## PHY356F: Quantum Mechanics I. Dec 7 2010 ; Lecture 11 notes. Rotations and Angular momentum.

Posted by peeterjoot on December 7, 2010

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## This time. Rotations (chapter 26).

Why are we doing the math? Because it applies to physical systems. Slides of IBM’s SEM quantum coral and others shown and discussed.

PICTURE: Standard right handed coordinate system with point . We’d like to discuss how to represent this point in other coordinate systems, such as one with the axes rotated to through an angle .

Our problem is to find in the rotated coordinate system from to .

There’s clearly a relationship between the representations. That relationship between and for a counter-clockwise rotation about the axis is

Treat and like vectors and write

Or

\paragraph{Q: Is a unitary operator?}

Definition is unitary if , where is the identity operator. We take Hermitian conjugates, which in this case is just the transpose since all elements of the matrix are real, and multiply

Apply the above to a vector and write . These are related as

Now we want to consider the infinitesimal case where we allow the rotation angle to get arbitrarily small. Consider this specific axis rotation case, and assume that is very small. Let and write

Define

which is the generator of infinitesimal rotations about the axis.

Our rotated coordinate vector becomes

Or

Many infinitesimal rotations can be combined to create a finite rotation via

For a finite rotation

Now think about transforming , an arbitrary function. Take is very small so that

\paragraph{Question: Why can we assume that is small.}

\paragraph{Answer: We declare it to be small because it is simpler, and eventually build up to the general case where it is larger. We want to master the easy task before moving on to the more difficult ones.}

Our function is now transformed

Recall that the coordinate definition of the angular momentum operator is

We can now write

For a finite rotation with angle we have

\paragraph{Question: somebody says that the rotation is clockwise not counterclockwise.}

I didn’t follow the reasoning briefly mentioned on the board since it looks right to me. Perhaps this is the age old mixup between rotating the coordinates and the basis vectors. Review what’s in the text carefully. Can also check by

If you rotate a ket, and examine how the state representation of that ket changes under rotation, we have

Or

Taking the complex conjugate we have

For infinitesimal rotations about the axis we have for functions

For finite rotations of a vector about the axis we have

and for functions

Vatche has mentioned some devices being researched right now where there is an attempt to isolate the spin orientation so that, say, only spin up or spin down electrons are allowed to flow. There are some possible interesting applications here to Quantum computation. Can we actually make a quantum computing device that is actually usable? We can make NAND devices as mentioned in the article above. Can this be scaled? We don’t know how to do this yet.

Recall that one description of a “particle” that has both a position and spin representation is

where we have a tensor product of kets. One usually just writes the simpler

An example of the above is

where is spin component one. For this would be .

Here we have also used

We can now ask the question of how this thing transforms. We transform each component of this as a vector. The transformation of

results in

Or with

Observe that this separates out nicely with the operation acting on the vector parts, and the operator acting on the functional dependence.

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