# Are there any problems that can only be tackled with density matrices and not with pure state evolution?

Say I have a state $|\Psi_0\rangle$.

I measure observable $\hat{A}$, the wavefunction collapses to one its eigenstates. I can write $|\Psi_0\rangle = \sum_j \alpha_j|\psi^A_j\rangle$, where $|\psi^A_j\rangle$ are the eigenstates of $\hat{A}$.

Suppose I do not have access to the results of the experiment though.

I now apply a second operator $\hat{B}$.

I can see two ways of proceeding to know how the system will evolve in time:

1) I can write $|\Psi^B_t\rangle = \hat{B}|\Psi_t\rangle = \sum_j \alpha_j\hat{B}|\psi^A_j(t)\rangle$, where I have to work out how each of the $|\psi^A_j\rangle$ evolves with time;

2) Defining the density matrix $\rho = \sum_i p_i|\psi^A\rangle\langle\psi^A|$, to then use $\frac{\partial \rho}{\partial t} = \frac{1}{i\hbar}[\rho, H]$.

I agree option #2 is easier, but option #1 is still feasible.

Is there any situation where I cannot use method 1 (pure state time evolution) but have to use use the density matrix approach?

• If your initial state is not pure... Like for instance a 50/50 mixture of $\vert +\rangle_z$ and $\vert +\rangle_y$. – ZeroTheHero Jan 15 '18 at 0:42
• Can I not write $|\Psi\rangle = 1/\sqrt{2} |+\rangle_z + 1/\sqrt{2} |+\rangle_y$? Or a similar combination with an orthogonal basis. – SuperCiocia Jan 15 '18 at 0:47
• $|\psi\rangle = \frac{1}{\sqrt{2}} |+\rangle_z + \frac{1}{\sqrt{2}} | + \rangle_y$ is a pure state, a superposition of two states. A 50/50 mixture cannot be describet by a pure state: 50% of the system is composed of $|+\rangle_z$ and 50% of $|+\rangle_z$. This is different from a superposition. – Patrick Jan 15 '18 at 0:59
• So if I did not want to use the density matrix method, I'd have to carry out 2 distinct calculations? – SuperCiocia Jan 15 '18 at 1:15
• I see your point though. This means that for my example, option #2 is wrong then? – SuperCiocia Jan 15 '18 at 1:21

Sorry, I have misunderstood your question in the comments. To be concrete, consider the Stern-Gerlach experiment. Assume you have an initial state $|\psi\rangle = c_0|z_+\rangle + c_1|z_-\rangle$ and pass it through a SG apparatus in the z direction. Then you know for sure that the end states are either $|z_+\rangle$ or $|z_-\rangle$. If you put your SG inside a black box so that you cannot know which of the end states it is, I believe you can treat this as a mixed state with $|c_0|^2$ probability of being $z_+$ and $|c_1|^2$ probability of being $z_-$: $$\rho = |c_0|^2|z_+\rangle\langle z_+| + |c_1|^2 |z_-\rangle\langle z_-|$$ So you can use this density matrix to evolve the system in time as you described.