What is the speed of light doing in the spin half hamiltonian?

In summary, the conversation discusses the introduction of a spin half hamiltonian in Sakurai's 'Modern Quantum Mechanics' (Revised Edition). The hamiltonian includes a term involving the speed of light, which is a result of spin-orbit coupling according to the Dirac equation. It is noted that this term is a matter of convention, similar to the choice between CGS and MKS in electromagnetism.
  • #1
center o bass
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I'm currently reading Sakurai's 'Modern Quantum Mechanics' (Revised Edition) and at page 76 he introduces a spin half hamiltonian

[tex] H = - (\frac{e}{mc}) \vec S \cdot \vec B.[/tex]

But what is c doing in this hamiltonian? Clasically the energy of a magnetic moment in a magnetic field is

[tex] E = - \vec \mu \cdot \vec B = - \frac{e}{2m} \vec S \cdot \vec B[/tex]

and as far as i know what one does to go to QM is to introduce the g-factor. What am I missing here?
 
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  • #2
Standard question, standard answer: the spin-orbit coupling is a specially relativistic effect, even though the spin notion itself can be derived without special relativity. The <c> comes from the Dirac equation.
 
  • #3
Say what? If this term is spin-orbit coupling, where is L??

The c comes about because according to Dirac the magnetic moment μ of the electron is -eħ/2mc, or -e/mc times the spin ħ/2.
 
  • #4
ah, thanks. I guess I have to be reading up on that then.
 
  • #5
Sorry, not paying attention. Disregard my post above.

If the <c> is there or not must be in the end a matter of convention, just like the as the case of CGS vs MKS in electromagnetism.
 
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Related to What is the speed of light doing in the spin half hamiltonian?

1. What is the spin half Hamiltonian?

The spin half Hamiltonian is a mathematical operator used in quantum mechanics to describe the behavior of a spin-1/2 particle, such as an electron. It takes into account the spin of the particle, which can be either up or down, and is used to calculate the energy and other properties of the particle.

2. What is the speed of light doing in the spin half Hamiltonian?

The speed of light, denoted by the symbol c, appears in the spin half Hamiltonian as a conversion factor between energy and frequency. This is due to the famous equation E=mc^2, where c is the speed of light. In quantum mechanics, energy and frequency are closely related, and the speed of light is used to convert between the two.

3. Why is the speed of light important in the spin half Hamiltonian?

The speed of light is important in the spin half Hamiltonian because it is a fundamental constant of the universe and plays a crucial role in the behavior of particles at the quantum level. It is also used to relate the energy and frequency of a spin-1/2 particle, which is essential in understanding its behavior.

4. Is the speed of light a variable in the spin half Hamiltonian?

No, the speed of light is not a variable in the spin half Hamiltonian. It is a constant value that is the same for all particles and does not change. This means that it is a fixed factor in the Hamiltonian and does not vary based on the properties or interactions of the particle.

5. How does the speed of light affect the behavior of spin-1/2 particles in the Hamiltonian?

The speed of light affects the behavior of spin-1/2 particles in the Hamiltonian by dictating the relationship between energy and frequency. As the speed of light is a constant, it determines the energy levels that a particle can have and how those levels correspond to different frequencies. This, in turn, influences the behavior and interactions of the particle.

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