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Chapter V notes for .
Obtain for spin 1 in the representation in which and are diagonal.
For spin 1, we have
and are interested in the states . If, like angular momentum, we assume that we have for
and introduce a column matrix representations for the kets as follows
then we have, by inspection
Note that, like the Pauli matrices, and unlike angular momentum, the spin states have not been considered. Do those have any physical interpretation?
That question aside, we can proceed as in the text, utilizing the ladder operator commutators
to determine the values of and indirectly. We find
Looking for equality between , we find
so we must have
Furthermore, from , we find
We must have . We could probably pick any
, and , but assuming we have no reason for a non-zero phase we try
Putting all the pieces back together, with , and , we finally have
A quick calculation verifies that we have , as expected.
Obtain eigensolution for operator . Call the eigenstates and , and determine the probabilities that they will correspond to .
The first part is straight forward, and we have
Taking we get
with eigenvectors proportional to
The normalization constant is . Now we can call these , and but what does the last part of the question mean? What’s meant by ?
Asking the prof about this, he says:
“I think it means that the result of a measurement of the x component of spin is . This corresponds to the eigenvalue of being . The spin operator has eigenvalue ”.
Aside: Question to consider later. Is is significant that ?
So, how do we translate this into a mathematical statement?
First let’s recall a couple of details. Recall that the x spin operator has the matrix representation
This has eigenvalues , with eigenstates . At the point when the x component spin is observed to be , the state of the system was then
Let’s look at the ways that this state can be formed as linear combinations of our states , and . That is
Letting , this is
We can solve the and with Cramer’s rule, yielding
It is and that are probabilities, and after a bit of algebra we find that those are
so if the x spin of the system is measured as , we have a $50\
Is that what the question was asking? I think that I’ve actually got it backwards. I think that the question was asking for the probability of finding state (measuring a spin 1 value for ) given the state or .
So, suppose that we have
or (considering both cases simultaneously),
Unsurprisingly, this mirrors the previous scenario and we find that we have a probability of measuring a spin 1 value for when the state of the operator has been measured as (ie: in the states , or respectively).
No measurement of the operator gives a biased prediction of the state of the state . Loosely, this seems to justify calling these operators orthogonal. This is consistent with the geometrical antisymmetric nature of the spin components where we have , just like two orthogonal vectors under the Clifford product.
Obtain the expectation values of for the case of a spin particle with the spin pointed in the direction of a vector with azimuthal angle and polar angle .
Let’s work with instead of to eliminate the factors. Before considering the expectation values in the arbitrary spin orientation, let’s consider just the expectation values for . Introducing a matrix representation (assumed normalized) for a reference state
Each of these expectation values are real as expected due to the Hermitian nature of . We also find that
So a vector formed with the expectation values as components is a unit vector. This doesn’t seem too unexpected from the section on the projection operators in the text where it was stated that , where was a unit vector, and this seems similar. Let’s now consider the arbitrarily oriented spin vector , and look at its expectation value.
With as the the rotated image of by an azimuthal angle , and polar angle , we have
The projections of this operator
are just the Pauli matrices scaled by the components of
so our expectation values are by inspection
Is this correct? While is a unit norm operator, we find that the expectation values of the coordinates of cannot be viewed as the coordinates of a unit vector. Let’s consider a specific case, with , where the spin is oriented in the plane. That gives us
so the expectation values of are
Given this is seems reasonable that from 3.43 we find
(since we don’t have any reason to believe that in general is true).
The most general statement we can make about these expectation values (an average observed value for the measurement of the operator) is that
with equality for specific states and orientations only.
Take the azimuthal angle, , so that the spin is in the
x-z plane at an angle with respect to the z-axis, and the unit vector is . Write
for this case. Show that the probability that it is in the spin-up state in the direction with respect to the z-axis is
Also obtain the expectation value of with respect to the state .
For this orientation we have
Confirmation that our eigenvalues are is simple, and our eigenstates for the eigenvalue is found to be
This last has unit norm, so we can write
If the state has been measured to be
then the probability of a second measurement obtaining is
Expanding just the inner product first we have
So our probability of measuring spin up state given the state was known to have been in spin up state is
Finally, the expectation value for with respect to is
Sanity checking this we observe that we have as desired for the case.
Consider an arbitrary density matrix, , for a spin system. Express each matrix element in terms of the ensemble averages where .
Let’s omit the spin direction temporarily and write for the density matrix
For the ensemble average (no sum over repeated indexes) we have
This gives us
and our density matrix becomes
We can easily find
So we can write the density matrix in terms of any of the ensemble averages as
Alternatively, defining , for any of the directions we can write
In equation (5.109) we had a similar result in terms of the polarization vector , and the individual weights , and , but we see here that this factor can be written exclusively in terms of the ensemble average. Actually, this is also a result in the text, down in (5.113), but we see it here in a more concrete form having picked specific spin directions.
If a Hamiltonian is given by where , determine the time evolution operator as a 2 x 2 matrix. If a state at is given by
then obtain .
Before diving into the meat of the problem, observe that a tidy factorization of the Hamiltonian is possible as a composition of rotations. That is
So we have for the time evolution operator
Does this really help? I guess not, but it is nice and tidy.
Returning to the specifics of the problem, we note that squaring the Hamiltonian produces the identity matrix
This allows us to exponentiate by inspection utilizing
Writing , and , we have
Note that as a sanity check we can calculate that as expected.
Now for , we have
It doesn’t seem terribly illuminating to multiply this all out, but we can factor the results slightly to tidy it up. That gives us
Consider a system of spin particles in a mixed ensemble containing a mixture of 25\
Note that we can also factor the identity out of this for
which is just:
Recall that the ensemble average is related to the trace of the density and operator product
But this, by definition of the ensemble average, is just
We can use this to compute the ensemble averages of the Pauli matrices
We can also find without the explicit matrix multiplication from 3.70
(where to do so we observe that for and , and .)
We see that the traces of the density operator and Pauli matrix products act very much like dot products extracting out the ensemble averages, which end up very much like the magnitudes of the projections in each of the directions.
Show that the quantity , when simplified, has a term proportional to .
Consider the operation
With , we have
which gives us the commutator
Insertion into the operator in question we have
With decomposition of the into symmetric and antisymmetric components, we should have in the second term our
where we expect . Alternately in components
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.