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 | ψ = dx p | x 〉〈 x | ψ 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: ψ ψ x x = ) ( ψ ψ z y x z y x , , ) , , ( = or x , y , z { } x , y , z ( x , y , z ) R 3 ψ ( p ) = p ψ
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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 =| ψ 1 (1) | ψ 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 ψ ψ ψ ψ 1 ) 1 ( H ψ 2 ) 2 ( H φ 1 12 H H H = ) 1 ( 1 ) 2 ( 1 ) 2 ( 1 ) 1 ( 1 ψ ϕ ϕ ψ
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Schmidt Basis The easiest way to find a good basis for a
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