(英文)量子力学-氢原子

Electron
The potential arises from the Coulomb interaction between the charged particles.
Ze 2 V 4 0 r Ze 2 4 0 ( x2 x1 ) ( y2 y1 ) ( z2 z1 )


m1 m2 m1 m2
2 [ ET V r , , ] T 0 2
reduced mass
Try solution
T ( x, y, z , r , , ) F ( x, y, z ) ( r , , )
Substitute and divide by T

This term only depends on center of mass coordinates. Other terms only on relative coordinates.
1 1 wk.baidu.com2 T r r2 r r
1 2 T 1 T r 2 sin 2 2 r 2 sin sin
The Hydrogen Atom
The only atom that can be solved exactly. The results become the basis for understanding all other atoms and molecules.
Nucleus charge +Ze mass m1 coordinates x1, y1, z1 charge –e coordinates x2, y2, z2 mass m2
relative position – polar coordinates
Substituting these into the Schrö dinger equation. Change differential operators.
2T 2T 2T 1 2 2 m1 m2 x y z2
x y z
m1 x1 m2 x2 m1 m2 m1 y1 m2 y2 m1 m2 m1 z1 m2 z2 m1 m2
center of mass coordinates
r sin cos x2 x1 r sin sin y2 y1 r cos z2 z1 .
Gives two independent equations
2 F 2 F 2 F 2( m1 m2 ) ETr F 0 2 2 2 2 x y z
Depends only on center of mass coordinates. Translation of entire atom as free particle. Will not treat further.
1 d 2 d R 1 d 2 1 d d 8 2 r sin 2 [ E V ( r )] 0 2 2 2 2 2 R r d r d r r sin d r sin d d
Multiply by r 2 sin 2 . Then second term only depends .
kinetic energy of nucleus kinetic energy of electron
potential
energy eigenvalues Can separate translational motion of the entire atom from relative motion of nucleus and electron. Introduce new coordinates x, y, z - center of mass coordinates r, , - polar coordinates of second particle relative to the first
sin 2 d 2 d R 1 d 2 sin d d 8 2 r 2 sin 2 r sin [ E V ( r )] 0 2 2 R d r d r d d d
Relative positions of particles. Internal “structure” of H atom.
In absence of external field V = V(r) Try ( r , , ) R( r ) ( ) ( )
Substitute this into the equation and dividing by R yields
1 2 1 2 1 r sin r 2 sin 2 2 r 2 sin r2 r r
2 [ E V ( r , , )] 0 2
With
ET ETr E
2 2 2 1 2
The Schrö dinger equation for the hydrogen atom is
1 2 T 2 T 2 T 1 2T 2T 2T 2 2 ( ET V ) T 0 2 2 2 2 2 m1 x12 m y1 z1 y2 z2 e x2