Understanding Spin Density Matrix Invariance

[baru]

Could anyone help me to understand how the spin density matrix is invariant under unitary transformation?

 

[tonyetc]

hi,
if you mean how a general density matrix is invariant under unitary transformation, the consider that density matrices are used to evaluate mean values of an operator A, <A R>= Tr(AR), where R is the density matrix and Tr denotes the trace operation. Then performing a unitary transformation yields the same result: Tr(SAS^-1 SRS^-1) = Tr(SARS^-1) = Tr(S^-1SAR) = Tr(AR) because of invariance under cyclic permutation of the trace.
If you mean something else, please forget my answer and post again.
thanks

 

[Entropia]

The spin density matrix is a mathematical representation of the spin state of a quantum mechanical system. It is a useful tool for studying the properties and behavior of particles with spin, such as electrons. One important property of the spin density matrix is its invariance under unitary transformations.

A unitary transformation is a mathematical operation that preserves the inner product and norm of a quantum state. In other words, it does not change the physical state of the system, but only changes the way we describe it mathematically. This means that if we apply a unitary transformation to a spin density matrix, the resulting matrix will still accurately represent the spin state of the system.

To understand why the spin density matrix is invariant under unitary transformations, we need to look at the mathematical definition of the matrix. The spin density matrix is a 2x2 matrix that contains information about the probabilities of finding a particle with spin up or spin down along different directions. These probabilities are represented by the elements of the matrix.

When we apply a unitary transformation to a spin density matrix, we are essentially changing the basis or reference frame in which we are describing the spin state. However, the probabilities of finding a particle with spin up or down along different directions do not change. This is because unitary transformations do not affect the physical state of the system, only the mathematical representation of it.

Therefore, the spin density matrix remains unchanged under unitary transformations, making it a useful and important tool in quantum mechanics. It allows us to study the properties of spin in a system without being affected by changes in the mathematical representation of the state. I hope this explanation helps you understand the invariance of the spin density matrix under unitary transformations.