# Relativistic Energy and Momentum

Previously, we have attempted to show how the *
Schrodinger time-dependent
wave equation *was derived in the form:

We might also highlight that the solution of this equation can be anchored in the form:

[3]

At this point, we will simply define [C] as a complex function of
spatial variables that is independent of time [t]. However, the more
important thing to point out is that we will initially assume energy
[E] to be positive, such that the exponential coefficient, in [3], must
be a negative multiple of [i]. This state of affairs is said to reflect
a ‘*positive frequency*’ that is a natural condition of non-relativistic
systems, where the Hamiltonian [H] is assumed to reflect positive kinetic
energy, e.g.

[4]

From classical physics, we can show that energy [E] can be described in terms of the rate of change of momentum [p] with time [t]:

[5a]

We might still perceive this relationship in Schrödinger’s equation,
as shown in [2], but note that momentum [p] now has a complex form,
which has been previously outlined in the discussion entitled '*Quantized
Wave Operators*':

[5b]

The Hamiltonian [H] for a relativistic particle conforming to deBroglie’s hypothesis has to combine [1a] and [5], such that [4] becomes:

Therefore, in quantum mechanics, the momentum [p^{2}] is
not only described in terms of an operator, which according to [6],
now involves a second-order differential expression, enclosed within
a square-root, but now suggests some form of negative component. Clearly,
such concepts will potentially require a different level of mathematical
‘*sophistication*’ to underpin any further physical interpretation
within the quantum world. Following this line of thought, the previous
outline of Schrodinger’s wave equation suggested that the variable [ψ]
might be subject to a different interpretation in comparison to classical
physics, such that we might have to consider [ψ] as the sum of
all possibilities, e.g. both positive and negative energy states.