A very powerful and useful instrument that will be discussed is the density matrix. Consider the outer product of a ket and a bra. i.e \( |\psi \rangle \langle \psi | \). This might look like a needless complication but as we shall see this entity tells us quite a bit of information.
Before we start with the wonders we shall start with the Stern Gerlach experiment. A beam of electron is sent into a magnetic field pointing in the \(\hat{z} \) direction. What was expected was that electron would be bent or deflected in all directions instead what was seen was that either the electrons were deflected up or down. This was the first indication of what physicists now call spin. I shall not go on about this experiment since it is one that is rehashed in almost every modern physics textbook or quantum mechanics text book. Instead what shall is consider the following scenario:
Supposing a beam of light is let go from a source and it is able to move through the Stern Gerlach instrunmentation by turning the instrument in some direction, then we shall call it a pure state. Note that we have not specified a direction, we have merely said a direction or orientation can be found in which the beam entirely goes through.
This is important because if particles in the beam are in a superposition of spin up and spin down, as long as all the particles are in this state then this beam is considered to be in a pure state. If this is the case we can then describe the system by a single state vector.
If one the other had we have beam of particles in one pure state and another beam of particles in another pure state, then this ensemble is considered to be in a mixed state. In order for this to occur the beams have to be prepared independently and by this we mean that no phase relation exists between these two beams.
Now we move to introduce the polarization vector which shall be defined as follows:
$$ P_i = \langle \sigma_i \rangle =\langle \chi |\sigma_i | \chi \rangle $$
If we are talking about a two level system then the sigma are the pauli spin matrices. Before we continue, let's get an inkling as to why it is called the polarization vector. If we consider the general two dimensional state vector \( | \chi \rangle = \cos \theta + e^{\delta}\sin \theta \) and then calculate \(P_i\), one gets the following column vector \( (\cos \delta \sin \theta, \sin \delta \sin \theta, \cos \theta)^{T} \) This should remind one of spherical co-ordinates. So we picture the bloch sphere the polarization is telling us the "composition" of our state vector. Hence how it is polarized, in very much the analogous way we might ask how the electric field for example is polarized. In fact there is an analogy to be made between filter EM waves and spins the Stern-Gerlach experiment.
The definition above should just remind one of a pure beam, well can we generalize it for mixtures and as a point in fact it can, namely:
$$ P_i = \sum _a W_a \langle \chi_a |\sigma_i | \chi _a\rangle $$
Here \( W_a \) are simply the statistical weights namely the proportion of particles in state \( \chi_a \)
The next step is to connect this with density matrix.
Before we start with the wonders we shall start with the Stern Gerlach experiment. A beam of electron is sent into a magnetic field pointing in the \(\hat{z} \) direction. What was expected was that electron would be bent or deflected in all directions instead what was seen was that either the electrons were deflected up or down. This was the first indication of what physicists now call spin. I shall not go on about this experiment since it is one that is rehashed in almost every modern physics textbook or quantum mechanics text book. Instead what shall is consider the following scenario:
Supposing a beam of light is let go from a source and it is able to move through the Stern Gerlach instrunmentation by turning the instrument in some direction, then we shall call it a pure state. Note that we have not specified a direction, we have merely said a direction or orientation can be found in which the beam entirely goes through.
This is important because if particles in the beam are in a superposition of spin up and spin down, as long as all the particles are in this state then this beam is considered to be in a pure state. If this is the case we can then describe the system by a single state vector.
If one the other had we have beam of particles in one pure state and another beam of particles in another pure state, then this ensemble is considered to be in a mixed state. In order for this to occur the beams have to be prepared independently and by this we mean that no phase relation exists between these two beams.
Now we move to introduce the polarization vector which shall be defined as follows:
$$ P_i = \langle \sigma_i \rangle =\langle \chi |\sigma_i | \chi \rangle $$
If we are talking about a two level system then the sigma are the pauli spin matrices. Before we continue, let's get an inkling as to why it is called the polarization vector. If we consider the general two dimensional state vector \( | \chi \rangle = \cos \theta + e^{\delta}\sin \theta \) and then calculate \(P_i\), one gets the following column vector \( (\cos \delta \sin \theta, \sin \delta \sin \theta, \cos \theta)^{T} \) This should remind one of spherical co-ordinates. So we picture the bloch sphere the polarization is telling us the "composition" of our state vector. Hence how it is polarized, in very much the analogous way we might ask how the electric field for example is polarized. In fact there is an analogy to be made between filter EM waves and spins the Stern-Gerlach experiment.
The definition above should just remind one of a pure beam, well can we generalize it for mixtures and as a point in fact it can, namely:
$$ P_i = \sum _a W_a \langle \chi_a |\sigma_i | \chi _a\rangle $$
Here \( W_a \) are simply the statistical weights namely the proportion of particles in state \( \chi_a \)
The next step is to connect this with density matrix.
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