Singlet/Triplet state mixing question

In summary, the book you are looking at does not skimp on the details, and it is necessary to have a good understanding of quantum mechanics in order to understand it. However, the book does not go into much detail on how the operator works. The author is looking for help from others and asks if anyone can help him out with understanding the operator. He is having difficulty understanding it himself and asks for help. After understanding the basics of the operator, the author is still having difficulty with evaluating it. He asks for help from those who are more knowledgeable about quantum mechanics and ends the summary with a request for help.
  • #1
pfollansbee
14
0
Hey guys!
I am trying to show how singlet and triplet states mix via spin-orbit coupling, but I am having some trouble.
I've gotten most of the way through, but I am stuck on how an operator works. Sadly, I think my poor basis in quantum is going to show right here.

My starting point...

[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle =\hbar \left|1,0\right\rangle [/itex]

The book I am looking at sort of skims past this and assumes that I have a clue what is going on here.


So I know
[itex]\left|0,0\right\rangle =\frac{1}{\sqrt{2}}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}[/itex]
[itex]s_z=\frac{\hbar }{2}\left\{\left|+\right\rangle \left\langle +\right|-\left|-\right\rangle \left\langle -\right|\right\}[/itex]

and I am thinking that
[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle = \frac{1}{\sqrt{2}}\left(s_{1z}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}-s_{2z}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}\right) [/itex]



After this it becomes a giant mess. Basically it hinges on my inability to evaluate things of the form
[itex]\left|+\right\rangle \left\langle +|-+\right\rangle[/itex]

I am thinking that this is heavily reliant on some other basics as well. In determining the s_z spinor I am not sure how to evaluate something like
[itex]\left\langle +|+\right\rangle \left\langle +|+\right\rangle - \left\langle +|-\right\rangle \left\langle -|+\right\rangle[/itex]
which is supposed to equal one and

[itex]\left\langle +|+\right\rangle \left\langle +|-\right\rangle - \left\langle +|-\right\rangle \left\langle -|-\right\rangle[/itex]
which is supposed to equal zero


I was hoping that someone here would be able to help me figure out how
[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle =\hbar \left|1,0\right\rangle [/itex]
and the above evaluations. Thanks a whole bunch!
 
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  • #2
You have two sub-systems. A convenient way to take care of this is the tensor product:
You write e.g. : [itex]| ++>=|+>\otimes|+>[/itex] and [itex]s_{1z}=s_z \otimes 1[/itex].
The action of the operator on the state then becomes:
[itex]s_{1z}|++>=s_z|+>\otimes 1|+>=\frac{\hbar}{2}|++>[/itex].
I hope that gives you an idea of how to proceed.
 
  • #3
Yes! That definitely will help me on my way. Unfortunately, I am still having some difficulty.

This is the part that is giving me trouble.
[itex] s_z|+\rangle ⊗ 1|+\rangle [/itex]

When I took a quantum class we never used the +/- notation, we were trained with α and β, but it seems that most modern books are using the +/-. Because of this, I am not very familiar with their mathematical meaning.

If you could be so kind as to help me solve: [itex]s_z|+\rangle[/itex]?

Here's my current process:

If [itex] s_z=\frac{\hbar }{2}\left[\left|+\right\rangle \left\langle +\right|-\left|-\right\rangle \left\langle -\right|\right] [/itex]
then [itex] s_z\left|+\right\rangle =\frac{\hbar }{2}[|+\rangle \langle +|+\rangle - |-\rangle \langle -|+\rangle] [/itex]

This is where I am stuck (If I am even correct to this point... for all I know, I may not be clear on the proper way to handle the operator)
Does <+|+> = 1 and <-|+> = 0? because that would make sense for getting [itex] \frac{\hbar }{2}|+\rangle [/itex]
(would this also mean that <-|-> = 1? or would it be -1?)

If so, how do I get the 0 and 1? If not, then how do I evaluate these correctly?

Thanks again!
 
  • #4
yes, + and - correspond one to one to alpha and beta.
They are defined to be the eigenstates of s_z.
The two states are orthogonal to each other (like any two eigenstates of a hermitian operator belonging to two different eigenvalues, namely that of s_z) whence <+|->=0 and <+|+>=1.
 
  • #5
I think one alternative way of seeing things is to realize that you're working within a finite state space (4 dimensional) and that you can write down these operators explicitly. Take a look at this page:

http://electron6.phys.utk.edu/qm1/modules/m10/twospin.htm

I don't get what your question has to do with SO coupling though.
 
  • #6
Thanks a bunch! After having a good think about it, it all makes perfect sense now and I now know exactly how the s_1z-s_2z operator works. I will post how it relates to spin orbit coupling and singlet triplet mixing a little later once I have written it out in an understandable manner.
 

1. What is singlet/triplet state mixing?

Singlet/triplet state mixing refers to the phenomenon where two electronic states, the singlet and triplet states, interact with each other and become energetically mixed. This occurs when the energy levels of the two states are similar.

2. How does singlet/triplet state mixing affect chemical reactions?

Singlet/triplet state mixing can affect chemical reactions by altering the energy barriers and pathways for the reaction. It can also change the products of a reaction, as the energetically mixed states can lead to different products than if the states were not mixed.

3. What factors influence singlet/triplet state mixing?

The main factors that influence singlet/triplet state mixing are the energy difference between the singlet and triplet states, the spin-orbit coupling strength, and the presence of a magnetic field. The strength of these factors can determine the extent of state mixing.

4. How is singlet/triplet state mixing studied in the lab?

Singlet/triplet state mixing can be studied in the lab using various spectroscopic techniques, such as electronic absorption spectroscopy or magnetic resonance spectroscopy. These techniques allow scientists to observe the changes in energy levels and interactions between the singlet and triplet states.

5. What are the potential applications of singlet/triplet state mixing?

Singlet/triplet state mixing has potential applications in fields such as organic electronics, where it can be used to control the properties of materials and devices. It can also be useful in understanding and designing chemical reactions, as well as in the development of new materials with unique properties.

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