4.4 Hydrogenic atoms

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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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4.4 Hydrogenic atoms	95

Harmonic oscillator

¯h

(Planck constant)/(2π)

Schrodinger¨

−

¯h2 ∂2ψn

mω2x2ψn

mass

equation

= Enψn

(4.67)

nth eigenfunction

∂x2

ψn

displacement

Energy

En = n+

(4.68)

integer ≥ 0

levelsa

2 ¯hω

total energy in nth state

angular frequency

ψn =

Hn(x/a)exp[−x2/(2a2)]

(4.69)

Eigen-

(n!2naπ1/2)1/2

Hermite polynomials

functions

a =

¯h

1/2

where

mω

Hermite

H0(y) = 1,

H1(y) = 2y,

H2(y) = 4y2 − 2

dummy variable

polynomials

Hn+1(y) = 2yHn(y) − 2nHn−1(y)

(4.70)

aE0 is the zero-point energy of the oscillator.

4.4 Hydrogenic atoms

Bohr modela

nth orbit radius

Quantisation

µr2Ω = n¯h

(4.71)

Ω

orbital angular speed

condition

principal quantum number

(> 0)

Bohr radius

0h2

Bohr radius

52.9pm

reduced mass ( me)

a0 =

(4.72)

πmee2

4πR

∞

−e

electronic charge

atomic number

Orbit radius

(4.73)

Planck constant

rn = Z a0 µ

¯h

h/(2π)

total energy of nth orbit

µe4Z2

Total energy

En =

(4.74)

permittivity of free space

− 8 02h2n2

−

∞

me n2

electron mass

Fine structure

µ0ce2

ﬁne structure constant

constant

α =

(4.75)

µ0

permeability of free space

4π 0¯hc

137

¯h2

Hartree energy

EH =

4.36 × 10−18 J

(4.76)

Hartree energy

mea02

Rydberg

cα2

Rydberg constant

constant

R∞ =

(4.77)

∞

8h3 2c

2hc

speed of light

Rydberg’s

photon wavelength

formulab

= R∞

−

(4.78)

mmn

λmn

integer > n

aBecause the Bohr model is strictly a two-body problem, the equations use reduced mass, µ = memnuc/(me +mnuc) me, where mnuc is the nuclear mass, throughout. The orbit radius is therefore the electron–nucleus distance.

bWavelength of the spectral line corresponding to electron transitions between orbits m and n.

96 Quantum physics

Hydrogenlike atoms – Schrodinger¨ solutiona

Schrodinger¨ equation

¯h2

Ze2

memnuc

−

2Ψnlm −

Ψnlm = EnΨnlm

with

µ =

(4.79)

2µ

4π 0r

me + mnuc

Eigenfunctions

Ψnlm(r,θ,φ) =

(n− l − 1)!

1/2

3/2 xle−x/2Ln2l+1l

(x)Ylm(θ,φ)

(4.80)

2n(n+ l)!

− −1

with

a =

x =

and Ln2l+1l

(x) = n−l−1

(l + n)!(−x)k

µ Z

k)!k!

− −1

(2l + 1 + k)!(n

−

k=0

Total energy

µe4Z2

(4.81)

total energy

− 8 02h2n2

permittivity of free space

− l(l

Planck constant

[3n

+ 1)]

(4.82)

mass of electron

Radial

a2n2

[5n2 + 1 − 3l(l + 1)]

(4.83)

¯h

h/2π

reduced mass (

me)

expectation

mnuc

mass of nucleus

values

1/r =

(4.84)

an2

Ψnlm

eigenfunctions

1/r2

(4.85)

charge of nucleus

−e

(2l + 1)n3a2

electronic charge

n = 1,2,3,...

(4.86)

Lpq

associated Laguerre

l = 0,1,2,... ,(n− 1)

(4.87)

polynomialsc

Allowed

classical orbit radius, n = 1

quantum

m = 0,±1,±2,... ,±l

(4.88)

electron–nucleus separation

numbers and

∆n = 0

(4.89)

harmonics

selection rulesb

spherical

0h2

∆l = ±1

(4.90)

Bohr radius =

± 1

πmee2

∆m = 0 or

(4.91)

a−3/2

r/a

a−3/2

r/2a

Ψ100 =

e−

Ψ200 =

2 −

!e−

π1/2

4(2π)1/2

a−3/2 r

r/2a

a−3/2 r

r/2a

iφ

Ψ210 =

e−

cosθ

Ψ21±1 =

e−

sin

θe±

4(2π)1/2

8π1/2

Ψ300 =

a−3/2

27 − 18

+ 2

e−

r/3a

Ψ310 =

21/2a−3/2

6 −

e−

r/3a

cosθ

81(3π)1/2

81π1/2

a−3/2

r/3a

iφ

a−3/2

r/3a

Ψ31±1 =

6 −

e−

sinθe±

Ψ320 =

e−

(3cos

θ − 1)

81π1/2

81(6π)1/2

a−3/2 r2

r/3a

iφ

a−3/2 r2

r/3a

2iφ

Ψ32±1 =

e−

sinθcosθe±

Ψ32±2 =

e−

sin

θe±

81π1/2

162π1/2

aFor a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free). bFor dipole transitions between orbitals.

cThe sign and indexing deﬁnitions for this function vary. This form is appropriate to Equation (4.80).

4.4 Hydrogenic atoms	97

Orbital angular dependence

				z
			(s)2	0.2	−0.4		(px)2	(py)2
−	0.4	−	0.2	−	−0.4
−			0.2		0.2

			0.2		0.2	y
			x	−0.2
				−0.2
(pz)2					(d 2	−y	2 )2	(dxz)2
				−0.4	x	−y
				−0.4

(d 2 )2	(d )2	(d )2
z	yz	xy
0	0

s orbital

= constant

(4.92)

Y m

spherical

(l = 0)

s = Y0

harmonics

p orbitals

px =

2−1/12

(Y11 − Y1−1) cosφsinθ

(4.93)

py =

(Y 1

+ Y

−1)

sinφsinθ

(4.94)

θ,φ

spherical polar

(l = 1)

coordinates

21/2

pz = Y10 cosθ

(4.95)

dx2 y2

(Y 2

+ Y −2)

sin2 θcos2φ

(4.96)

−

21/2

d orbitals

dxz =

2−1/12

(Y21 − Y2−1) sinθcosθcosφ

(4.97)

(3cos

θ −

x φ

(l = 2)

dz2 = Y2

(4.98)

dyz =

(Y21 + Y2−1) sinθcosθsinφ

(4.99)

21/2

dxy =

2−1/i2

(Y22 − Y2−2) sin2 θsin2φ

(4.100)

aSee page 49 for the deﬁnition of spherical harmonics.

98 Quantum physics

4.5 Angular momentum Orbital angular momentum

(4.101)

angular

L = r×pˆ

¯h

∂

momentum

Angular

(4.102)

position vector

Lz = i

∂y − y ∂x

linear momentum

¯h

∂

momentum

(4.103)

xyz

Cartesian

∂φ

coordinates

operators

ˆ2

ˆ 2

rθφ

spherical polar

= Lx

+ Ly

+ Lz

(4.104)

coordinates

∂

1 ∂2

¯h

(Planck

= −¯h2

sinθ

(4.105)

constant)/(2π)

sinθ

∂θ

sin2 θ

∂φ2

Ladder

L± = Lx ± iLy

∂

(4.106)

Lˆ±ml

ladder operators

iφ

spherical

operators

= ¯he±

icotθ

∂φ

∂θ

(4.107)

harmonics

Lˆ±Ylml

= ¯h[l(l + 1) − ml(ml ± 1)]1/2Ylml ±1

(4.108)

l,ml

integers

ˆ2

= l(l + 1)¯h

(l ≥ 0)

(4.109)

Eigen-

= ml¯hYl

(|ml| ≤ l)

(4.110)

functions and

LzYl

eigenvalues

(θ,φ)

(4.111)

[L±Yl

(θ,φ)] = (ml ± 1)¯hL±Yl

l-multiplicity = (2l + 1)

(4.112)

Angular momentum commutation relationsa

							L	angular momentum
Conservation of angular		ˆ	ˆ		(4.113)		p	momentum
momentumb		[H,Lz] = 0			(4.113)		H	Hamiltonian
momentumb							H	Hamiltonian
							Lˆ	ladder operators
							±
					ˆ	ˆ		ˆ	(4.120)
ˆ		(4.114)			[Lx,Ly] = i¯hLz				(4.120)
ˆ					ˆ	ˆ		ˆ
[Lz,x] = i¯hy					ˆ	ˆ		ˆ	(4.121)
ˆ					[Lz	,Lx] = i¯hLy			(4.121)
[Lz,y] = −i¯hx		(4.115)			[Lˆy,Lˆz] = i¯hLˆx				(4.122)
ˆ		(4.116)			ˆ	ˆ		ˆ
[Lz,z] = 0		(4.116)			ˆ	ˆ		ˆ
[Lˆz,pˆx] = i¯hpˆy		(4.117)			[L+,Lz] = −¯hL+				(4.123)
ˆ					ˆ	ˆ		ˆ	(4.124)
ˆ					[L−,Lz] = ¯hL−				(4.124)
[Lz,pˆy] = −i¯hpˆx		(4.118)			[Lˆ+,Lˆ		] = 2¯hLˆz		(4.125)
ˆ		(4.119)				−
[Lz,pˆz] = 0		(4.119)			ˆ2	ˆ			(4.126)
					[L	,L±] = 0			(4.126)
ˆ2	ˆ	ˆ2	ˆ	ˆ2 ˆ			(4.127)
[L ,Lx] = [L ,Ly] = [L ,Lz] = 0							(4.127)

aThe commutation of a and b is deﬁned as [a,b] = ab− ba (see page 26). Similar expressions hold for S and J. bFor motion under a central force.

4.5 Angular momentum	99

Clebsch–Gordan coe cientsa

−

l1,

−

m1;l2,

−

= (

−

1)l1+l2−j

j,mj

l1,m1;l2,m2

+3/2

1/2 1/2

1 1/2

3/2

+1/2

/ +1/2

+1 +1/2

3/2 1/2

+1 2×

+1/2 −1/2

1/2

+1 −1/2

1/3

2/3

...

1/2 +1/2

1/2 1/2

0 +1 / 2

2/3

1/3

−

m1×

coe cients

−

j,mj |l1,m1;l2,m2

m1 m2

+5/2

. .

3/2 1/2

. .

2 1/2

5/2

+3/2

. .

/ +1/2

+2 +1/2

5/2 3/2

+3 2×

+3/2 −1/2

1/4 3/4

+2 −1/2

1/5 4/5

+1/2

+1/2 +1/2

3/4 −1/4

+1 +1/2

4/5 −1/5

5/2 3/2

+1/2 −1/2

1/2 1/2

+1 −1/2

2/5 3/5

−1/2 +1/2

1/2 −1/2

+5/2

0 +1/2

3/5 −2/5

1 1

3/2 1

5/2

+3/2

+1 +1

5/2 3/2

+3 2×

+1 0

1/2 1/2

+3/2 0

2/5 3/5

+1/2

0 +1

1/2 −1/2

+1/2 +1

3/5 −2/5

5/2 3/2 1/2

+1 −1

1/6 1/2 1/3

+3/2 −1

1/10 2/5 1/2

0 0

2/3 0 −1/3

1/2 0

3/5 1/15 −1/3

−1 +1

1/6 −1/2 1/3

3/2 3/2

−1/2 +1

3/10 −8/15 1/6

/ +3/2

+3 2×

+2 +1

3 2

+3/2 +1/2

1/2 1/2

+2 0

1/3 2/3

+1/2 +3/2

1/2 −1/2

+1 +1

2/3 −1/3

+3/2 −1/2

1/5 1/2 3/10

+2 −1

1/15 1/3 3/5

+1/2 +1/2

3/5 0 −2/5

+1 0

8/15 1/6 −3/10

−1/2 +3/2

1/5 −1/2 3/10

0 +1

6/15 −1/2 1/10

+3/2 −3/2

1/20 1/4 9/20 1/4

+1 −1

1/5 1/2 3/10

+1/2 −1/2

9/20 1/4 −1/20 −1/4

0 0

3/5 0 −2/5

−1/2 +1/2

9/20 −1/4 −1/20 1/4

−1 +1

1/5 −1/2 3/10

−3/2 +3/2

1/20 −1/4 9/20 −1/4

+7/2

2 3/2

7/2

+5/2

+2 +3/2

7/2 5/2

+2 +1/2

3/7 4/7

+3/2

+1 +3/2

4/7 −3/7

7/2 5/2 3/2

+2 +2

+2 −1/2

1/7 16/35 2/5

+2 +1

1/2

+1 +1/2

4/7 1/35 −2/5

+1/2

+1 +2

1/2 −1/2

0 +3/2

2/7 −18/35 1/5

7/2

5/2 3/2 1/2

+2 0

3/14 1/2 2/7

+2 −3/2

1/35 6/35 2/5 2/5

+1 +1

4/7 0 −3/7

+1 −1/2

12/35 5/14

0 −3/10

0 +2

3/14 −1/2 2/7

0 +1/2

18/35 −3/35 −1/5 1/5

+2 −1

1/14 3/10 3/7

1/5

−1 +3/2

4/35 −27/70 2/5 −1/10

+1 0

3/7 1/5 −1/14 −3/10

0 +1

3/7 −1/5 −1/14 3/10

−1 +2

1/14 −3/10 3/7 −1/5

+2 −2

1/70 1/10 2/7 2/5 1/5

+1 −1

8/35 2/5 1/14 −1/10 −1/5

0 0

18/35 0 −2/7 0

1/5

−1 +1

8/35 −2/5 1/14 1/10 −1/5

−2 +2

1/70 −1/10 2/7 −2/5 1/5

aOr “Wigner coe cients,” using

the

Condon–Shortley

sign

convention. Note that

square

root is assumed

over all coe cient digits, so

that

“−3/10” corresponds

to −

≥ 0 are

3/10.

Also for clarity,

only values of mj

listed here. The coe cients

for

0 can be obtained

from

the

symmetry relation

j, m

(−1)l1+l2−j j,mj |l1,m1;l2,m2 .

(

−

100 Quantum physics

Angular momentum additiona

J = L + S

(4.128)

J ,J

total angular momentum

= Lz + Sz

(4.129)

L,L

orbital angular

ˆ2

momentum

+ 2L S

(4.130)

Total angular

= L

+ S

S ,S

spin angular momentum

= mj ¯hψj,m/j ·

momentum

Jˆzψj,mj

(4.131)

eigenfunctions

Jˆ2ψj,mj

= j(j + 1)¯h2ψj,mj

(4.132)

magnetic quantum

number |mj | ≤ j

j-multiplicity = (2l + 1)(2s+ 1)

(4.133)

(l + s) ≥ j ≥ |l − s|

Mutually

,Jz,L · S }

(4.134)

commuting

{}

set of mutually

,Lz,Sz,Jz}

(4.135)

commuting observables

sets

Clebsch–

|j,mj =

j,mj |l,ml;s,ms |l,ml |s,ms

|·

eigenstates

Gordan

·|·

Clebsch–Gordan

coe cientsb

coe cients

+m =m

(4.136)

bSumming spin and orbital angular momenta as examples, eigenstates |s,ms and |l,ml .

Or “Wigner coe cients.” Assuming no L–S interaction.

Magnetic moments

µB

Bohr magneton

µB =

e¯h

(4.137)

−e

electronic charge

2me

¯h

(Planck constant)/(2π)

electron mass

Gyromagnetic

γ =

orbital magnetic moment

(4.138)

gyromagnetic ratio

ratioa

orbital angular momentum

Electron orbital

γe =

−¯hµB

(4.139)

gyromagnetic

−e

γe

electron gyromagnetic ratio

ratio

(4.140)

2me

Spin magnetic

µe,z = −geµBms

(4.141)

µe,z

z component of spin

¯h

magnetic moment

moment of an

= ±geγe

(4.142)

electron g-factor (

2.002)

electronb

gee¯h

= ±

(4.143)

spin quantum number (±1/2)

4me

µJ

total magnetic moment

µJ = gJ

J(J + 1)µB

(4.144)

J,z =

(4.145)

µJ,z

z component of µJ

magnetic quantum number

Lande´ g-factorc

−

g µ

J(J + 1) + S(S + 1) − L(L+ 1)

total, orbital, and spin

gJ = 1 +

J,L,S

quantum numbers

2J(J + 1)

(4.146)

Lande´ g-factor

aOr “magnetogyric ratio.”

bThe electron g-factor equals exactly 2 in Dirac theory. The modiﬁcation ge = 2 + α/π + ..., where α is the ﬁne structure constant, comes from quantum electrodynamics.

cRelating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming ge = 2.

4.5 Angular momentum	101

Quantum paramagnetism

		1
		0.8	B∞(x) = L(x)
		0.6	B∞(x) = L(x)
		0.4	B4(x)
			B1(x)
		0.2	B1/2(x) = tanhx
		0
−10	−5	−0.2	5	x	10
		−0.4

−0.6

−0.8

−1

BJ (x) =

2J + 1

(2J + 1)x

coth

−

coth

(4.147)

Brillouin

J + 1

BJ (x)

Brillouin function

(x 1)

total angular momentum

J (x)

(4.148)

quantum number

function

L(x)

(x)

Langevin function

(4.149)

= cothx

−

1/x (see page 144)

x = tanhx

1/2(

)

mean magnetisation

number density of atoms

Lande´ g-factor

Mean

µBB

(4.150)

magnetic ﬂux density

magnetisationa

M = nµBJgJ BJ JgJ kT

Bohr magneton

Boltzmann constant

M for isolated

µBB

temperature

spins

(J = 1/2)

M 1/2 = nµB tanh

(4.151)

M 1/2 mean magnetisation for

J = 1/2 (and gJ = 2)

aOf an ensemble of atoms in thermal equilibrium at temperature T , each with total angular momentum quantum number J.

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