Quantum physics problem- Bra-Ket notation and operators

[max_jammer]

Hello all,

Homework Statement

I’m trying to derive a result from a book on quantum mechanics but I have trouble with bra-ket notation and operators…
Let’s say we have a photon moving along the cartesian z-axis. It is polarized and its state is
Psi(theta) = cos (theta) x1 + sin(theta) x1
Here, x1 and x2 are the base vectors.
The book states that a rotation about z axis is represented by an operator U, which has the matrix (respective to x1 and x2 base):
cos(fi) sin(fi)
-sin(fi) cos(fi)
It is the next step I have trouble with, the book states that by applying a rotation to psi(theta) you will get psi(theta+fi).
When I use simple matrix multiplication of U and psi, I don’t get this result but rather Psi(fi-theta)…
I did manage to produce the correct result when I used hermetian conjugate od U… Why is this so?

Homework Equations

What is the correct procedure and why? What am I doing wrong?

The Attempt at a Solution

Simple matrix multiplication...

 

[diazona]

Hm... well, if you take a look at this Wikipedia article, I think it may suggest an explanation

 

[max_jammer]

Thanks for your replay.

I did actually do quite extensive search on the web, but I did not find what I was looking for.

Maybe I should explain...

The matrix (for the base $\chi_{1}$ and $\chi_{2}$) of the rotation operator is (according to the book):
$U_{\phi, \textbf{k}}$ =
cos $\phi$ sin $\phi$
-sin $\phi$ cos $\phi$

the state of the photon is

$\Psi_{\theta}$ = cos $\theta$ $\chi_{1}$ + sin $\theta$ $\chi_{2}$

The book also states that when rotating this state by the rotation matrix above, you will get

$\Psi_{\theta + \phi}$

But this is not what I get unless I transpose the rotation matrix.

If I do that, the matrix looks exactly like the one in the wikipedia article.

So is the book wrong?

The most confusing part of it all is when I look at the lecture notes. According to those, the professor first calculated the effect of the rotation on the base, like this:

$\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{1}$ > = cos $\phi$ | $\chi_{1}$ > + sin $\phi$ | $\chi_{2}$ >
and
$\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{2}$ > = -sin $\phi$ | $\chi_{1}$ > + cos $\phi$ | $\chi_{2}$ >

then he writes

$\widehat{U_{\phi, \textbf{k}}}$ | $\Psi_{\theta}$ > = $\widehat{U_{\phi, \textbf{k}}}$ (cos $\theta$ | $\chi_{1}$ > + sin $\theta$ | $\chi_{2}$ > ) =

then he simply rearanges:

= cos $\theta$ $\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{1}$ > + sin $\theta$ $\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{2}$ > )

and then he substitues the result $\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{1}$ > and $\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{2}$ > from above and that gives the result from the book.

My question is:

1) why is he using this procedure instead of simple matrix multiplication?
2) why is Matrix multiplication wrong?
3) why does the rotation matrix in the book look exactly like the transposed version of the one in wikipedia article (and any other article I could find)
4) How did the professor calculate the effect of the operator on the base (i.e. $\widehat{U_{\phi, \textbf{k}}}$ | $\chi_{1}$ >)

I've been tormented by this for a week now, any input is appreciated...

 

[M Quack]

This is most probably a problem of active and passive coordinate transformations.

Basically, if you rotate your coordinate axes one way, then the new coordinates of a vector are obtained by rotating the old coordinates in the opposite direction. This is a constant pain in the rear that creeps up every time you do coordinate transformations.

This explains why the rotation matrix you give in the first post has the opposite sign of theta than the one on the Wiki page linked in the second post.

http://en.wikipedia.org/wiki/Active_and_passive_transformation

http://arxiv.org/abs/1106.4446 section 1.5 (very brief)

 

[paranormal]

I understand your frustration with bra-ket notation and operators. Quantum mechanics can be quite complex and the use of abstract mathematical notation can make it even more challenging. However, bra-ket notation and operators are essential tools in quantum mechanics as they allow us to represent and manipulate quantum states and operators in a concise and elegant way.

In this specific problem, the book is using the bra-ket notation to represent the state of the photon as a vector in a two-dimensional vector space spanned by the base vectors x1 and x2. The operator U represents a rotation about the z-axis, and its matrix representation is given in terms of the base vectors.

When applying the rotation operator to the state of the photon, we need to use the Hermitian conjugate of the operator, denoted by U†, which is the conjugate transpose of U. This is because the rotation operator is unitary, meaning it preserves the inner product of vectors. In other words, when we apply the rotation operator to the state vector, we want the resulting vector to have the same inner product with any other vector in the space. This is achieved by using the Hermitian conjugate instead of the operator itself.

Therefore, the correct procedure is to use the Hermitian conjugate of the operator U, which in this case is the same as the inverse of the matrix representation of U, to rotate the state vector psi(theta). This will give you the desired result of psi(theta+fi).

In summary, the use of bra-ket notation and operators may seem confusing at first, but they are powerful tools that allow us to work with quantum states and operators in a concise and elegant way. In this problem, the Hermitian conjugate is necessary to preserve the inner product and obtain the correct result.