H Atom resume

H Atom resume

𝐷 = 𝐴 ⃗

2𝑚|𝐸

| = 1

2𝑚|𝐸

|

1

2 ⃗𝑝×𝐿 − 𝐿× ⃗𝑝 − mk ̂𝑟

[𝐻

𝐷] = [𝐻

𝐿] = 0; [𝐿

𝐷

] = 𝑖𝜀

𝐷

; [𝐿

𝐷

] = 𝑖𝜀

𝐷

; [𝐿

𝐿

] = 𝑖𝜀

𝐿

𝐽

=

(𝐿 − 𝐷 ), 𝐽

=

(𝐿 + 𝐷 ); 𝐽

, 𝐽

= 0 𝐽

𝐽, 𝑚1, 𝑚2 = 𝐽 𝐽 + 1 𝐽, 𝑚

, 𝑚

𝐽

𝐽, 𝑚1, 𝑚2 = 𝑚

𝐽, 𝑚

, 𝑚

𝐿

𝐽, 𝑚1, 𝑚2 = (𝑚

+𝑚

) 𝐽, 𝑚

, 𝑚

𝐷

𝐽, 𝑚1, 𝑚2 = (𝑚

−𝑚

) 𝐽, 𝑚

, 𝑚

. 𝐽

+ 𝐽

= 𝐿

Possible basis choices

• Directions: instead of projections on 2 arbitrary directions 𝑛

• But also (𝐿

,𝐿

) corresp. to choice: (1,2,3) – real space, rot. in (1,2)- 𝐿

• Other possible simple rotations in real space (1,3) - 𝐿

, (2,3) - 𝐿

• But also we can take (1,2,4) with (1,2)- 𝐿

, (1,4)- 𝐷

, (2,4)- 𝐷

the “square”= 𝐿

+ 𝐷

+ 𝐷

and one of 3 choices

Similarly (1,3,4) with the “square”= 𝐿

+ 𝐷

+ 𝐷

and one of 3 possible choices

Similarly (1,2,4) 𝐷

+ 𝐷

+ 𝐿

-- the so called Lambda basis

𝐽_!", 𝐽_#"

• H atom in an external electric field

In dipole approximation an external electric field perturbes by:

V = e ⃗ 𝐹. ⃗𝑟

The perturbation analysed using the Pauli Replacement.

Degenerate n-manifold – for weak field we restrict to it For an operator 𝑇 = ⃗𝑟 ⃗𝑟 ⋅ ⃗𝑝 − ⃗𝑝 ⃗𝑟 ⋅ ⃗𝑟 + 𝑖ℏ⃗𝑟, we have

2𝐻_!⃗𝑟m = ^"_# 𝐴 +⃗ _%ℏ^$ [𝐻_!, 𝑇]

The expectation value.

2𝐸_'𝑚 ⃗𝑟 = ^"

#⟨ ⃗𝐴⟩

Thus, in atomic units we have

⃗𝑟 = − 3

2 𝑛^#⟨ ⃗𝐴⟩

The expectation value of The commutator with n manifold is zero.

• H in an external electric field (Linear Stark- Lo Surdo effect)

Field along z: On making the replacement we have 𝑉 = − ^"

#𝑛^#𝐹𝐴₍ = − ^"

#𝑛 𝐹 𝐷₍ In terms of the state |𝑗 𝑚_$𝑚_#⟩ we have

𝐿₍ 𝑗 𝑚_$𝑚_# = 𝑚_$ + 𝑚_# 𝑗 𝑚_$𝑚_# =M ^{𝑗 𝑚}_$^𝑚_# 𝐷₍ 𝑗 𝑚_$𝑚_# = 𝑚_# − 𝑚_$ |𝑗 𝑚_$𝑚_#⟩

Where, −𝑗 ≤ 𝑚_$, 𝑚_# ≤ 𝑗 = ^')$

# .

The energy eigenvalues are then given by 𝐸 𝑛, 𝑘 = − _#'^$_! + ^"_# 𝑛𝑘𝐹. Where, 𝑘 = 𝑚_$ − 𝑚_# which take the values − 𝑛 − 1 ≤ 𝑘 ≤ (𝑛 − 1) For a given 𝑘 we have 𝑛 − |𝑘| degenerate values.

If we have the electric field in a general direction, we have 𝑉_' = 𝜔_* ⋅ 𝐷 𝜔_* = ^"_#𝑛𝐹 in a.m.u and 𝜔_* = ^"_#^+,-_./^" 𝑛𝐹 in M.K.S units.

For given M Stark multiplet – 𝑛 − |𝑀| levels separated by 2𝜔_* Circular states 𝑀 = 𝑛 − 1; 𝑘 = 0 not perturbed by F

H in F

𝐸 𝑛, 𝑘 = − 1

2𝑛^# + 3

2 𝑛𝑘𝐹

𝑛 − |𝑘| degenerate values

− 𝑛 − 1 ≤ 𝑘 ≤ (𝑛 − 1)

• H atom in presence of external magnetic field.

In presence of an external magnetic field the Hamiltonian for a 2-particle system (with opposite charges 𝑒) is given as ℋ = ^{0#)/ ⃗2 3#}

!

#._# + ^{0!4/ ⃗2 3!}

!

#._! − ^$

+,-_"

/^! 3_# )3_!

Define Π_% ≡ 𝑝_% ∓ 𝑒 ⃗𝐴 𝑟_% = 𝑚𝑣_% - the kinetic momentum.

For 𝐵 along the z-direction we have Π_$5, Π_$6 = 𝑖ℏ𝑒𝐵 Π_#5, Π_#6 = −𝑖ℏ𝑒𝐵 , Assume gauge 𝐴 =⃗ ^$

# 𝐵×⃗𝑟

A change of coordinates Π₇₈ = Π_$ + Π_#, p = ^.!_.^9#).#9!

#4._! and ⃗𝑟 = ⃗𝑟_$- ⃗𝑟_# Leads to a Hamiltonian of the form ℋ = ^9$%!

#8 + _#.^:! − ^$

+,-_"

/^! 3

with, 𝑀 = 𝑚_$ + 𝑚_# & 𝑚 = _.^.#.!

#4._!

Defining an operator 𝐶 ≡ Π⃗ ₇₈ − 𝑒 ⃗𝑟×𝐵 = 𝑃 − ^/_# ⃗𝑟×𝐵 one finds [𝐻, ⃗𝐶] = 0 with the commutation relation 𝐶_% , 𝐶_; = 0

• Atom in presence of external magnetic field.

The operator ⃗𝐶 can be written as 𝐶 = 𝑃 −⃗ ^/_# ⃗𝑟×𝐵 thus giving 𝑃 = ⃗𝐶 + ^/_# ⃗𝑟×𝐵 Write the wavefunction in the form Ψ 𝑅, ⃗𝑟 = exp _ℏ^% 𝐶 +⃗ ^<_# ⃗𝑟×𝐵 𝑅 𝜙 ⃗𝑟 . We get ℎ ⃗𝑟 𝜙 ⃗𝑟 = 𝐸𝜙 ⃗𝑟

ℎ ⃗𝑟 = ⃗𝑝^#

2𝑚 − 1 4𝜋𝜀_!

𝑒^#

𝑟 + 𝑒 2

𝑚_# − 𝑚_$

𝑚_$𝑚_# 𝐵 ⋅ 𝐿 + 𝑒

𝑀 𝐶×𝐵 ⋅ ⃗𝑟⃗ + ^/!

=. ⃗𝑟×𝐵 ^# + ^7⃗!

#8 /

#

._!)._#

._#._! 𝐵 ⋅ 𝐿 -- Linear Zeeman effect

/^!

=. ⃗𝑟×𝐵 ^#- quadratic, “diamagnetic” Zeeman term

/

8 𝐶×𝐵 ⋅ ⃗𝑟⃗ - electric field in the moving frame

Simple:

We can also have a simple semiclassics for the interaction in magnetic fields ignoring the diamagnetic part. Assuming an 𝑒

moving in an orbit gives an electric current 𝐼 = −

/

we then have the magnetic moment ⃗𝜇 = 𝐼 ⋅ ∮ 𝑑 ⃗𝑎 .

∮ ⃗𝑎 =

(

∫

𝑟

𝑑𝜙 =

(2

∫

𝑑𝜙 =

(2

𝑇 Thus, we have ⃗𝜇 = −

(2

ℓ , 𝜇

∼ ℓ(ℓ + 1)

This is usually represented as ⃗𝜇 = −

ℓ where, 𝜇

→ Bohr Magneton. And we define

=

ℏ

= 𝛾

• Atom in presence of external magnetic field.

Linear Zeeman effect: ℋ ⃗𝑟 =

"#

−

%&'_"

(^!

)

+

"#

ℓ ⋅ 𝐵 For field along z 𝐸 𝑛, 𝑀 = −

+ 𝛾

𝐵𝑀

similar to the E.F case defining 𝜔

= 𝛾

𝐵 .

For arbitrary direction, the energy 𝐻 = 𝐸

+ 𝑉 , 𝑉 → 𝜔

⋅ 𝐿 .

• Both Magnetic and Electric Field (𝐵, ⃗𝐹).

Leads to 𝑉

= 𝜔

⋅ 𝐿 + 𝜔

⋅ 𝐷 with 𝜔

= −

𝐵 & 𝜔

=

𝐹 ⃗ In 𝐽

=

"

(𝐿 − 𝐷) , 𝐽

=

"

(𝐿 + 𝐷) 𝑉

= 𝜔

⋅ 𝐽

+ 𝜔

⋅ 𝐽

Basis of 𝐽

, 𝐽

, 𝐽

, 𝐽

⇒ |𝐽, 𝑚

, 𝑚

⟩ is wrong but 𝑛

= 𝜔

/||𝜔

| In new basis defined by these directions |𝐽, 𝑚

, 𝑚

⟩

𝐸 = −

"*^!

+ 𝑚

𝜔

+ 𝑚

𝜔

. where, −

"

≤ 𝑚

, 𝑚

≤

"

• Atom in presence of external fields.

In case 𝐹 ⊥ 𝐵⃗ , we have 𝜔_$ = 𝜔_# = 𝜔_>^# + 𝜔_*^# with energy given as 𝐸_'? = − 1

2𝑛^# + 𝑘 𝜔_>^# + 𝜔_*^#

Where, − 𝑛 − 1 ≤ 𝑘 = 𝑚_$ + 𝑚_# ≤ 𝑛 − 1 with degeneracy 𝑛 − 𝑘 On defining an operator ⃗𝜆 = 𝐷₅, 𝐷₆, 𝐿₍

𝜆₅ = 𝐷₅ ⟼ 𝑆𝑡𝑎𝑟𝑘 𝐸𝑓𝑓𝑒𝑐𝑡 ⟼ ⃗𝐹 along 𝑥 𝜆₍ = 𝐿₍ ⟼ 𝑍𝑒𝑒𝑚𝑎𝑛 𝐸𝑓𝑓𝑒𝑐𝑡 ⟼ 𝐵 along 𝑧

We then have 𝑉 = ⃗𝜆 ⋅ 𝜔_# = ⃗𝜆 ⋅ 𝜔_> + 𝜔_* , giving the basis states |𝑛, 𝜆^#, ⃗𝜆 ⋅ 𝜔_#⟩ . For a given 𝑘, values of 𝜆 are such that 𝑘 ≤ 𝜆 ≤ 𝑛 − 1

Zeeman and linear Stark effects are both contained as limiting cases.

𝜆₅ ⃗𝜆 ⋅ 𝜔_# 𝜆₍

Stark Zeeman

The eigenfunctions of 𝜆^#, 𝜆_@ continuously evolve from one limit to other.

From right to left and n around 60

Large electric field - separation still possible

E.g. the semi-parabolic coordinates 𝑢 = 𝑟 + 𝑧, 𝑣 = 𝑟 − 𝑧

@^!4A^!

# = 𝑟 , ^@!)A!

# = z.

We get the Schr. Eqn. _{# @}^0&!_!⁴⁰_4A^'!_! − _@!4A^# _! + 𝐹 ^@!)A_# ^! 𝜓 = 𝐸𝜓 with,

𝑝_@^# = − _B@^B!_! + _@^$ _B@^B + _#@^8!_! , 𝑝_A^# = − _BA^B!_! + ^$_{A BA}^B + _#A^8!_! for the ansatz. Ψ = 𝜓 𝑢, 𝑣 𝑒^%8C

For 𝐹 = 0, ^0&!40'!

# − E u^# + v^# Ψ = 2Ψ

𝐸 is related to 𝜔 as ^D_#^! = −𝐸 ⇒ 𝜔 = −2𝐸, and thus we have, 2𝑛_@ + 𝑀 + 1 −2𝐸 + 2𝑛_A + 𝑀 + 1 −2𝐸 = 2 and Energy

𝐸 = − 1

2 𝑛_@ + 𝑛_A + 𝑀 + 1 ^#

n M (𝟐𝒏 + 𝑴 + 𝟏) 𝝎 Degeneracies

0 0 𝝎 1

0 ± 1 2𝝎 2

0 ± 2 3𝝎

1 0 3𝝎

0 ± 3 4𝝎

0 ± 1 4𝝎

3

4 2𝑛_@ + 𝑀 + 1 −2𝐸 + 2𝑛_A + 𝑀 + 1 −2𝐸 = 2 with

𝐸 = − 1

2 𝑛_@ + 𝑛_A + 𝑀 + 1 ^#

H in strong electric field

𝑝_@^# + 𝑝_A^#

2 − E u^# + v^# + 𝐹 𝑢⁺ − 𝑣⁺

2 Ψ = 2Ψ