130_notes.dvi

12.2 Quantum Mechanics in Three Dimensions

We have generalized Quantum Mechanics to include more than one particle. We now wish to include
more than one dimension too.

Additional dimensions are essentially independent although they maybe coupled through the po-
tential. The coordinates and momenta from different dimensions commute. The fact that the
commutators are zero can be calculated from the operators thatwe know. For example,

[x,py] = [x,

̄h

i

∂

∂y

] = 0.

The kinetic energy can simply be added and the potential now depends on 3 coordinates. The
Hamiltonian in 3Dis

H=

p^2 x

2 m

+

p^2 y

2 m

+

p^2 z

2 m

+V(~r) =

p^2

2 m

+V(~r) =−

̄h^2

2 m

∇^2 +V(~r).

This extension is really very simple.

12.3 Two Particles in Three Dimensions

The generalization of the Hamiltonian to three dimensions is simple.

H=

~p^21

2 m

+

~p^22

2 m

+V(~r 1 −~r 2 )

We define the vectordifference between the coordinatesof the particles.

~r≡~r 1 −~r 2

We also define the vectorposition of the center of mass.

R~≡m^1 ~r^1 +m^2 ~r^2

m 1 +m 2

We will use the chain rule to transform our Hamiltonian. As a simple example, if we were working
in one dimension we might use the chain rule like this.

d

dr 1

=

∂r

∂r 1

∂

∂r

+

∂R

∂r 1

∂

∂R

In three dimensions we would have.

∇~ 1 =∇~r+ m^1

m 1 +m 2

~∇R