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Homework Statement
Hi. This isn't really a 'problem' as such but it is to do with coursework and seems a little too basic to be posted in the maths section. I need to know the result of an arbitrary bra, <an|, say, representing an eigenvector of a system, acting on a compound ket such as |0>a|1>b.
The reason I'm doing this is, I need to find the time dependence of my compound system. I solved the eigenvalues and eigenvectors of my Hamiltonian which gave me the basis {|a>} upon which I need to expand my initial state vector, which is in compound form, i.e. |n>a|m>b |n>a and |m>b belong to different Hilbert spaces and represent different objects.
Homework Equations
The Attempt at a Solution
Since the eigenvectors I found were derived by considering the interaction part of the Hamiltonian of the total system (my two systems are magnetically coupled), I assume they span the Hilbert spaces of both a and b, so an element from any of my eigenvectors can multiply an element in either an a or b vector. Also I know that |0>a|1>b represents the tensor product of the a and b vectors. So for example
|a>=[1;1], say, then; <a||0>a|1>b= [1 1]*[1;0]a[1 0]b
Is this correct so far? But then I am at a loss, I don't know how to compute the tensor product of two vectors in different Hilbert spaces, and don't know what kind of object this would be once all is said and done. I've looked around the internet but haven't found anything helpful. Any help would be greatly appreciated!
(p.s, sorry about the lack of latex, it was doing weird things when I previewed the post. The notation for vectors is meant to be MATLAB notation, so
[1;1] represents a column vector:
1
1
and [1 0] represents a row vector (1, 0).
Thanks!)