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.

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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[domainofscience]

This was brilliant! Perfect explanation.

[risingredstone]

minor error at 2:50 both the arrows in the first two spin states upwards, one has to point downwards

[sagnikdey]

I have been so confused with the concepts of pure and mixed states and the density matrix for so long, and watched numerous videos on YouTube. But finally I got it clear after watching your video. Thanks Parth, it was a brilliant explanation!!

[kyozpsycho]

EXTREMELY USEFUL, starting to see the path.

[triviumdaniel]

At 2:48 the arrow in the spin down ket is upside down. Great video overall keep it up dude!!

[upsal.nanayakkara]

wow man . its not about the mixed states but this is the simplest yet most detailed explanation of density matrix . you can find tons of videos people are trying to explain density metrices yet couldnt make a single sense even with math. Brilliant how you summered an entire 2 hour lecture to a 7 minute video. just brilliant .

[shreyaspradhan]

Damn that was super intuitive. Perfect explanation. I wish all sources were this clear and intuitive.

[DavidAspden]

Great length and good content density.

[Draginx]

Parth, you're my favorite youtuber! You make amazing content and you're a great teacher and you motivate me to do better at physics! You make me remember why I love physics and science so much. Bet you won't reply to this...

[nirmalbk]

Finally ...i get it ❤the concept of mixed state

[bapanbiswas]

Sir please one details video on gradient 🙏

[MysteriousSlip]

I am giving 'thumb's up' to every video of yours that I find well presented and useful. So far I am just spamming upthumbs to you. I wish the best for your channel because I truly believe you deserve it.

[rsbenari]

Brilliant. An essential and much needed, intuitive intro. Thanks.

[stevenschilizzi]

Brilliantly crystal clear! Please keep doing these! Many thanks.

[dasmolduck]

Hi Parth,
Thanks for making the video. Have a few questions that I am not quite sure
- 1:47 Should it be 1/2 rather than 1 / sqrt (2)
- 2:49 Should the middle equation be spin down

[prashantlale]

That was astonishingly great and i like the change in your background.

[portugalforme]

Terrific explanation....saved me a good few very much for posting.

[SomethingImpromptu]

In your example about an electron source you said that it sometimes spits electrons out with spin up, sometimes with spin down, & sometimes a superposition of the two... I thought the way it worked is that the spin state will always be a superposition until observed, & after observation it will always either be spin up or down. Is that incorrect? My impression was that the collapse of the wave-function occurred because when you observe it, you bounce photons off of the wave-particle, which interferes with the wave-function & forces it to settle in a particular state, so you wouldn’t have an electron source that just straight up spits particles out sometimes in a particular state & sometimes in a superposition.

At any rate, extremely interesting & well articulated video. I had never heard of these mixed states & quantum theory just gets more mind-blowing with every new thing you learn about it. The nature of our reality is truly just about the strangest & most amazing of all possibilities.

[MusicEngineeer]

thanks for clarifying this. the terminology is really confusing here. i initially thought, a "pure" state is something like "100% up" or "100% down" - as in: no superposition. and "mix" would be kinda synonymous to "superposed". so, what are these "100%" states then called? basis states?

[moondo]

Parth You are an answer to many prayers, namely that of bridging the gap between those of us who are rudimentary or Ancient calculus Education, but I rather more profound Interest in the subject at hand

Thank you kindly and I’m going to patrion