Lecture Notes on Density Matrix and Quantum Statistics

Introduction to Density Matrix

• Density Matrix: Classical analog of the density distribution function.
• Denoted by: ( \rho ) (rho)
• Function of: Position coordinate (Q), conjugate momentum (p), probability time (t).
• In stationary systems, time (t) is absent in the function.
• Quantum Mechanical Operator: Expressed in matrix form, also known as statistical operator.

Importance of Density Matrix

• Powerful Tool: Analyzes statistical behavior of quantum mechanical systems.
• Applications:
  • Finding expectation values of observables.
  • Analyzing time-dependent Hamiltonians.
  • Calculating entropy and entanglement measures.

Properties of Density Matrix

• Hermitian Operator: Positive and semi-definite.
• Defined in terms of the state vector (( \psi )).
• Types of States:
  • Pure State: Represented by a single eigenvector (state vector).
  • Mixed State: Combination of different quantum states.

Pure State Density Matrix

• Definition:
  • A pure quantum state is represented by a single state vector, ( | \alpha \rangle ).
  • Density matrix ( \rho ) is the outer product:
    [ \rho = | \alpha \rangle \langle \alpha | ]
• Rank 1 Projection Operator: Density matrix acts as a rank 1 projection operator.

Mixed State Density Matrix

• Definition: A mixed state is described by a statistical ensemble of different pure states.
• Density Matrix Calculation:
  • Outer product of various pure states weighted by their probabilities.
  • [ \rho = \sum P_{\alpha} | \alpha \rangle \langle \alpha | ]
• Weighted Sum: Not a simple sum due to varying probabilities of quantum states._

Representation of State Vectors

• State Vector Expansion:
  • A state vector ( | \alpha \rangle ) can be expressed as a linear combination of orthonormal eigenfunctions (( | \phi_n \rangle )).
  • [ | \alpha \rangle = \sum c_{n \alpha} | \phi_n \rangle ]
• Density Matrix Elements:
  • [ \rho_{mn} = \sum P_{\alpha} c_{m \alpha}^* c_{n \alpha} ]*_

Key Properties of Density Matrix

1. Hermitian: ( \rho_{mn} = \rho_{mn}^* )
2. Probability Interpretation:
   • Diagonal elements indicate the probability of the system being in a particular state.
   • Off-diagonal elements capture coherence and quantum correlations.
3. Trace:
   • Trace of the density matrix (sum of diagonal elements) is always equal to 1, indicating normalization of probability.
4. Eigenvalues: All eigenvalues of the density matrix are non-negative.*

Next Steps in Lecture

• Upcoming discussion on the application of the density matrix in Time Evolution under time-dependent Hamiltonians.
• Introduction to the quantum mechanical version of Liouville’s theorem.