Lect7_TensorProd - Tensor-Product of state spaces Clearly...

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Tensor-Product of state spaces Clearly the Hilbert space of square-integrable functions ! ( x,y,z ) in three dimensions is not the same as for the one dimension states ( x ) . What happens to the Hilbert space of a particle when you add spin on top of its motional degree’s of freedom? How do you describe the combined state of two quantum systems, especially if they start to interact? Definition of Tensor product Suppose I have a particle with 10 possible states, and I add ten more states to its Hilbert space. The dimensionality of the Hilbert space increases from 10 to 20 The particle can now be found in one of 20 possible states This is a sum of two Hilbert spaces Instead, suppose I add a second particle to the system, which also has 10 possible states The first particle can be in 1 of 10 possible states The second particle can also be in 1 of 10 possible states, independent of the state of the first particle The dimensionality of the Hilbert space thus goes from 10 to 10x10=100 Specifying the state of the system now requires two quantum numbers, the state of particle one (1- 10) and the state of particle two (1-10) so there are 100 possible distince states This combined Hilbert space is a Tensor-Product of the two Hilbert spaces
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Formalism Let ! 1 and ! 2 be two Hilbert spaces of dimension N 1 and N 2 , respectively. Vectors and operators with subscript 1 belong to ! 1 , and those with subscript 2 belong to ! 2
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Lect7_TensorProd - Tensor-Product of state spaces Clearly...

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