Raising and lowering operators for spin

In summary, when setting the raising and lowering operators for spin to be S_{\pm} = S_x \pm i S_y, we are following the right hand convention. This convention is based on the mathematical definition of angular momentum, where L=R x P, and is also supported by the commutation relations [L_z, L \pm] = \pm \hbar (L_x \pm i L_y). Defining the operators in a left hand convention would result in the signals being converted, which is not desirable.
  • #1
lion8172
29
0
When we set the raising and lowering operators for spin to be [tex]S_{\pm} = S_x \pm i S_y[/tex], what convention are we following (i.e. why is the first term taken to be S_x and the second taken to be S_y)?
 
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  • #2
If we label the eigenstates of [itex]S_z[/itex] as [itex]|{+}\rangle[/itex] and [itex]|{-}\rangle[/itex], so that [itex]S_z|{\pm}\rangle=\pm\frac12 |{\pm}\rangle[/itex], then
[tex]S_+|{-}\rangle=|{+}\rangle[/tex]
[tex]S_-|{+}\rangle=|{-}\rangle[/tex]
Also,
[tex]S_+|{+}\rangle=0[/tex]
[tex]S_-|{-}\rangle=0[/tex]
That is, [itex]S_+[/itex] raises the value of [itex]S_z[/itex], and [itex]S_-[/itex] lowers it. That is how the raising and lowering operators are defined, and [itex]S_x \pm i S_y[/itex] is just what they work out to be.
 
  • #3
lion8172 said:
When we set the raising and lowering operators for spin to be [tex]S_{\pm} = S_x \pm i S_y[/tex], what convention are we following (i.e. why is the first term taken to be S_x and the second taken to be S_y)?

It is the right hand convention.
think about the angular momentum...
it is defined mathematically L=R x P... it is a "right handed representation".
We can of sure define that in left hand repr...
 
  • #4
I think that is a Math method.
Example:
angular momentum: After "operate" with Math signals, Lz will be raised or lowered. (with operation L+=Lx+iLy)

--
If define [tex]L \pm = L_x \pm iL_y [/tex]
When comute them:
[tex][L_z ,L \pm ] = ... = \pm \hbar (L_x \pm iL_y )[/tex]
(Griffiths D.J_Quantum Mechanics...)
If define "left hand" the signal will become to convert.
+- become -+
That is not good!
 
Last edited:

What are raising and lowering operators for spin?

Raising and lowering operators for spin are mathematical operators used in quantum mechanics to describe the angular momentum of a particle. They are denoted as S+ and S- and are used to raise or lower the spin state of a particle.

How do raising and lowering operators for spin work?

Raising and lowering operators for spin work by acting on a spin state to change its value. The S+ operator increases the spin by one unit, while the S- operator decreases the spin by one unit.

What is the commutation relationship between raising and lowering operators for spin?

The commutation relationship between the raising and lowering operators for spin is given by [S+, S-] = Sz, where Sz is the z-component of the spin angular momentum operator. This relationship is a fundamental property of spin operators in quantum mechanics.

What is the significance of raising and lowering operators for spin?

Raising and lowering operators for spin are significant in quantum mechanics because they allow us to describe and manipulate the spin state of a particle. They also play a crucial role in the understanding of quantum systems, such as atoms and nuclei, which have intrinsic spin.

How are raising and lowering operators for spin used in experiments?

In experiments, raising and lowering operators for spin are used to measure the spin of a particle and to manipulate its state. They are also used to study the behavior of quantum systems and to develop new technologies, such as quantum computing.

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