Understanding Quantum Physics: An Advanced Guide for the Perplexed

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 mechanics.
    Now let’s analyze
the implications of this derivation for the meaning of the conservation of
energy and momentum. First of all, we point out that the linearity of evolution
is an indispensable presupposition in the derivation. As we have stressed in
the derivation of the free Schrödinger equation, spacetime translation
invariance does not lead to dH/dt = 0 and [P, H] = 0 without assuming the
linearity of evolution. Therefore, the common wisdom that invariance or
symmetry implies laws of conservation only holds true for linear evolutions;
spacetime translation invariance no longer leads to the conservation of energy
and momentum for any nonlinear evolution, and the invariance imposes no restriction
for the nonlinear evolution either. Moreover, for a general nonlinear evolution
H(ψ), energy and momentum will be not conserved by Ehrenfest’s theorem [41] :

    We can see the
violation of the conservation of energy and momentum more clearly by analyzing
the nonlinear evolution of momentum eigenstates e i(px−Et) and their
superpositions. If a nonlinear evolution can conserve energy and momentum for
momentum eigenstates, then the momentum eigenstates must be the solutions of
the nonlinear evolution equation; otherwise the evolution will change the
definite momentum eigenvalues or energy eigenvalues or both, and thus the
conservation of energy and momentum will be violated. Some nonlinear evolutions
can satisfy this requirement. For example, when H(ψ) = P 2 /2m + α|ψ| 2 ,
the solutions still include the momentum eigenstates e i(px−Et) ,
where E = p 2 /2m + α, and thus energy and momentum are conserved for
such nonlinear evolutions of momentum eigenstates. However, even if a nonlinear
evolution can conserve energy and momentum for momentum eigenstates, it cannot
conserve energy and momentum for the superpositions of momentum eigenstates.
The reason is obvious. Only for a linear evolution the momentum eigenstates and
their superpositions can both be the solutions of the evolution equation. For
any nonlinear evolution H(ψ), if the momentum eigenstates are already its
solutions, then their linear superpositions cannot be its solutions. This means
that the coefficients of the momentum eigenstates in the superposition will
change with time during the evolution. The change of amplitudes of the
coefficients directly leads to the change of the probability distribution of
momentum eigenvalues and energy eigenvalues, while the change of phases of the
coefficients leads to the change of the momentum eigenvalues or energy
eigenvalues, which also leads to the change of the probability distribution of
momentum eigenvalues or energy eigenvalues. In fact, a nonlinear evolution may
not only change the probability distributions of energy and momentum
eigenvalues, but also change the energy-momentum relation in general cases
(e.g. in the above example) [42] .
These results are understandable when considering the fact that a nonlinear
evolution of the spatial wave function will generally introduce a
time-dependent interaction between its different momentum eigenstates, which is
equivalent to introducing a time-dependent external potential for its free
evolution in some sense. Therefore, it is not beyond expectation that a
nonlinear evolution violates the conservation of energy and momentum in
general.
    Two points needs
to be stressed here. First, energy and momentum are still defined as usual for
nonlinear evolutions in the above discussions. One may object that they should
be re-defined for a nonlinear evolution. However, this may be not the case. The
reason is as follows. Momentum is defined as the generator of space
translation, and this definition uniquely determines that its eigenstates are e ipx .
Similarly, energy is defined as the generator of time translation, and this
definition uniquely determines that its eigenstates satisfy H(ψ)ψ(x) = Eψ(x).
Since these definitions are independent of