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Is it possible that we have entanglement in different degrees of freedom of a singe particle. like spin and linear momentum .

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I wish nobody had used the word "entaglement" in physics when it really means a "coherent functional dependence". Of course the QM solutions of the one particle state will have a coherent functional correlation with all the variables and quantum numbers entering the problem. –  anna v Mar 12 at 14:13

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Even though you think of it as a single particle -- each of it's different properties like momentum, spin, etc (corresponding to each valid quantum number) sits in a Hilbert space of their own and the possible configurations of a particle sits in a tensor product of those Hilbert spaces.

$$\mathcal{H_{particle}} = \mathcal{H_{momentum}} \otimes \mathcal{H_{spin}} \otimes \ldots $$

Like any other case, when we have a state in some space which can be broken up into subsystems tensored together, we can talk about correlations in those subsystems. If those correlations are quantum mechanical, then of course we will have entanglement -- like any other system you know about.

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Yes it is possible and in fact such states also have a fully classical description (coherence theory). Due to this fact this type of coherence is sometimes refered to as "classical entanglement."

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Yes, we can have entanglement between different degree of freedom of same particle or system. That is known as ''hybrid entanglement'' and that is experimentally demonstrated also. http://arxiv.org/pdf/1007.1322v1.pdf

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