Understanding Quantum Physics: An Advanced Guide for the Perplexed

is essentially the free particle Schrödinger
equation in quantum mechanics.
    By using the
definition of classical potential and requiring an appropriate expectation
value correspondence, d < P > /dt =< F >=< ∂V/∂x >, we can
further obtain the Schrödinger equation under an external potential [40] :

    The general form
of a classical potential may be V (x,∂/∂x,t) and its concrete form is
determined by the non-relativistic approximation of the quantum interactions
involved, which are described by the relativistic quantum field theory. Since
the potential V (x, t) is real-valued, the Hamiltonian H = P 2 /2m + V
(x, t) is Hermitian, and as a result, the time translation operator or
evolution operator U(t) is also unitary.
    3.4 Further discussions
    We have derived
the free Schrödinger equation in quantum mechanics based on spacetime translation
invariance and relativistic invariance. The derivation may not only make the
equation more logical and understandable, but also shed some new light on the
physical meaning of the wave function ψ(x, t) in the equation.
    The free
Schrödinger equation is usually "derived" in textbooks by analogy and
correspondence with classical physics. There are at least two mysteries in such
a heuristic "derivation". First, even if the behavior of microscopic
particles is like wave and thus a wave function is needed to describe them, it
is unclear why the wave function must assume a complex form. Indeed, when
Schrödinger originally invented his equation, he was very puzzled by the
inevitable appearance of the imaginary unit "i" in the equation.
Next, one doesn’t know why there are the de Broglie relations for momentum and
energy and why the non-relativistic energy-momentum relation must be E = p 2 /2m.
Usually one can only resort to experience and classical physics to answer these
questions. This is unsatisfactory in logic as quantum mechanics is a more
fundamental theory, of which classical mechanics is only an approximation.
    As we have argued
above, the key to unveil these mysteries is to analyze the origin of momentum
and energy. According to the modern understanding, spacetime translation gives
the definitions of momentum and energy. The momentum operator P is defined as
the generator of space translation, and it is Hermitian and its eigenvalues are
real. Moreover, the form of momentum operator can be uniquely determined by its
definition. It is P = −i∂/∂x , and its eigenstate is e ipx , where p
is a real eigenvalue. Similarly, the energy operator H is defined as the
generator of time translation. But its form is determined by the concrete
situation. Fortunately, for an isolated system the form of energy operator,
which determines the evolution equation, can be fixed by the requirements of
spacetime translation invariance and relativistic invariance (when assuming the
evolution is linear). Concretely speaking, time translational invariance
requires that dH/dt = 0, and the solution of the evolution equation i∂ψ(x,t)/∂t=Hψ(x,
t) must assume the form ψ(x, t) = ϕ E (x)e −iEt .
Space translational invariance requires [P, H] = 0, and this further determines
that ϕ E (x) is the eigenstate of P , namely ϕ E (x) = e ipx . Thus spacetime translation
invariance entails that the state of an isolated system with definite momentum
and energy assumes the plane wave form e i(px−Et) . Furthermore, the
relation between p and E or the energy-momentum relation can be determined by
the relativistic invariance of the momentum eigenstate e i(px−Et) ,
and its non-relativistic approximation is just E = p 2 /2m. Then we
can obtain the form of energy operator for an isolated system, H = P 2 /2m,
and the free Schrödinger equation, Eq.(3.26). To sum up, this analysis may
answer why the wave function must assume a complex form in general and why
there are the de Broglie relations and why the non-relativistic energy-momentum
relation is what it is.
    So far so good.
But how does the wave function ψ(x, t) in the thus-derived free Schrödinger
equation relate to the actual physical state of the system? Without answering
this question the above analysis seems vacuous in physics. This leads us to the
problem of interpreting the wave function. According to the standard
probability interpretation, the wave function in quantum mechanics is a
probability amplitude, and its modulus square gives the probability density of
finding a particle in certain locations. Notwithstanding the success of