Tensor Products

Quantum Mechanics · Part 1

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Table of Contents

  1. 1.Motivation: Composite Systems
  2. 2.Definition: Tensor Product
  3. 3.Product States vs. Entangled States
  4. 4.Inner Product in Tensor Product Spaces
  5. 5.Operators on Tensor Product Spaces
  6. 6.Partial Trace and Reduced Density Matrices
  7. 7.Schmidt Decomposition
  8. 8.Physical Examples
  9. 9.Summary
  10. 10.Dimension

Key Equations

$$|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B \quad \Rightarrow \quad |\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j\rangle$$
$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\uparrow\rangle + |\downarrow\downarrow\rangle\right)$$
$$(\hat{A} \otimes \hat{B})(|\phi\rangle \otimes |\chi\rangle) = (\hat{A}|\phi\rangle) \otimes (\hat{B}|\chi\rangle)$$
$$\hat{\rho}_A = \text{Tr}_B(|\psi\rangle\langle\psi|) = \sum_{i,i'} \left(\sum_j c_{ij} c_{i'j}^*\right) |a_i\rangle\langle a_{i'}|$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Quantum Mechanics course on CoursesHub.World. Master quantum mechanics from mathematical foundations to advanced topics. Covers Hilbert spaces, wave functions, angular momentum, perturbation theory, scattering, and path integrals with 450+ pages ...

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