^2)

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17
Q

to show something is orthonormal

A

take the conjugate

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18
Q

hydrogen atom wavefunction

A

Ψnlm on formula sheet

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19
Q

virial theorem

A

<V> = -2<T>
</T></V>

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20
Q

Write the Hamiltonian in the form of its kinetic and potential energy

A

<H> = <T> + <V>
</V></T></H>

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21
Q

how are the relativstic corrections and the spin-orbit coupling related

A

En^(0) = E^(1) rel + E^(1) SO

on the formula sheet = En (Zα)^2/n^2 …

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22
Q

in the stark effect z can be written as

A

z = rcosθ

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23
Q

energy shift in the Paschen-back regime

A

strong magnetic field

∆E = ωlℏ(m+2ms) on formula sheet

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24
Q

how is m related to l

A

-l ≤ m ≤ l

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25
Q

what does >JJ mean

A

j,mj>

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26
Q

what does >LS mean

A

ml,ms>

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27
Q

how is >JJ related to >LS

A

|jmax,jmax>JJ = l,s>LS

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28
Q

how does n relate to l

A

n = 1,2,3,…
l = 0,1,2,…

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29
Q

the total energy of two electrons to first order pertubation

A

Etot = E(0) + E(1)

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30
Q

binding energy

A

En ~ - Ry Z^2/n^2

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31
Q

transition energy

A

E(gamma) = Ry(Z^2/n1^2 - Z^2/n2^2)

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32
Q

labelling of shells and subshells

A

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

33
Q

[p,x] =

A

iℏ

34
Q

quantum harmonic oscillator ladder operators

A

a-|n> = √n|n-1>
a+|n> = √n+1|n+1>

35
Q

harmonic oscillator restoring force

A

F = -kx = - mω^2x

36
Q

force constant

A

k = mω^2

37
Q

what is the first order perturbation

A

H(1)

38
Q

what is the energy of the first order perturbation

A

E(1) = <n|H(1)|n>

39
Q

rotational energy

A

E = 1/2ℏω

40
Q

reduced mass for hydrogen

A

µ = 1/2 mp

41
Q

Operator L^2 and S^2 and J^2

A

J = h(bar)^2 j(j+1) similarly for L and S

42
Q

Zeroth order perturbation

A

H(0)|n> = En(0)|n>

43
Q

<n|m>

A

δnm

44
Q

Standard deviation

A

Δr = (<r^2> -<r>^2)^(1/2)</r>

45
Q

Stark energy shift

A

ΔEstark = eEz.z
Hstark = εE.z

eε ∫ z dr^3 |psi|^2

46
Q

Diagonalisation of the matrix of eigenvalues

A

det|H(1) - λI|

gives the energy splitting

47
Q

Applying the lowering operator to the ground state

A

a-|n0> = 0

48
Q

K alpha

A

n = 2 to n= 1

49
Q

K beta

A

n = 3 to n = 1

50
Q

Energy levels for harmonic oscillators.

A

En = ℏω(n+1/2)

51
Q

The number of spin states for para spin states

A

N = (s+1)(2s+1)

52
Q

The number of spin states for ortho spin states

A

N = s(2s+1)

53
Q

In a weak magnetic field

A

(L . 2S) . B is small

54
Q

Re write Lz + 2Sz in terms of Jz and Sz

A

Lz + Sz + Sz

=> Jz + Sz

55
Q

x(hat)

A

x(hat) = i (ℏ/2mω)^1/2 (A- - A+)

56
Q

p(hat)

A

p(hat) = (mℏω/2)^1/2 (a+ + a-)

57
Q

Screening factors

A

Z1 ~ (Z-a) and Z2 ~ (Z-b)

58
Q

Parity

A

Pr = r
Pθ = π - θ
PΦ = Φ + π

59
Q

the orbital magnetic moment

A

mL = -e/2me L

60
Q

L x L

A

iℏL

61
Q

S x S

A

iℏS

62
Q

spin magnetic moment

A

ms = -eg/2me S

63
Q

nuclear magneton

A

μN = eℏ/2mN

64
Q

{A,B}

A

AB + BA

65
Q

energy of a multielectron states

A

Ex(0) = -Z^2 Ry (1/n1^2 - 1/n2^2)

66
Q

state labelling the the L-S coupling scheme (diatomic molecules)

A

2S+1 XJ where X = S,P,D,F

67
Q

S1 . S2

A

S(1,+)S(2,-) + S(1,-)S(2,+) + 2S(1,z)S(2,z)

68
Q

L . S

A

L+S- + L-S+ + 2LzSz

69
Q

[Lz,z]

A

=0

70
Q

[Lz,L±]

A

±ħL±

71
Q

[L^2,L ±]

A

=0

72
Q

[L,S]

A

=0

73
Q

A probability distribution is maximised

A

When x tends to infinity

74
Q

Energy shift to First order perturbation of the stark effect

A

En < psi | eεz | psi >

= eε ∫ dr^3 z |psi|^2

75
Q

Probability density

A

|Rnl|^2 |Yml|^2 r^2 sin theta d theta d phi dr

76
Q

Property of factorials

A

n! = n x (n-1)!

77
Q

|jmax, -jmax> JJ

A

= | -l, -s>LS

78
Q

|jmax,jmax>JJ

A

|l,s>

Atomic Systems (3 decks)

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