130_notes.dvi

∇~ 2 =−~∇r+ m^2

m 1 +m 2

∇~R

Putting this into theHamiltonianwe get

H=

− ̄h^2

2 m 1

[

~∇^2 r+

(

m 1

m 1 +m 2

) 2

∇~^2 R+^2 m^1

m 1 +m 2

∇~r·∇~R

]

+

− ̄h^2

2 m 2

[

∇~^2 r+

(

m 2

m 1 +m 2

) 2

~∇^2 R−^2 m^2

m 1 +m 2

~∇r·~∇R

]

+V(~r)

H=− ̄h^2

[(

1

2 m 1

+

1

2 m 2

)

∇~^2 r+ m^1 +m^2

2(m 1 +m 2 )^2

∇~^2 R

]

+V(~r).

Defining thereduced massμ

1

μ

=

1

m 1

+

1

m 2

and the total mass

M=m 1 +m 2

we get.

H=−

̄h^2

2 μ

∇~^2 r− ̄h

2

2 M

∇~^2 R+V(~r)

The Hamiltonian actuallyseparates into two problems: the motion of thecenter of massas a
free particle

H=

− ̄h^2

2 M

∇~^2 R

and theinteraction between the two particles.

H=−

̄h^2

2 μ

∇~^2 r+V(~r)

This is exactly the same separation that we would make in classical physics.