Chapter 6
Key topics
- Time development
- Wave packets
- Conservation laws
Time development
This chapter covers the time evolution of a quantum system.
One important way that quantum systems are characterized is by their observables.
Find a set of commuting operators that label all of the unit vectors that span the space. The eigenvalues of these operators represent the possible outcomes of a measurement on the system.
Thus the observables and their operators have a natural role to play as the relationship between measurements and quantum systems.
Let us also not that the operators are also linked in a special way to the transformation properties of systems.
Consider the following transformations:
- Translation
o Translation through space
§ Same experiment is carried out at two different locations
o Translation through time
§ Same experiment is carried out at two different times
- Rotations
§ Observe a system from using two different coordinate axes related by a rotation
- Boosts
§ You analyze the requirements of special relativity. What is the relationship between moving systems?
In mechanics we discover that the observable quantities: momentum, energy, and angular momentum are related to these transformations. In quantum mechanics this relationship takes on a special form. Let us examine this relationship for time-translation.
This is the operator that takes a state at time t and evolves it to time t’ under the appropriate action of the forces in the problem.
Where the meaning of the operator in the
numerator is clarified by a
Remember the discussion on translating a function
Momentum and Energy are referred to as the generators for translation.
The Schrödinger equation tells us that the sum of the kinetic and potential energy evolves the system in time.
Examine the propogation of an energy eignestate.
using
We can examine
the general states evolution.
let us examine
Wave packets
For free particle
the energy eigenstates can be found that are also momentum eigenstates.
Thus the above result for the time evolution implies that a solution for a state
In terms of the wavefunction has a known time evolution if you can decompose the original state into its momentum components.
Combining the two equations
Start with some general state at t=0. In order to find the time evolution one can evolve the state with the time development operator. The explicit application in general is complicated because for a general state
so the
- Find the state in terms of b(k) by finding b(k)
- Explicitly add in the time dependence as it is straight forward since the states in the sum are energy eigenstates.
- Knowing the spatial dependence of each of the energy/momentum eigenstates explicitly put in the spatial dependence and sum up (integrate) to get a new wave function.
Notice
The time evolved system looks almost the same as the
original. You are adding two functions E1(x) and E2(x). The only difference is
each time adds the functions together with a different phase and the phase of
each term changes differently. You can consider two sin functions. If they have
a different frequency then the addition of the two will result in regions where
the waves are in phase and adding and regions of the where they are canceling.
If you shift both waves by the same phase then these regions don’t change. For
example if the two cosine waves with zero phase are added then there is constructive interference
at x=0. If both waves are shifted by 2π then nothing has changed. If one
wave is shifted by π the waves will be canceling and the overall sum will
be diminished at x=0. There can be some
confusion on this point due to the nature of the cosine function. Certainly
shifting the cosine function by π/2 results in a function that has changed
from 1 to 0 at x=0. However the phase
factor included above is 
which always has an amplitude of 1
and merely changes the real and imaginary ratio. Mathematically a constant phase factor can be
factored out of the sum and then this factor will not have an impact on the
probability. This example is meant to illustrate that changing the relative
phase of two waves changes their sum which can be seen
in the case of two cosine functions that are changed or shifted relative to
each other while changing each wave by the same amount does not result in a
different probability for an event. The
book uses beats as an exmple. Here waves added change
from loud to soft. The times that are
loud and those that are soft are determined by the relative phase not the
absolute phases.
If one examine the probability density for both cases given above t=0, t=later
So the wave function changes wrt time even though the relative amounts of the energy eigenstates remain the same.
For the wave packet the most instructive view is to break the wavefunction into thre pieces the amplitude, mean and sigma
Each one of these has an explicit time dependence and the impact of the time evolution can be seen in terms of how the standard parameters that describe a Gaussian change in time.
The translation of a system in space, time and through some angle (rotation) can be performed using operators of the general form
where the operator is referred to as a generator.
Time translation has the Hamiltonian.
Space translation has the momentum.
Rotation has the angular momentum operator.
Specifying that the system should not change behavior when moves to a transformed frame means the operators must commute with the Hamiltonian. This then results in a conservation law.
Parity
Parity is similar to a mirror reflection where xè-x, yè-y, zè-z . (A mirror just inverts z). Two actions of the parity operator should return you to initial state. This can be used to show that the eigenvalues for the parity operator are 1, -1.
Parity is an important symmetry. It is actually not a symmetry that every physics interaction possesses.