# K^2 = J^2 + 1/4 for the central force problem of the Dirac equation

• Kamikaze_951
In summary, the conversation is about understanding the central force problem of the Dirac equation, specifically the operator K and its commutation with the hamiltonian H and total angular momentum J. The last equality involving K^2 is discussed and clarified as a shorthand notation for the commutator of L_i and L_j. The person seeking help expresses gratitude for the clarification.
Kamikaze_951

## Homework Statement

To whom it may concern,

I am trying to understand the central force problem of the Dirac equation. In particular, I am following Sakurai's Advanced Quantum Mechanics book. There (section 3.8, p.122), it is shown that there is an operator

$K = \beta(\Sigma . L + \overline{h})$
where $\Sigma_i = \left( \begin{array}{cc} \sigma_i && 0 \\ 0 && \sigma_i \\ \end{array}\right)$

This operator commutes with the hamiltonian H and the total angular momentum J. At the top of page 123, it is shown that $K^2 = J^2 + \overline{h}/4$. At some point, we get

$K^2 = L^2 + i\Sigma . (L × L) + 2\overline{h}\Sigma . L + \overline{h} = L^2 + \overline{h}\Sigma . L + \overline{h}^2$

I don't understand this last equality. Why is $i \Sigma . (L × L) = -\overline{h}\Sigma.L$?

## The Attempt at a Solution

I looked a lot in the literature, but I didn't find any more precise calculation. I don't seem to grasp what L × L means. I know that $\vec{L} = \vec{x}× (-i\overline{h}\nabla)$. If we had vectors instead of operators, we would have $\vec{L} × \vec{L} = 0$, but what does the vector product of 2 identical operators mean?

Thanks for the help.

Kami

The misleading/confusing notation L x L c/should be a shorthand from $[L_i, L_j]$, i.e. from the commutator.

Hi dextercioby,

Thank you a lot for your reply, that was exactly what I needed. I did the calculation with that in mind and it worked. In fact, I am ashamed not having seen this by myself.

## 1. What is the significance of the equation K^2 = J^2 + 1/4 in the central force problem of the Dirac equation?

The equation K^2 = J^2 + 1/4 is known as the Klein-Gordon equation and it is a fundamental equation in quantum mechanics. It describes the energy of a particle in a central force field, which is a force that depends only on the distance from the particle to the center of the force. This equation is important because it allows us to calculate the energy levels and wavefunctions of particles in this type of system.

## 2. How is the central force problem of the Dirac equation different from the non-relativistic case?

The central force problem of the Dirac equation is different from the non-relativistic case because it takes into account the effects of special relativity. In the non-relativistic case, the energy of a particle is given by the Hamiltonian operator, H = p^2/2m + V, where p is the momentum, m is the mass, and V is the potential. However, in the relativistic case, the energy is given by the Klein-Gordon equation, K^2 = J^2 + 1/4, which includes the spin operator, J. This means that the energy levels and wavefunctions of particles in a central force field will be different in the relativistic case compared to the non-relativistic case.

## 3. What is the role of the spin operator in the central force problem of the Dirac equation?

The spin operator, J, is a fundamental concept in quantum mechanics that describes the intrinsic angular momentum of a particle. It is important in the central force problem of the Dirac equation because it affects the energy levels and wavefunctions of particles in a central force field. The spin operator is included in the Klein-Gordon equation, K^2 = J^2 + 1/4, because it is a relativistic equation that takes into account the effects of special relativity.

## 4. How does the central force problem of the Dirac equation relate to the Bohr model of the atom?

The central force problem of the Dirac equation is closely related to the Bohr model of the atom, which was proposed by Niels Bohr in 1913. In the Bohr model, the energy levels of electrons in an atom are quantized, meaning they can only have certain discrete values. This is similar to the energy levels of particles in a central force field, which are also quantized. However, the Bohr model is a non-relativistic model, while the central force problem of the Dirac equation takes into account the effects of special relativity.

## 5. What is the physical significance of the solutions to the central force problem of the Dirac equation?

The solutions to the central force problem of the Dirac equation have important physical significance because they describe the energy levels and wavefunctions of particles in a central force field. These solutions allow us to understand the behavior and properties of particles, such as electrons, in these types of systems. They also have implications for various phenomena in quantum mechanics, such as the spin-orbit interaction and the Zeeman effect.

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