Understanding Quantum Physics: An Advanced Guide for the Perplexed

form
    ψ(x, t) = ϕ E (x)e −iEt , (3.9)
    where E is a
constant, and ϕ E (x) is the eigenstate of H and satisfies the
time-independent equation:
    H ϕ E (x) = E ϕ E (x).
(3.10)
    The commutative
relation [ P, H] = 0 further implies that P and H have common eigenstates. This
means that ϕ E (x) is also the eigenstate of P . Since the eigenstate
of P = −i∂/∂x is e ipx , where p is a real eigenvalue, the solution of
the evolution equation Eq.(3.5) for an isolated system will be e i(px−Et) .
In quantum mechanics, P and H, the generators of space translation and time
translation, are also called momentum operator and energy operator, respectively.
Correspondingly, e i(px−Et) is the eigenstate of both momentum and
energy, and p and E are the corresponding momentum and energy eigenvalues,
respectively. In other words, the state e i(px−Et) describes an
isolated system (e.g. a free microscopic particle) with definite momentum p and
energy E.
    3.2 Relativistic invariance
    The relation
between momentum p and energy E can be determined by the relativistic
invariance of the momentum eigenstate e i(px−Et) , and it turns out to
be E 2 = p 2 c 2 + m 2 c 4 ,
where m is the mass of the system, and c is the speed of light [36] . In the nonrelativistic domain, the energy
momentum relation reduces to E = p 2 /2m.
    Now we will derive
the relation between momentum p and energy E in the relativistic domain.
Consider two inertial frames S 0 and S with coordinates x 0 ,
t 0 and x, t. S 0 is moving with velocity v relative to S.
Then x, t and x 0 , t 0 satisfy the Lorentz transformations:

    Suppose the state
of a free particle is ψ = e i(p0x0−E0t0) , an eigenstate of P , in S 0 ,
where p 0 , E 0 is the momentum and energy of the particle
in S 0 , respectively. When described in S by coordinates x, t, the
state is

    This means that in
frame S the state is still the eigenstate of P , and the corresponding momentum
p and energy E is [37]

     
    4 We can also get this result from
the definition Eq. (3.16) by using the above transformations of momentum and
energy Eq.(3.14) and Eq.(3.15).
    We further suppose
that the particle is at rest in frame S 0 . Then the velocity of the
particle is v in frame S 4 . Considering that the velocity of a
particle in the momentum eigenstate e i(px−Et) or a wavepacket
superposed by these eigenstates is defined as the group velocity of the
wavepacket, namely
    u = dE/dp, (3.16)
    we have
    dE 0 /dp 0 = 0, (3.17)
dE/dp = v. (3.18)
    Eq.(3.17) means
that E 0 and p 0 are independent. Moreover, since the
particle is at rest in S 0 , E 0 and p 0 do not
depend on v. By differentiating both sides of Eq.(3.14) and Eq.(3.15) relative
to v we obtain

    Dividing Eq.(3.20)
by Eq.(3.19) and using Eq.(3.18) we obtain
    This means that p 0 = 0. Inputting this important result into Eq.(3.15) and Eq.(3.14), we
immediately obtain

    Then the
energy-momentum relation is:

    where E 0 is the energy of the particle at rest, called rest energy of the particle, and
p and E is the momentum and energy of the particle with velocity v. By defining
m = E 0 /c 2 as the (rest) mass of the particle [38] , we can further obtain the familiar
energy-momentum relation
    E 2 = p 2 c 2 + m 2 c 4 (3.25)
    In the
nonrelativistic domain, this energy-momentum relation reduces to E = p 2 /2m.
    3.3 Derivation of the free Schrödinger
equation
    The relation
between energy E and momentum p for momentum eigenstates in the nonrelativistic
domain implies that the operator relation is H = P 2 /2m for an
isolated system, where H is the free Hamiltonian of the system. Note that since
the value of E is real by Eq.(3.24), H is Hermitian and U(t) is unitary for
free evolution. By inputting this operator relation into the evolution equation
Eq.(3.5), we can obtain the free evolution equation, which assumes the same
form as the free particle Schrödinger equation [39] :

    It is worth noting
that, unlike the free particle Schrödinger equation, the reduced Planck
constant with dimension of action is missing in this equation. However, this is
in fact not a problem. The reason is that the dimension of can be absorbed into
the dimension of the mass m. For example, we can stipulate the dimensional
relations as p = 1/L, E = 1/T and m = T/L 2 , where L and T represents
the dimensions of space and time, respectively (see Du ff , Okun and Veneziano 2002 for more
discussions). Moreover, the value of can be set to the unit of number 1 in
principle. Thus the above equation