Formula of STR and Sommerfeld (SO)

This Paper

Kinetic Energy

K r e = m c 2 − m 0 c 2 . (18)

K r e = p r e 2 m 0 + m . (23)

K r e , n = m e c 2 − m n c 2 . (26)

K r e , n = p r e , n 2 m e + m n . (29)

Energy-Momentum Relationship

( m 0 c 2 ) 2 + p 2 c 2 = ( m c 2 ) . (17)

Holds in isolated systems in free space

( m e c 2 ) 2 − p n 2 c 2 = ( m n c 2 ) 2 . (25)

Applicable to an electron in a hydrogen atom

Energy Levels of a Hydrogen Atom

E S O = − K r e = m 0 c 2 − m c 2 . (18)

E SO , n = m e c 2 [ 1 − ( n 2 n 2 − α 2 ) 1 / 2 ] , n = 1 , 2 , ⋯ . *

E SO , n ≈ − α 2 m e c 2 2 n 2 ( 1 + 3 α 2 4 n 2 + 5 8 α 4 n 4 ) . *

E re , n = − K re , n = m n c 2 − m e c 2 . (34)

E re , n = m e c 2 [ ( n 2 n 2 + α 2 ) 1 / 2 − 1 ] , n = 0 , 1 , 2 , ⋯ . (40b)

E re , n ≈ − α 2 m e c 2 2 n 2 ( 1 − 3 α 2 4 n 2 + 5 α 4 8 n 4 ) . (41c)

E 1

E SO , 1 = − 13.60624   eV . *

E re , 1 = − 13.60515   eV . *