What is the difference between pure and mixed states in quantum mechanics?

In quantum mechanics, the state of a system refers to all the information we have about its properties. This information can be complete (pure state) or incomplete (mixed state). Here's a breakdown of the key differences:

Pure State:

* Represents a system with complete knowledge about its state.
* Mathematically described by a wavefunction (ψ), which is a complex-valued function containing all the information about the system's probabilities.
* The wavefunction's norm (square root of the sum of the absolute values of its components squared) is equal to 1, signifying complete knowledge.
* Pure states exhibit quantum superposition, meaning the system can exist in multiple states simultaneously until measured.
* Example: A coin spinning in the air before landing is in a pure state, as it has an equal probability of being heads or tails.

Mixed State:

* Represents a system with incomplete knowledge about its state. This could be due to:
* Lack of information: We may not have complete knowledge about the initial state.
* Ensemble of states: The system could be a statistical mixture of pure states, and we know the probability of finding it in each pure state.
* Mathematically described by a density matrix (ρ), which is a positive semidefinite operator that encodes the probabilities of different pure states within the mixture.
* Unlike pure states, the trace (sum of the diagonal elements) of the density matrix squared (ρ²) is less than 1, reflecting the incomplete knowledge.
* Mixed states do not exhibit true superposition. While the system can exist in multiple states probabilistically, it's in a definite state after a measurement is performed.
* Example: A coin that has already landed heads or tails, but we haven't looked at it yet. The system is in a mixed state because we know it's in one of the two definite states (heads or tails) but don't know which one.

Here's a table summarizing the key points:

| Feature | Pure State | Mixed State |
|-------------------------|---------------------------------------------------|---------------------------------------------------|
| Knowledge | Complete | Incomplete |
| Mathematical Representation | Wavefunction (ψ) | Density Matrix (ρ) |
| Norm/Trace of Square | ||ψ||² = 1 | Tr(ρ²) < 1 |
| Superposition | Yes | No (appears probabilistic after measurement) |
| Example | Spinning coin (before landing) | Coin (landed, but not observed) |

Understanding pure and mixed states is crucial in quantum mechanics, as they represent the different ways a system can exist and how our knowledge about it influences its behavior.