Oct 19, PHY356F lecture notes.
Posted by peeterjoot on October 20, 2010
Last time, we started thinking about angular momentum. This time, we will examine orbital and intrinsic angular momentum.
Orbital angular momentum in classical physics and quantum physics is expressed as
where and are quantum mechanical operators corresponding to position and momentum
\item a) Determine the commutators and
With , we have
Since the operators commute, all the first terms cancel, leaving just
\item b) in spherical coordinates.
The answer is
Work through this.
Part of the task in this intro QM treatment is to carefully determine the eigenfunctions for these operators.
The spherical harmonics are given by such that
The z-component is quantized since, is an integer . The total angular momentum
is also quantized (details in the book).
The eigenvalue properties here represent very important physical features. This is also important in the hydrogen atom problem. In the hydrogen atom problem, the particle is effectively free in the angular components, having only dependence. This allows us to apply the work for the free particle to our subsequent potential bounded solution.
Note that for the above, we also have the alternate, abstract ket notation, method of writing the eigenvalue behavior.
For the total angular momentum our spherical harmonic eigenfunctions have the property
Alternately in plain old non-abstract notation we can write this as
Now we can introduce the Raising and Lowering Operators, which are
respectively. These are abstract quantities, but also physically important since they relate quantum levels of the angular momentum. How do we show this?
Last time, we saw that
Note that it is implied that we are operating on ket vectors
Question: What is ?
So is another spherical harmonic, and we have
This lowering operator quantity causes a physical change in the state of the system, lowering the observable state (ie: the eigenvalue) by .
Now we want to normalize , via .
We can use
So, knowing (how exactly?) that
we have from 2.26
and can normalize the functions as
Abstract notation side note:
Generalizing orbital angular momentum.
To explain the results of the Stern-Gerlach experiment, assume that there is an intrinsic angular momentum that has most of the same properties as . But has no classical counterpart such as .
This experiment is the classic QM experiment because the silver atoms not only have the orbital angular momentum, but also have an additional observed intrinsic spin in the outermost electron. In turns out that if you combine relativity and QM, you can get out something that looks like the the operator. That marriage produces the Dirac electron theory.
We assume the commutation relations
Where we have the analogous eigenproperties
Electrons and protons are examples of particles that have spin one half.
Note that there is no position representation of . We cannot project these states.
This basic quantum mechanics end up applying to quantum computing and cryptography as well, when we apply the mathematics we are learning here to explain the Stern-Gerlach experiment to photon spin states.
(DRAWS Stern-Gerlach picture with spin up and down labeled , and with the magnetic field oriented in along the axis.)
Silver atoms have and , where is the quantum number associated with the z-direction intrinsic angular momentum. The angular momentum that is being acted on in the Stern-Gerlach experiment is primarily due to the outermost electron.
What about ? We can leave the detector in the plane, but rotate the magnet so that it lies in the direction.
We have the correspondence
but this is perhaps more properly viewed as the matrix representation of the less specific form
Where the translation to the form of 2.41 is via the matrix elements
We can work out the same for using and , or equivalently for using and , where
The operators are the Pauli operators, and avoid the pesky factors.
And from , we have eigenvalues for the operator.
The corresponding eigenkets in column matrix notation are found
which can be normalized as
We see that this is different from
We will still end up with two spots, but there has been a projection of spin in a different fashion? Does this mean the measurement will be different. There’s still a lot more to learn before understanding exactly how to relate the spin operators to a real physical system.