Quantum Mechanics
Probability Rules
For a wave function
Where
The wave function
After a measurement, the wave function instantly collapses to the measured state. With time, the wave function evolves back to a widely distributed state. The wave function can be solved with the Schrodinger Equation.
An example for calculating with the probability density. (I'm not very smart about physics, so this example in my mind, is so smart and witty.)
Suppose I drop a rock off a cliff of height
Solution:
The velocity is $$ v(t) = \frac{dx}{dt} = gt $$ and the total flight times is
The probability of the camera flashing is $ dt / T$ (if the camera flashes for a short period of time
So the probability density is
The average distance is
Because the wave function have a relationship with probability, the wave function must be normalized. The normalization condition is:
The normalization condition with a constant (complex constant)
Because the integral is independent of
Thus with the Schrodinger Equation:
So the integral:
Because the wave function must be zero at
Interestingly, problem 1.17 in the book gave an example of a particle that is not stable, and have a half life. In that case, the probability function
Where
In the quantum mechanic's world, particles don't follow the normal laws of mechanics, but interestingly there is the Ehrenfest's theorem. Which states that the expectation values obey the classical laws.
Here we also introduces another new operator, for any quantity related to
For a given particle, it is impossible to know it's position and momentum at the same time. This is called the Heisenberg Uncertainty Principle. The uncertainty principle is a direct consequence of the wave nature of matter. The uncertainty principle is:
Where