10.3 The energy-momentum relation and the Sch-
r ̈odinger equation for a free particle
We will review this subject in chapter 40 but for now we just need the rela-
tionship special relativity posits between energy and momentum. Space and
time are put together in “Minkowski space”, which isR^4 with indefinite inner
product
((u 0 ,u 1 ,u 2 ,u 3 ),(v 0 ,v 1 ,v 2 ,v 3 )) =−u 0 v 0 +u 1 v 1 +u 2 v 2 +u 3 v 3Energy and momentum are the components of a Minkowski space vector (p 0 =
E,p 1 ,p 2 ,p 3 ) with norm-squared given by minus the mass-squared:
((E,p 1 ,p 2 ,p 3 ),(E,p 1 ,p 2 ,p 3 )) =−E^2 +|p|^2 =−m^2This is the formula for a choice of space and time units such that the speed of
light is 1. Putting in factors of the speed of lightcto get the units right one
has
E^2 −|p|^2 c^2 =m^2 c^4
Two special cases of this are:
- For photons,m= 0, and one has the energy momentum relationE=|p|c
- For velocitiesvsmall compared toc(and thus momenta|p|small com-
pared tomc), one has
E=
√
|p|^2 c^2 +m^2 c^4 =c√
|p|^2 +m^2 c^2 ≈
c|p|^2
2 mc+mc^2 =
|p|^2
2 m+mc^2In the non-relativistic limit, we use this energy-momentum relation to
describe particles with velocities small compared toc, typically dropping
the momentum-independent constant termmc^2.In later chapters we will discuss quantum systems that describe photons,
as well as other possible ways of constructing quantum systems for relativistic
particles. For now though, we will just consider the non-relativistic case. To
describe a quantum non-relativistic particle we choose a Hamiltonian operator
Hsuch that its eigenvalues (the energies) will be related to the momentum
operator eigenvalues (the momenta) by the classical energy-momentum relation
E=|p|
2
2 m:H=
1
2 m(P 12 +P 22 +P 32 ) =
1
2 m|P|^2 =
−~^2
2 m(
∂^2
∂q 12+
∂^2
∂q^22+
∂^2
∂q^23)
The Schr ̈odinger equation then becomes:i~∂
∂tψ(q,t) =−~^2
2 m(
∂^2
∂q 12+
∂^2
∂q^22+
∂^2
∂q^23)
ψ(q,t) =−~^2
2 m∇^2 ψ(q,t)