How does Quantum Mechanics predict the behavior of a particle?
In classical physics we can know everything we want to know about a particle, such as its position, velocity, momentum, angular momentum, energy, etc. At t = 0 we can make measurements to determine the initial values for these properties. Newton's laws then give us information about the subsequent motion of the particle. They predict the outcome of future measurements.
In quantum mechanics we can know a wave function for a particle. From this wave function we can extract all the information we are allowed to know about the particle. At t = 0 we can make measurements to determine the initial wave function. The Schroedinger equation then predicts the evolution of this wave function until the next measurement. The wave function at time t lets us make probabilistic predictions about the outcome of future measurements.
Schroedinger equation:
(-ħ2/(2m))∂2ψ(x,t)/∂x2 + U(x)ψ(x,t) = iħ∂ψ(x,t)/∂t
Schroedinger equation for a free particle:
(-ħ2/(2m))∂2ψ(x,t)/∂x2 = iħ∂ψ(x,t)/∂t
Solutions:
Wave packets build
from plane waves of the form exp(i(kx - ωt))
with
a range of amplitudes and k-values with ħ2k2/(2m)
= ħω. These wave packets are called
wave functions
ψ(x,t).
Interpretation:
We interpret |ψ(x,t)|2
as a probability density. The probability of finding the particle at time t
in an interval ∆x
about the position x is proportional to
|ψ(x,t)|2∆x.
Consequences:
The wave function must be single valued. continuous, and normalizable.
A measurement determines the wave function at the time of the measurement. The Schroedinger equation determines its evolution until another measurement is made. The wave function contains all the information an observer has about the particle before the observer makes another measurement.
We have to "operate" on the wave function. Every measurable quantity is associate with an operator, something we have to do to the wave function.