Free Understanding Quantum Physics: An Advanced Guide for the Perplexed by Shan Gao

from the aspect of the laws of motion. Continuous motion can
be regarded as a very special form of discontinuous motion, for which the
position density of a particle is ρ(x, t) = δ 2 (x − x(t)) and its
velocity is v(t) = dx(t)/dt, where x(t) is the continuous trajectory of the
particle. However, such states are not solutions of the free Schrödinger
equation, though they do satisfy the continuity equation. According to the free
Schrödinger equation, an initial local state like δ(x − x 0 ) cannot
sustain its locality during the evolution, and it will immediately spread
throughout the whole space. Thus the law of free motion, which is derived based
on the requirements of spacetime translation invariance etc, seems to imply
that the motion of a particle cannot be continuous but be essentially
discontinuous. Note that our derivation of the free Schrödinger equation does
not depend on the picture of discontinuous motion, and thus this argument for
the non-existence of continuous motion is not a vicious circle.
    As noted above,
our derivation of the free Schrödinger equation relies on the presupposition
that the Hamiltonian H is independent of the evolved state, i.e., that the
evolution is linear. It can be reasonably assumed that the linear evolution and
nonlinear evolution both exist, and moreover, they satisfy spacetime
translation invariance respectively because their effects cannot counteract
each other in general. Then our derivation only shows that the linear part of
free evolution, if satisfying spacetime translation invariance and relativistic
invariance, must assume the same form as the free Schrödinger equation in the
non-relativistic domain. Obviously, our derivation cannot exclude the existence
of nonlinear quantum evolution. Moreover, since a general nonlinear evolution
can readily satisfy spacetime translation invariance, the invariance
requirement can no longer determine the concrete form of possible nonlinear
evolution.
    3.5 On the conservation of energy-momentum
    The conservation
of energy and momentum is one of the most important principles in modern
physics. In this section, we will analyze the basis and physical meaning of
this principle, especially its relationship with the linearity of quantum
dynamics.
    As we have noted
in the above derivation of the free Schrödinger equation, the origin of
momentum and energy is closely related to spacetime translation; the momentum
operator P and energy operator H are defined as the generators of space
translation and time translation, respectively. Moreover, it is well known that
the conservation of energy and momentum results from spacetime translation
invariance. The usual derivation is as follows. The evolution law for an
isolated system satisfies spacetime translation invariance due to the
homogeneity of space and time. Time translational invariance requires that H
has no time dependence, namely dH/dt = 0, and space translational invariance
requires that the generators of space translation and time translation are
commutative, namely [P, H] = 0. Then by Ehrenfest’s theorem for an arbitrary
observable
    d< A
>/dt=<∂A/∂t> − i<[A, H]>, (3.28)
    where = ∫ψ ∗ (x, t)Aψ(x, t)dx is defined as the
expectation value of A, we have
    d< H >/dt=0,
(3.29)
    and
    d< P >/dt=0.
(3.30)
    This means that
the expectation values of energy and momentum are conserved for the evolution
of an isolated system. Moreover, for arbitrary functions f(H) and f(P ), we
also have
    d< f(H)
>/dt= 0, (3.31)
and
d< f(P) >/dt= 0. (3.32)
    This is equivalent
to the constancy of the expectation values of the generating functions or
spacetime translation operators U(a) ≡ e −iaH and T (a) ≡ e −iaP
    d < U(a)
>/dt = 0, (3.33)
and
d< T (a) >/dt = 0. (3.34)
    By these two
equations it follows that the probability distributions of energy eigenvalues
and momentum eigenvalues are constant in time. This statement is usually defined
as the conservation of energy and momentum in quantum