Hi Steve,
Right out of my Modern Physics and Quantum mechanics book by Anderson, the definition of a wave amplitude is "such that its modulus squared is proportional to the probability of finding the electron (or particle) at position x at time t. We can write a wave function in the form,
[math]\psi (x,t) = \psi_0 e^{i(kx-\omega t)}[/math]
Although this complex wave function is not directly observable (that is, measurable) it's physical significance rests on the assumption that the quantity,
[math]|\psi(x_1,t_1)|^2=\psi^*(x_1,t_1) \psi(x_1,t_1)[/math]
The whole point I'm trying to make is that I think the wave function is a real phenomena that makes physics work. In your own words "A wavefunction is simply a generic term for amplitude versus mass (or equivalent frequency).", you think the wave function is nothing special. Your not alone. A lot of physicists refuse to acknowledge the existence of the wave function. That's why we can't get to the cool physics! In my opinion, I think that the quantum wave states, using the operator,
[math]p_x = -i \hbar \frac{\partial}{\partial x}[/math]
can be used to define the momentum quantum states. Generally, momentum wave states are measured in a random way. That is, other than knowing what all of the eigenstates are, you can't predict which momentum state will be measured first, second, third, etc.
But I think there is a way to put those quantum states in a linear order. If that could be verified with an experiment, which I can describe, you could have a frequency term of the form,
[math]\omega(x) = \frac{\Delta \omega}{\Delta x}x \omega_0[/math]
You would have a frequency term that is linear with distance, that is, it increases with distance. What does this show? Well, take the equation for a wave function,
[math]\psi(x,t) = Ae^{i(k_x x - \omega(x)t)}[/math]
Watch this! Let's use the momentum operator to calculate the momentum expectation value . We get,
[math] = =-i^2(k_x - \frac{\partial \omega (x)}{\partial x})[/math]
The momentum is,
[math]p_x = k_x - \frac{\Delta \omega}{\Delta x}t[/math]
Then something interesting happens. The equation for a force is,
[math]\vec F = \frac{\partial \vec p}{\partial t}[/math]
What is the force?
[math]\vec F = \frac{\partial \vec p}{\partial t} = \frac{\partial}{\partial t}(h k_x - h \frac{\Delta \omega}{\Delta x}t)[/math]
The force caused by a linear distribution of frequency quantum states is,
[math]\vec F = h \frac{\Delta \omega}{\Delta x}[/math]
My idea is that a graviton has quantum momentum states already built into it. When they are captured by quantum systems, they behave like wave functions. When they are allowed to expand at the speed of light for many seconds, they overlap and become spacetime. Spacetime is made of quantum position/momentum states.