Tensor Product Space and Dimensionality

Given two vector spaces $H_1$ and $H_2$, both of which are 2-dimensional, the tensor product space $H_1 \otimes H_2$ has a dimensionality equal to the product of the dimensions of the individual spaces. In this case:

$\text{dim}(H_1) = 2, \quad \text{dim}(H_2) = 2$

So, the dimensionality of the tensor product space $H_1 \otimes H_2$ is:

$\text{dim}(H_1 \otimes H_2) = \text{dim}(H_1) \times \text{dim}(H_2) = 2 \times 2 = 4$

Basis of the Tensor Product Space

The basis of the tensor product space $H_1 \otimes H_2$ can be constructed from the bases of $H_1$ and $H_2$.

Given:

• $B(1) = \{|1(1)\rangle, |2(1)\rangle\}$ is the basis of $H_1$,
• $B(2) = \{|1(2)\rangle, |2(2)\rangle\}$ is the basis of $H_2$,

The basis of $H_1 \otimes H_2$ consists of all possible tensor products of the basis vectors of $H_1$ and $H_2$. These basis vectors are written as:

$|1(1)\rangle \otimes |1(2)\rangle, \quad |1(1)\rangle \otimes |2(2)\rangle$ $|2(1)\rangle \otimes |1(2)\rangle, \quad |2(1)\rangle \otimes |2(2)\rangle$

Thus, the basis for $H_1 \otimes H_2$ is:

$\left\{ |1(1)\rangle \otimes |1(2)\rangle, |1(1)\rangle \otimes |2(2)\rangle, |2(1)\rangle \otimes |1(2)\rangle, |2(1)\rangle \otimes |2(2)\rangle \right\}$

This is a 4-dimensional space with the given basis.

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Would you like more details on tensor products?

Here are 5 related questions to explore further:

1. How do you calculate the tensor product of two specific vectors from $H_1$ and $H_2$?
2. What are the properties of the tensor product in terms of linearity?
3. Can the tensor product be generalized to vector spaces of different dimensions?
4. How does the dimensionality of the tensor product space change for higher dimensions?
5. What is the role of the tensor product in quantum mechanics?

Tip: When working with tensor products, always verify the dimensions and construct the basis systematically by taking all combinations of basis vectors.