# Is it correct to represent a mixed state by a wavefunction?

Many sources (such as The Physics of Quantum Mechanics and the answers to this Physics.SE question) warn against conflating

• a mixed state where $|\psi\rangle$ is $|n\rangle$ with probability $p_n$, and
• a pure state with a wavefunction $|\psi\rangle = \sum_n \sqrt{p_n}\;|n\rangle$

But in this lecture, it seems that this very mistake is being made. The lecturer assigns (3m20s) a density operator to a single wavefunction $|\psi\rangle$; and later (24m23s) he considers the possibility that this wavefunction may represent a pure or a mixed state.

Is this interpretation a legitimate one, or is the lecture erroneous, or do I misunderstand the lecture?

• You might be asking too much if you expect people to watch a whole video lecture ... It might help if you could summarize the relevant part of the lecture in the post. Also, this would be more beneficial for the Q+A format of the site. – Norbert Schuch Dec 13 '15 at 0:10
• There is no problem assigning a density matrix to a single state $|\psi\rangle$, since in this case $\rho=|\psi\rangle\langle \psi|$. – Meng Cheng Dec 13 '15 at 3:54
• @r-c What Meng said, with emphasis on "the lecturer assigns a density matrix to a pure state, not a wavefunction to a mixed state". A mixed state cannot be represented as a single wavefunction, unless it reduces to a pure state. – udrv Dec 13 '15 at 7:14
• @NorbertSchuch I don't expect people to watch the whole lecture; I give two pointers to specific moments of the lecture. – user84106 Dec 13 '15 at 8:35
• @udrv but that's a (pretty boring) special case, right? I don't think that whole lecture is devoted to defining the density operator of a pure state. Besides, see 24m23s in the lecture. – user84106 Dec 13 '15 at 8:37