A linear function follows two properties:
- If a function is a solution and each variable is scaled or multiplied by the same number, then this is also a solution.
- If two solutions to a function are found, then a third solution of the function is the summation of each variable in the function.
Given a linear operator L and an unknown variable u, the following properties apply:
Here is an example:
Linearity as related to Quantum Mechanics
A linear system is far less complicated than a non-linear system.
Maxwell’s equation is linear, for instance. Newton’s equations are not linear.
Consider the example below that explains a particular scenario in which Newton’s equations are shwon to be non-linear:
Quantum Mechanics is linear. Schroedinger’s Equation, devised in 1925 for a dynamic variable, ψ termed the wavefunction.
H_hat is the Hamiltonian, a linear operator as was L in the previous example for Linearity [link]. This means that in Quantum Mechanics, solutions can easily be scaled and added together. Thus, it is proven that Quantum Mechanics is actually simpler than classical mechanics. i is the complex number operator equal to the square root of negative one and h_bar is Planck’s constant.
You might ask what the wavefunction is about, if there are any units, for instance. Interestingly, Schroedinger was not sure what the wavefunction referred to exactly. Max Born later proposed that it had to do with probability.
Complex Numbers in Quantum Physics
The complex operator i at the front of the formula notes a significant departure from classical mechanics in which almost all systems are primarily real. In the case of Quantum Mechanics, complex numbers are essential.
Euler’s formula, e^(i*x) = cos(x) + i*sin(x) also proves useful in Quantum Mechanics.
Below is some review relevant to Quantum Mechanics: