Breaking Quantum Physics (But Not Really): Mixed States + Density Operators | Parth G

Pure quantum states have wave function representations, but the same is not true for mixed states. Find out why density matrices are needed to correctly represent mixed states.

Hey everyone! Popular science discussions of quantum mechanics often focus on the wave function - and rightly so. It's an important entity in the mathematics of quantum physics. But a wave function is not always enough to represent any given quantum state.

If we happen to be studying a system where we know what state it's in, for example the spin state of an electron (even if this is a superposition of multiple possible states) then we are said to be dealing with a pure state because we have as much knowledge of it as is possible to have. A superposition is interesting because when we make a measurement on the system, it collapses into one of the superposed states - but before the measurement, it's genuinely in a blend of all possible experimental results. A pure state has a simple wave function representation. It can therefore also be represented by a single vector.

However if we are dealing with a system where we do not have complete information about it, such as which pure state it's in, then we are dealing with a mixed state. If we know the likelihood of our system being in a particular psi (pure) state, then we can find its density matrix. To do this, we first find the density matrices of each possible pure state that it could be in, by multiplying the bra and the ket representing the pure state (column vector x row vector). Then we add up each of the possible density matrices, weighted by the likelihood of our system being in each pure state. This gives us the final density matrix, or density operator, for our mixed state system.

In this video, we discuss some quantum concepts that may not be easy to understand if you've never seen them before - please check out my quantum mechanics playlist for lots of videos covering them all! https://www.youtube.com/playlist?list=PLOlz9q28K2e4Yn2ZqbYI__dYqw5nQ9DST

If you are familiar with the mathematics of quantum mechanics, please check out Professor M does Science's video on this topic, as it covers the maths in great detail. https://www.youtube.com/watch?v=LR5kfhrs4Cc

Timestamps:
0:00 - Wave functions in terms of electron spin states
2:01 - Pure states in quantum mechanics - represented by a single wave function
2:30 - Mixed states - when we don't know enough about our system, not related to quantum probabilities
4:19 - Density operators, density matrices, and the vector representation of wave functions

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