for a single particle state, its wave function can only be a superposition of eigrnstates:

where is the probability of the corresponding eignstate. the eigen equation is:

Notice that the state need not be the eigen state. The expectation of operator Q for a pure state is straight forward:

However, for many particles state, the wave function is a sum of individual wave function of each particle.

where N is the total number of state, such that the individual wave functions are independent.

the expectation for operator Q will be:

due to the in dependency of each particle wave function, the double sum reduce to a single sum.

by using the identity,

rearrange by taking out the eigen ket out:

therefore, we can write it as the trace of matrix under eigen ket bais:

such that the density matrix:

Under rotation, like Rabi π-pulse, if we using the same ket as basis, the expectation value can be found by apply the rotation on the density matrix.

=============Example=================

Pure state:

if we use the eigen basis and donate:

Thus the trace is 0.

Mixed state:

notices that the eigen state from each particle cannot be added.

For a general 2 states system:

the is population of state k. and k is either up-state, down-state, transverse-state or random state. since the random state will cancel itself on both 3 axis, thus, it make no contribution.

the meaning of the off-diagonal value can be understood by finding the expectation of and . If we we have some mixed state with 40% up-state , 20% down-state and 40 % in random state.

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