What's the difference between a fock state and a mixed ensemble?

[kof9595995]

I have trouble distinguishing the two, what's the physical difference between a fock state|p1;p2> and a mixed ensemble described by density matrix 0.5|p1><p1|+0.5|p2><p2|?

 

[kof9595995]

What confuses me is that both cases we have determined portions of particles in different states, and both have completely unknown phases, so I don't see the difference physically. However, mathematically a single vector in Hilbert space describes a pure state, doesn't it? So there has to be some differences between Fock state and a mixed state, but what are they?

 

[vanhees71]

A Fock state is a symmetrized (bosons) or anti-symmetrized (fermions) N-particle-product state. If you have a single-particle basis, you can build the N-particle basis from such (anti-)symmetrized product states. The Fock space is the orthogonal sum of all (anti-)symmetric N-particle states with $N \in \{0,1,2,3,\ldots\}$.

A system is in a quantum-theoretically completely determined state, if this state is described by a state vector, which can be an arbitrary superposition of (anti-)symmetrized product state and even a superposition of states of different particle number (except if you have a superselection rule which forbids such superpositions). To prepare a system in a pure state, you have to measure a complete set of compatible observables of the system precisely.

A mixed state is given by a positive semidefinite hermitean operator with trace 1. It describes a system, about which you only have incomplete knowledge. E.g., if you know only the total energy of a many-particle system (microcanonical ensemble) or even only the average energy (canonical ensemble).

 

[kof9595995]

To prepare a system in a pure state, you have to measure a complete set of compatible observables of the system precisely.

Ah.. I see, |p1;p2> can only be interpreted as 1 particle is in |p1> and the other particle is in |p2>, while 0.5|p1><p1|+0.5|p2><p2| could have different interpretation if we use another basis, in this sense |p1;p2> gives more information than 0.5|p1><p1|+0.5|p2><p2|, right？

 

[Physics Monkey]

$$|p_1 p_2 \rangle$$ is a pure state of two particles describing a particle with momentum p1 and a particle with momentum p2.

$$\frac{1}{\sqrt{2}}( |p_1 \rangle + |p_2 \rangle )$$ is a pure state of one particle in a quantum superposition of momentum p1 and momentum p2.

$$\frac{1}{2} |p_1 \rangle \langle p_1 | + \frac{1}{2} |p_2 \rangle \langle p_2 |$$ is a mixed state of one particle having momentum p1 with probability .5 and momentum p2 with probability .5 (with no interference between the states).

Hope this helps.

 

[kof9595995]

$$\frac{1}{2} |p_1 \rangle \langle p_1 | + \frac{1}{2} |p_2 \rangle \langle p_2 |$$ is a mixed state of one particle having momentum p1 with probability .5 and momentum p2 with probability .5 (with no interference between the states).

How can a mixed state describe 1 particle? I thought mixed state necessarily describe a many-particle system, with a lot of information loss.

 

[A. Neumaier]

How can a mixed state describe 1 particle? I thought mixed state necessarily describe a many-particle system, with a lot of information loss.

No. MIxedness and compositeness are two independent properties of a system.

Pure and mixed 1-particle states both describe the state of a single particle considered as a subsystem of a big system consisting of the pareticle and its environment (i.e., the rest of the universe).

Pure and mixed N-particle states both describe the state of N particles considered as a subsystem of a big system consisting of the pareticle and its environment (i.e., the rest of the universe).

The only difference between a 1-particle system and an N-particle system is in the complexity of the system: an N-particle states has N times more coordinates as a 1-particle state.

And the only difference between pureness and mixedness is that the pureness is an idealization that can often be used if interaction with the environment doesn't affect the system significantly. It simplifies the descriprion and makes a number of calculations more tractable.

 

[kof9595995]

The only difference between a 1-particle system and an N-particle system is in the complexity of the system: an N-particle states has N times more coordinates as a 1-particle state.

I have trouble understanding this, I'm now quite confused by the concepts 1-particle state, N-particle state and N-particle system, is there any detail instruction on this?

And the only difference between pureness and mixedness is that the pureness is an idealization that can often be used if interaction with the environment doesn't affect the system significantly. It simplifies the descriprion and makes a number of calculations more tractable.

Do you mean that for a pure entangled state, if we look only at part of the system, we'll see an mixed state?How can I prove this?

 

[A. Neumaier]

I have trouble understanding this, I'm now quite confused by the concepts 1-particle state, N-particle state and N-particle system, is there any detail instruction on this?

Classically and quantummechanically, a system of N particles has 3N position coordinates.
A 1-particle system is the special case N=1.

Do you mean that for a pure entangled state, if we look only at part of the system, we'll see an mixed state?How can I prove this?

Yes, usually. For example, if you have an entangled state 1/sqrt(2)(|1>x|1'>+2>x|2'>) and you look only at the unpimed particle, it appears as having the mixed state
rho = diag (1/2, 1/2).

 

[kof9595995]

Classically and quantummechanically, a system of N particles has 3N position coordinates.
A 1-particle system is the special case N=1.

Hmm, consider this, if we identically prepared N particles and put them together, shall we use 1-particle state or N-particle state to describe them? Is there any difference at all?

 

[A. Neumaier]

Hmm, consider this, if we identically prepared N particles and put them together, shall we use 1-particle state or N-particle state to describe them? Is there any difference at all?

A big difference.

N identically prepared particles may be described by a single particle state only if they don't coexist at the same time in the region of interest.

On the other hand, in an N-particle state all N particles are present simultaneously.

In particular, if you manage to put N particles together (by arranging a scattering experiment) then you need to describe them by an N-particle state.

 

[kof9595995]

A big difference.

N identically prepared particles may be described by a single particle state only if they don't coexist at the same time in the region of interest.

Why? I'm talking about non-interacting particles, and I'm not thinking about the indistinguishabilty effects(e.g. exclusion principle), then what's the reason we can't use 1-particle state to describe "N particle put together"?

 

[A. Neumaier]

Why? I'm talking about non-interacting particles, and I'm not thinking about the indistinguishabilty effects(e.g. exclusion principle), then what's the reason we can't use 1-particle state to describe "N particle put together"?

What do you mean by ''put together''?

In typical experiments, N identically prepared particles are produced one after another, and hence cannot be put together in any meaningful way.

The only way to put particles together is to have them interact.

 

[kof9595995]

I see, so the reason is interaction?

 

[A. Neumaier]

I see, so the reason is interaction?

Yes.

 

[kof9595995]

Thanks a lot~