Relativistic Energy and Momentum

1Previously, we have attempted to show how the Schrodinger time-dependent wave equation was derived in the form:

[2]      2

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

[3]      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]      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]    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]    5b

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

[6]      6

Therefore, in quantum mechanics, the momentum [p2] 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.