Advanced Visual Quantum Mechanics by Bernd Thaller

Author:Bernd Thaller [Thaller, Bernd]
Language: eng
Format: epub, pdf
Published: 2007-08-26T13:59:00+00:00

The meaning of the density operator:

A mixed state p of a physical system is a statistical ensemble of pure states {Wi, 02, ...} (forming an orthonormal set), where each pure state ,4,j occurs with a probability p3. The probabilities pj are just the eigenvalues of the density operator p, and the pure states VJj are the corresponding eigenvectors.

Whenever one of the eigenvalues of p is degenerate, then the canonical form of p is not unique. Consider, for example, a qubit in a mixed state described by the density matrix

P=2IV,+)(,+I+2 v->("4,-I=21. (5.94)

Here, the eigenvalue pi = P2 = 1/2 is degenerate with degree 2. But any orthonormal basis {0i, z('2} in C2 has the completeness property

'I)("'iI + I'/'2)(/'2I = 1, (5.95)

and hence the density matrix (5.94) can be represented in a multitude of ways.

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