Raising and Lowering Operators

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In summary, the raising and lowering operators in a spin 1/2 system have a factor of $\hbar$ because they are defined in terms of the angular momentum components, which have dimensions of angular momentum. This is why the physical interpretation of $S_+$ is that it raises the component by one unit of $\hbar$. However, when looking at the specific example of $S_z S_+ |-\rangle$, the factor of $\hbar$ does not seem to raise the spin component by one unit. This is due to the normalization of the eigenvectors, which cancels out one factor of $\hbar$.
  • #1
aliens123
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Why is it that the raising and lowering operators in a spin 1/2 system have a factor of $\hbar ?$

From Sakurai:
$$S_+ \equiv \hbar | + \rangle \langle - |, S_- \equiv \hbar | - \rangle \langle + |$$

"So the physical interpretation of $S_+$ is that it raises the component by one unit of $\hbar ? $ " (Page 22, 2017 edition).

But it seems like this does not raise the spin component by one unit of $\hbar. $ If I have a vector in the state $|- \rangle $ then
$$S_z |- \rangle = (-\hbar /2) | - \rangle$$
But now if I look at:
$$S_z S_+ |- \rangle = (\hbar^2 /2) | + \rangle$$
This didn't raise the spin component by one unit of $hbar.$ But if $S_+$ weren't defined with that factor of $\hbar,$ then it would have raised the factor by one unit of $hbar.$
 
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  • #2
Just divide by the constant then, it’s defined that way since the raising and lowering operators take simple forms.
 
  • #3
Usually the raising- and loweing-operators of angular momentum operators are defined in terms of the angular-momentum components,
$$\hat{S}_{\pm}=\hat{S}_x \pm \mathrm{i} \hat{S}_y.$$
Thus it has dimensions of angular momentum, which is where the factors ##\hbar## come from in the OP.

Of course, you have to normalize the eigenvectors, which cancels one factor ##\hbar## again.
 

Related to Raising and Lowering Operators

1. What are raising and lowering operators?

Raising and lowering operators are mathematical operators used in quantum mechanics to describe the behavior of particles. They are used to raise or lower the energy levels of a quantum system.

2. How do raising and lowering operators work?

Raising and lowering operators work by acting on a quantum state, changing its energy level. The raising operator increases the energy level by a fixed amount, while the lowering operator decreases it by the same amount.

3. What are the properties of raising and lowering operators?

Raising and lowering operators have several important properties, including hermiticity, commutation relations, and eigenvalue properties. These properties allow them to accurately describe the behavior of particles in quantum systems.

4. What is the significance of raising and lowering operators in quantum mechanics?

Raising and lowering operators are essential tools in quantum mechanics as they allow us to describe the behavior of particles in a quantum system. They are used to calculate probabilities, determine energy levels, and predict the behavior of particles.

5. How are raising and lowering operators used in practical applications?

Raising and lowering operators have many practical applications, including in quantum computing, quantum cryptography, and quantum information processing. They are also used in various fields of physics, such as quantum optics and solid state physics, to study the behavior of particles at the quantum level.

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