1. Hydrogen relativistic effects II
a. relativistic mass effect
Total Energy
𝑯= ( 𝒑 𝟐 𝒄 𝟐 + 𝒎 𝟐 𝒄 𝟒 ) 𝟏/𝟐 + 𝑽(𝒓)
Kinetic Energy
𝑬 𝒌𝒊𝒏
= ( 𝒑 𝟐 𝒄 𝟐 + 𝒎 𝟐 𝒄 𝟒 ) 𝟏/𝟐 − 𝒎 𝒄 𝟐
~ 𝒑 𝟐 𝟐𝒎 − 𝟏 𝟖 𝒑 𝟒 𝒎 𝟑 𝒄 𝟐 +…
Correction Term
− 𝟏 𝟖 𝒑 𝟒 𝒎 𝟑 𝒄 𝟐
=− 𝟏 𝟐𝒎 𝒄 𝟐 𝒑 𝟐 𝟐𝒎 𝟐
=− 𝟏 𝟐𝒎 𝒄 𝟐 𝑬 𝒌𝒊𝒏 𝟎 𝟐
=− 𝟏 𝟐𝒎 𝒄 𝟐 𝑬 𝒏 −𝑽 𝒓 𝟐

2. = − 𝟏 𝟐𝒎 𝒄 𝟐 𝑬 𝒏 𝟐 −𝟐 𝑬 𝒏 − 𝒁 𝒆 𝟐 𝟒𝝅 𝜺 𝟎 𝒓 + 𝒁 𝟐 𝒆 𝟒 𝟒𝝅 𝜺 𝟎 𝟐 𝒓 𝟐
Energy Shift
∆ 𝑬′ 𝒏
=− 𝟏 𝟐𝒎 𝒄 𝟐 𝑬 𝒏 −𝑽 𝒓 𝟐
= − 𝟏 𝟐𝒎 𝒄 𝟐 𝑬 𝒏 𝟐 −𝟐 𝑬 𝒏 − 𝒁 𝒆 𝟐 𝟒𝝅 𝜺 𝟎 𝒓 + 𝒁 𝟐 𝒆 𝟒 𝟒𝝅 𝜺 𝟎 𝟐 𝒓 𝟐
with
𝟏 𝒓 = 𝟏 𝒏 𝟐 ∙ 𝒁 𝒂 𝟎 , 𝟏 𝒓 𝟐 = 𝟏 ℓ+ 𝟏 𝟐 𝒏 𝟑 𝒁 𝒂 𝟎 𝟐 , 𝑬 𝒏 =− 𝟏 𝟒𝝅 𝜺 𝟎 ∙ 𝒁 𝟐 𝒆 𝟐 𝟐 𝒂 𝟎 ∙ 𝟏 𝒏 𝟐
∆ 𝑬 𝒏 ′ =− 𝜶 𝟐 𝒁 𝟐 𝒏 𝟐 𝑬 𝒏 𝟑 𝟒 − 𝒏 ℓ+𝟏/𝟐
The relativistic mass effect depends on l and n. It is ~
𝜶 𝟐 𝒁 𝟐
or ~ 𝑣 2 / 𝑐 2 smaller than 𝑬 𝒏 .Therefore larger for heavy atoms with large

3. Spin-orbit interaction
Correction term depends on spin
The electron moves in the electric of 𝐸 field of the nucleus .Therefore, in the rest frame of the electron exists a magnetic field 𝐸 = 1 𝐶 2 𝑣 × 𝐸
with
The name spin-orbit is due to 𝑙∙𝑠
Energy shift
with
<𝑙∙𝑠> is not diagonal in 𝑙, 𝑚 𝑙 ,𝑠, 𝑚 𝑠 ,but in 𝑗, 𝑚 𝑗 , 𝑙 2 , 𝑠 2

4. New coupled total angular momentum j
𝑗=ℓ+𝑠
𝑗 2 = ℓ 2 +2ℓ∙𝑠+ 𝑠 2
ℓ∙𝑠= 1 2 ( 𝑗 2 − ℓ 2 − 𝑠 2 )
ℓ∙𝑠 = ℎ 2 2 {𝑗 𝑗+1 −ℓ ℓ+1 −𝑠 𝑠+1 }
The relativistic mass effect and the spin-orbit interaction can be put together

5. c.Darwin-Term − ℎ 2 4 𝑚 2 𝑐 2 𝑒𝑬∙𝛻 ℓ=0
− ℎ 2 4 𝑚 2 𝑐 2 𝑒𝑬∙𝛻
ℓ=0
This term has no classical analogue
∆𝐸′′′
= 𝜋 ℎ 2 2 𝑚 2 𝑐 2 𝑍 𝑒 2 4𝜋 𝜀 0 Ψ 0 2
=− α 2 𝑍 2 𝑛 2 𝐸 𝑛 𝑛

6. Total relativistic energy shift note: for l=0 there is no spin –orbit interaction, put together in one expression which is also valid for l=0 we have
∆𝐸=∆ 𝐸 ′ +∆ 𝐸 ′′ +∆ 𝐸 ′′′ =− 𝛼 2 𝑍 2 𝑛 2 𝐸 𝑛 − 𝑛 𝑗+1/2
𝑗=ℓ± 1 2
note: Total energy shift does not depend on l(even though the individual terms do). Therefore terms with the same j are degenerate. The lambshift(discussed next removes this degeneracy

7. Fine-structure of the n=2 and n=3 levels of hydrogen according to the Dirac theory for the Balmer – α line at 656 nm (red)

9. Fine-structure of the n=2 terms with Lamb-shift note: degeneracy in 𝑙 is removed The indicated transition frequency is 1058 MHz

10. Lamb-shift The electromagnetic field is quantized in QED
Lamb-shift The electromagnetic field is quantized in QED. A quantum mechanical harmonic oscillator has fluctuations in the ground state=Zitterbewegung =vacuum fluctuations ,

11. g-factor of the electron