Lect24_TensorProduct - Lecture 24: Tensor Product States...

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Lecture 24: Tensor Product States Phy851 Fall 2009
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Basis sets for a particle in 3D Clearly the Hilbert space of a particle in three dimensions is not the same as the Hilbert space for a particle in one-dimension In one dimension, X and P are incompatible – If you specify the wave function in coordinate- space, x | ψ , its momentum-space state is completely specified as well: p | = dx p | x 〉〈 x | – You thus specify a state by assigning an amplitude to every possible position OR by assigning and amplitude to every possible momentum In three dimensions, X, Y, and Z, are compatible. – Thus, to specify a state, you must assign an amplitude to each possible position in three dimensions. – This requires three quantum numbers – So apparently, one basis set is: x x = ) ( z y x z y x , , ) , , ( = or x , y , z { } x , y , z ( x , y , z ) R 3 ( p ) = p
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Definition of Tensor product Suppose you have a system with 10 possible states Now you want to enlarge your system by adding ten more states to its Hilbert space. – The dimensionality of the Hilbert space increases from 10 to 20 – The system can now be found in one of 20 possible states – This is a sum of two Hilbert sub-spaces – One quantum number is required to specify which state Instead, suppose you want to combine your system with a second system, which has ten states of its own – The first system can be in 1 of its 10 states – The second system can be in 1 of its 10 states The state of the second system is independent of the state of the first system – So two independent quantum numbers are required to specify the combined state The dimensionality of the combined Hilbert space thus goes from 10 to 10x10=100 – This combined Hilbert space is a ( Tensor ) Product of the two Hilbert sub-spaces
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Formalism Let H 1 and H 2 be two Hilbert spaces – We will temporarily ‘tag’ states with a label to specify which space the state belongs to Let the Hilbert space H 12 be the tensor- product of spaces H 1 and H 2 . The Tensor product state | ψ 12 =| 1 (1) | 2 (2) belongs to H 12 . The KEY POINT TO ‘GET’ IS: Bras and kets in the same Hilbert space ‘attach’ . – BUT, Bras and kets in different Hilbert spaces do not ‘attach’ 2 1 ) 1 ( 2 ) 1 ( 1 1 ) 1 ( H 2 ) 2 ( H φ 1 12 H H H = ) 1 ( 1 ) 2 ( 1 ) 2 ( 1 ) 1 ( 1 ϕ
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Schmidt Basis The easiest way to find a good basis for a tensor product space is to use tensor products
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Lect24_TensorProduct - Lecture 24: Tensor Product States...

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