Physical Chemistry Third Edition

17.1 The Hydrogen Atom and the Central Force System 727

wherevcandvare the velocity of the center of mass and the relative velocity:

vcx

dxc

dt

and vx

dx

dt

(17.1-7)

with similar equations forvcy,vcz,vy, andvz. Thereduced massμis defined by

μ

mnme

mn+me



mnme

M

(17.1-8)

Since the nucleus is much more massive than the electron, the reduced mass is nearly

equal to the electron mass and the center of mass is close to the nucleus.

In order to construct the Hamiltonian operator for the hydrogen atom we must

express the kinetic energy in terms of momentum components. The center of mass

momentum components are

pcxMvcx, pcyMvcy, pczMvcz (17.1-9)

The components of the relative momentum are

pxμvx, pyμvy, pzμvz (17.1-10)

The classical Hamiltonian is

Hcl

1

2 M

(

p^2 cx+p^2 cy+p^2 cz

)

+

1

2 μ

(

p^2 x+p^2 y+p^2 z

)

+V(r) (17.1-11)

whereris now expressed in terms ofx,y, andz:

r(x^2 +y^2 +z^2 )^1 /^2 (17.1-12)

The Hamiltonian operator is obtained by the replacements specified in Eq. (16.3-8) and

its analogues:

Ĥ−h ̄

2

2 M

(

∂^2

∂x^2 c

+

∂^2

∂y^2 c

+

∂^2

∂z^2 c

)

− ̄

h^2

2 μ

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

+V(r)

−

h ̄^2

2 M

∇^2 c−

h ̄^2

2 μ

∇^2 +V(r) (17.1-13)

The two operators∇c^2 and∇^2 areLaplacian operators, as defined in Eq. (15.2-26) and

in Eq. (B-45) of Appendix B.

The first term in the Hamiltonian operator is thecenter-of-mass Hamiltonian

Ĥc−h ̄

2

2 M

∇^2 c (17.1-14)

and the other two terms are therelative Hamiltonian

Ĥrel−h ̄

2

2 μ

∇^2 +V(r) (17.1-15)

The time-independent Schrödinger equation is

(

̂Hc+̂Hrel

)

ΨEΨ (17.1-16)