To introduce the density operator, let us first consider a

|ψ

The system is in a well defined state, but if more than one of the c

Let <ψ

The mean value of an observable A at time t is

<A>(t) = <ψ_{k}(t)|A|ψ_{k}(t)> = ∑_{np}c_{n}^{k}*(t)c_{p}^{k}(t)A_{np}, where A_{np} = <u_{n}|A|u_{p}>.

We now introduce the **pure state density operator** ρ^{k}(t)
= |ψ_{k}(t)><ψ_{k}(t)|,

i.e. the projector onto the state |ψ_{k}(t)>,
and the **density matrix** with **matrix
elements
**ρ

We will now show that knowing the density operator (or the elements of the density
matrix) completely suffices to characterize the state of the system. All of the
predictions of Quantum Mechanics that can be made knowing |ψ_{k}(t)>
can also be made knowing ρ^{k}(t).

In terms of the density operator ρ^{k}(t)
we express:

- (a)
**conservation of probability**,

<ψ_{k}|ψ_{k}> = ∑_{n}<ψ_{k}|u_{n}><u_{n}|ψ_{k}> = ∑_{n}<u_{n}|ψ_{k}><ψ_{k}|u_{n}> = ∑_{n}ρ_{nn}^{k}(t) = Tr{ρ^{k}(t)} = 1.

(d/dt)Tr{ρ^{k}(t)} = 0. - (b)
**the mean value of the observable A**,

<A> = ∑_{np}c_{n}^{k}*(t)c_{p}^{k}(t)A_{np}= ∑_{np}ρ_{pn}^{k}(t)A_{np}= ∑_{np}<u_{p}|ρ^{k}(t)|u_{n}><u_{n}|A|u_{p}>

= ∑_{p}<u_{p}|ρ^{k}(t)A|u_{p}> = Tr{ρ^{k}(t)A}. - (c)
**the time evolution of the system**,

(d/dt)ρ^{k}(t) = (d|ψ_{k}>/dt) <ψ_{k}| + |ψ_{k}>(d<ψ_{k}|/dt)

= (1/(iħ)H|ψ_{k}><ψ_{k}| - (1/(iħ)|ψ_{k}><ψ_{k}|H

= (iħ)^{-1}[H,ρ^{k}(t)]. - (d)
**the probability that a measurement of A will yield the eigenvalue a**._{n}

P_{k}(a_{n}) = <ψ_{k}(t)|P_{n}|ψ_{k}(t)> = Tr{ρ^{k}(t)P_{n}},

where P_{n}is the projector into the eigensubspace of A associated with the eigenvalue a_{n}. The probability is the mean value of P_{n}.

We have therefore shown that knowing ρ^{k}(t) suffices to characterize the quantum state of the system.

It is a

**Why is the density matrix formulation useful?
**The important equations,

Tr{ρ

<A> = Tr{ρ

P

(d/dt)ρ

are all

d|ψ(x)|

<A> = ∫ψ*(x) A ψ(x)dx

P

The linear equations can easily be generalized to describe a statistical mixture of states and express all physical predictions about such a system in terms of a density operator.

Note:

- If {|u
_{n}>} is an eigenbasis of H and |ψ_{k}(t)> is an eigenstate of H, then the matrix ρ_{np}^{k}(t) of the pure state density operator is diagonal. All off-diagonal elements are zero.

|ψ_{k}> = |u_{k}>, ρ_{k }= |u_{k}><u_{k}|. - If |ψ
_{k}(t)> is a coherent state then ρ_{np}^{k}(t) has off-diagonal elements.

| ψ_{k}> = ∑_{n}c_{n}|u_{n}>, ρ^{k}= ∑_{nn'}c_{n}*c_{n'}|u_{n'}><u_{n}|. - We can always choose {|u
_{n}>} to be an eigenbasis of the density operator itself.

Then ρ_{np}^{k}(t) is diagonal.

In experiments, we are often dealing with systems about which we have **incomplete
information**.

- A C
^{4+}beam emerges from an accelerator. The C ions have been stripped of most of their electrons by having been past through a thin foil. Most ions are in their ground state, but some are in an excited, metastable state. For each ion we only know that it may be in a state |ψ_{k}> with probability p_{k}, ∑_{k}p_{k}= 1. We are dealing with a**statistical mixture of states**.

For such a system we introduce the density operator ρ(t) = ∑_{k}p_{k}ρ^{k}(t), and the density matrix with matrix elements

ρ_{pn}(t) = ∑_{k}p_{k}<u_{p}|ρ^{k}(t)|u_{n}>
= = ∑_{k}p_{k}c_{p}^{k}(t)c_{n}^{k}*(t).

In terms of the density operator we express:

- (a)
**conservation of probability**,

Tr{ρ(t)} = Tr{∑_{k}p_{k}ρ^{k}(t))} = ∑_{k}p_{k}Tr{ρ^{k}(t)) = ∑_{k}p_{k}= 1. - (b)
**the probability that a measurement of the observable A will yield the eigenvalue a**,_{n}

P(a_{n}) = ∑_{k}p_{k}P_{k}(a_{n}) = ∑_{k}p_{k}Tr{ρ^{k}(t)P_{n}} = Tr{∑_{k}p_{k}ρ^{k}(t)P_{n}} = Tr{ρ(t)P_{n}},

where P_{n}is the projector into the eigensubspace of A associated with the eigenvalue a_{n}. - (c)
**the mean value of the observable A**,

<A>(t) = ∑_{n}a_{n}P(a_{n}) = ∑_{n}a_{n}Tr{ρ(t)P_{n}} = Tr{ρ(t)∑_{n}a_{n}P_{n}} = Tr{ρ(t)A},

since ∑_{n}a_{n}P_{n}= A, A|ψ> = ∑_{n}A|a_{n}><a_{n}|ψ> = ∑_{n}a_{n}|a_{n}><a_{n}|ψ> = ∑_{n}a_{n}P_{n}|ψ>,

where {|a_{n}>} is an eigenbasis of A. - (d)
**the time evolution of the system**.

The linearity of (d/dt)ρ^{k}(t) = (iħ)^{-1}[H,ρ^{k}(t)] implies that (d/dt)ρ(t) = (iħ)^{-1}[H,ρ(t)].

**Properties of ρ(t)
**ρ(t) is not a projector in general; ρ(t)

- Let |ψ
_{k}> = |u_{k}> with probability p_{k}, ρ = ∑_{k'}p_{k'}|u_{k'}><u_{k'}|,

ρ_{kk}= <u_{k}|∑_{k'}p_{k'}|u_{k'}><u_{k'}|u_{k}> = p_{k}, ρ_{k}_{n}(k ≠ n) = 0, Tr{ρ} = ∑_{k}p_{k}= 1, Tr{ρ^{2}} = ∑_{k}p_{k}^{2}≤ 1.

ρ(t) is a Hermitian operator (ρ^{† }= ρ from the definition of ρ)
and

<ψ_{a}|ρ||ψ> = ∑_{k}p_{k}<ψ|ψ_{k}><ψ_{k}|ψ>
= ∑_{k}p_{k}|<ψ|ψ_{k}>|^{2} ≥ 0.

ρ(t) is a **positive operator**.