I am currently in quantum chemistry, and in class one day my professor spent some time talking about Maxwell's equations. I am looking at my notes, trying to piece together Maxwell's equations, differential equations, and the principle of superposition, since this is not in my book. I haven't taken dif eq either, so that might be part of the problem.
I was told (if I understand my notes) that the solutions to Maxwell's equations are differential equations that have the quality of being homogenous and linear, therefore the superposition principle will work. 1. What does it mean for a function to be homogenous and linear?
Also in my notes, I wrote: E1 = a solution to Maxwell's eq
E2 = a solution to Maxwell's equations
For any constants, λ1 and λ2 the linear combinations λ1E1+λ2E2 will be a solution to Maxwell's equations. 2. This may be outside the realm of the pure math people, but why does this combination describe a light field?
I don't think there are any.
The Attempt at a Solution
1. Well I was told that a homogenous linear equation takes this form:
g(x) is the probe function
The fact the it is equal makes it homogenous - but can't you make any equation equal to zero by subtracting it from both sides?
There must be no powers in the differential, what would that look like? Must g(x) just be a linear function, then?
Also, the power of the coefficient doesn't matter.
So I know these things, but I do not know what the mean, really, I don't know how to recognize a linear homogenous (LH) function, and what the significance of a LH function is.
2. I don't know if E stands for energy or electric field - I think electric field, since in class he spoke about how E2 is proportional to intensity. Either way, I don't see a light field.