In quantum mechanics, particles and systems can exhibit wave-particle duality, which means they can behave both as particles and as waves. When we consider the wave-like behavior of particles, we describe them using wave functions, which are mathematical functions that can represent the probability distribution of finding a particle in different states.
In the context of wave functions, the concept of "coherent waves taking all virtual paths" is related to the principle of superposition. According to this principle, when multiple possible paths exist for a particle to travel from one point to another, the wave function of the particle can be described as a combination or superposition of all these possible paths. Each path contributes to the overall wave function with a certain amplitude, and these amplitudes interfere with each other, leading to the characteristic wave-like phenomena.
The probabilities associated with different paths in quantum mechanics are determined by the squared magnitudes of the amplitudes. The more likely paths have higher probabilities, meaning that particles are more likely to be found in regions where the amplitudes reinforce each other.
As for the randomness in the chosen landing position or measurement outcome, it is a fundamental aspect of quantum mechanics known as quantum randomness or indeterminacy. When a measurement is made on a quantum system, the wave function "collapses" into one specific state corresponding to the measurement outcome. The exact outcome is inherently unpredictable, and we can only assign probabilities to different possible outcomes based on the wave function's properties.
The randomness in quantum mechanics is not due to our lack of knowledge or information about the system but is an inherent characteristic of the theory itself. It introduces an irreducible element of uncertainty at the quantum level. This randomness is often described by the Born rule, which states that the probability of obtaining a particular measurement outcome is given by the squared magnitude of the corresponding component of the wave function.
In summary, coherent waves in quantum mechanics can be described by wave functions that consider all possible paths. The most likely paths have higher probabilities, but the ultimate measurement outcome is subject to inherent quantum randomness, where the exact landing position or outcome cannot be precisely predicted but can only be assigned probabilities.