Tensor Product of Hilbert Spaces

More complicated systems require “larger” Hilbert spaces.
We can build these by taking tensor products

Let and let and be orthonormal bases.
Then

is the tensor product that contains all elements of form

The inner product is defined on basis elements

In we’ll use the shorthand notation

where and

Examples

• Hydrogen atom is described by the Hamiltonian

𝟙

• A 2-Qubit system
  is the onb basis of
  is the onb of
  and to calculate them we use:

We will also use .