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galvin452
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Is there a connection between the Dirac four spinor and "spin up", i.e one of the four spinor states is spin up or are these two separate unconected things.
Sonderval said:Yes there is a connection - depending on how you write the spinor, one of the four components can be interpreted as the up-component of the electron, one as the up-component of the positron (the other two are the down components).
galvin452 said:Can't one use all of the Dirac four spinor in relativistic calculation of, e.g. the quantum states of the electron in hydrogen? Penrose in `The Road to Reality' (pg 629) specifically uses the four spinor to describe the electron only.
The Dirac Equation has both positive frequency and negative frequency solutions. The negative frequency ones are sometimes interpreted as negative energy ("hole") states. But in the normal way of writing a four-spinor, the four solutions do not one-to-one correspond to the four components of the spinor. Each solution involves all four components.galvin452 said:Can't one use all of the Dirac four spinor in relativistic calculation of, e.g. the quantum states of the electron in hydrogen? Penrose in `The Road to Reality' (pg 629) specifically uses the four spinor to describe the electron only.
Bill_K said:The Dirac Equation has both positive frequency and negative frequency solutions. The negative frequency ones are sometimes interpreted as negative energy ("hole") states. But in the normal way of writing a four-spinor, the four solutions do not one-to-one correspond to the four components of the spinor. Each solution involves all four components.
If you write the four-spinor as a pair of two-spinors, Ψ1, Ψ2 and put this into the Dirac Equation, you find they are coupled together:
(E + eφ)Ψ1 = c σ·p Ψ2
(E + eφ + 2mc2)Ψ2 = c σ·p Ψ1
where E is kinetic plus potential energy (relativistic energy minus mc2) and φ is the electrostatic potential. Putting φ = e/r, you can solve this pair of equations to find the bound states of the hydrogen atom. As Penrose said, all four components of the spinor are involved in the solution.
Yes, in the nonrelativistic limit. In the second equation, (E + eφ + 2mc2)Ψ2 = c σ·p Ψ1, the rest energy mc2 is the largest energy, so we can approximategalvin452 said:Does this mean one can not associate a "spin up" with anyone of the spinors? Is there any interpretaion of the two-spinor Ψ1?
A Dirac four spinor is a mathematical object used to describe the spin of a particle in quantum mechanics. It is a four-component complex vector that contains information about the particle's spin and its interactions with other particles.
A Dirac four spinor is related to spin up through the Pauli spin matrices, which are used to transform the spinor's components from the spin up basis to the spin down basis. The spinor's first two components correspond to spin up, while the last two components correspond to spin down.
The connection between Dirac four spinor and spin up is significant because it provides a mathematical framework for understanding the intrinsic spin of particles and how they behave in quantum mechanics. It also allows for the prediction and calculation of a particle's spin in various situations, such as in an external magnetic field.
The Dirac equation, which describes the behavior of fermions in quantum mechanics, incorporates the concept of spin by using Dirac four spinors to represent the spin states of particles. The equation also includes terms for the spin angular momentum and the spin-orbit interaction.
Yes, there are several practical applications of understanding the connection between Dirac four spinor and spin up. For example, it is essential for understanding the behavior of electrons in materials, which is crucial for the development of new technologies such as transistors and computer memory. It is also necessary for understanding the behavior of particles in particle accelerators and in the study of fundamental particles and forces.