Two identical spin 1/2 particles

In summary: To construct a state, one would need to take linear combinations of these eigenstates. In this case, it would be written as ##|\psi\rangle = \sum_{n,l,m,S,M} c_{n,l,m,S,M} | \textbf{P} \rangle \otimes | n,l,m \rangle \otimes | S,M \rangle##, where ##c_{n,l,m,S,M}## are coefficients.
  • #1
Lebnm
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I am studying identical particles and I have some doubts. Considerer two identical spin 1/2 particles interacting through a central potential ##V##. In the rest of CM, the hamiltonian is $$ H = \frac{\textbf{P}^{2}}{2M} + \frac{\textbf{p}^{2}}{2\mu} + V(r),$$ where ##\textbf{P}## is the momentum of CM, ##\textbf{p}## is the momentum associated with the relative coordinate ##\textbf{r}##, ##M## is the total mass and ##\mu## is the reduced mass. The text I am reading write the state of the system as $$| \psi \rangle = | \textbf{P} \rangle \otimes | n,l,m \rangle \otimes | S,M \rangle.$$ Here, ##| \textbf{P} \rangle## is an eigenstate of the operator ##\textbf{P}##, ## | n,l,m \rangle ## is the solution of the central potential problem and ##| S,M \rangle## are the eigenstates of ##\textbf{S}^{2}## and ##S_{z}##, being ##\textbf{S}## the total spin of the system. My questions are: it's not exactly a state, but a bases, isn't it? To construct a state,do I need to take linear combinations of it? In this case, How can I write this?
 
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  • #2
The ##|\psi\rangle## you wrote there represents an eigenstate of the Hamiltonian (in addition to being a spin eigenstate).

It is a state. The set of all such eigenstates forms a complete basis, such that any state can be written as a sum of these eigenstates.
 
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1. What is the spin of two identical spin 1/2 particles?

The spin of two identical spin 1/2 particles is 1.

2. How does the spin of two identical spin 1/2 particles affect their behavior?

The spin of two identical spin 1/2 particles affects their behavior by determining how they interact with each other and with other particles in their surroundings.

3. Can two identical spin 1/2 particles have different spins?

No, two identical spin 1/2 particles always have the same spin. This is a fundamental property of identical particles in quantum mechanics.

4. What is the significance of the spin 1/2 value for particles?

The spin 1/2 value for particles is significant because it is a fundamental property of particles in quantum mechanics. It determines how they behave and interact with other particles, and is a key factor in understanding the structure and behavior of matter.

5. How is the spin of two identical spin 1/2 particles measured?

The spin of two identical spin 1/2 particles can be measured using various experimental techniques, such as Stern-Gerlach experiments or nuclear magnetic resonance (NMR) spectroscopy. These techniques involve manipulating the particles' spin and observing the resulting behavior to determine their spin value.

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