TensorProduct of state spaces
•
Clearly the Hilbert space of squareintegrable
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 (110) so there
are 100 possible distince states
–
This combined Hilbert space is a
TensorProduct
of
the two Hilbert spaces
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View Full DocumentFormalism
•
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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 Fall '08
 MichaelMoore
 mechanics, Vector Space, Hilbert space, tensor product

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