1.16 Two Particles in 3 Dimensions
So far we have been working with states of just one particle in one dimension. The extension to two
different particles and to three dimensions (See section 12) is straightforward. The coordinates and
momenta of different particles and of the additional dimensionscommute with each otheras we
might expect from classical physics. The only things that don’t commute are a coordinate with its
momentum, for example,
[p(2)z,z(2)] =̄h
i
while
[p(1)x,x(2)] = [p(2)z,y(2)] = 0.
We may write states for two particles which are uncorrelated, likeu 0 (~x(1))u 3 (~x(2)), or we may write
states in which the particles are correlated. The Hamiltonian for twoparticles in 3 dimensions simply
becomes
H=
− ̄h^2
2 m(1)(
∂^2
∂x^2 (1)+
∂^2
∂y^2 (1)+
∂^2
∂z^2 (1))
+
− ̄h^2
2 m(2)(
∂^2
∂x^2 (2)+
∂^2
∂y^2 (2)+
∂^2
∂z^2 (2))
+V(~x(1),~x(2))H=
− ̄h^2
2 m(1)∇^2 (1)+
− ̄h^2
2 m(2)∇^2 (1)+V(~x(1),~x(2))If two particles interact with each other, with no external potential,
H=
− ̄h^2
2 m(1)∇^2 (1)+
− ̄h^2
2 m(2)
∇^2 (1)+V(~x(1)−~x(2))the Hamiltonian has atranslational symmetry, and remains invariant under the translation~x→
~x+~a. We can show that this translational symmetry impliesconservation of total momentum.
Similarly, we will show that rotational symmetry implies conservation of angular momentum, and
that time symmetry implies conservation of energy.
For two particles interacting through a potential that depends only on difference on the coordinates,
H=~p^21
2 m+
~p^22
2 m+V(~r 1 −~r 2 )we can make the usual transformation to the center of mass (Seesection 12.3) made in classical
mechanics
~r≡~r 1 −~r 2
R~≡m^1 ~r^1 +m^2 ~r^2
m 1 +m 2
and reduce the problem to the CM moving like a free particle
M=m 1 +m 2H=− ̄h^2
2 M∇~^2 R
plus one potential problem in 3 dimensions with the usual reduced mass.
1
μ=
1
m 1+
1
m 2H=−̄h^2
2 μ∇~^2 r+V(~r)So we are now left with a 3D problem to solve (3 variables instead of 6).