(a) What are the reduced density matrices in position and...

(a) What are the reduced density matrices in position and momentum spaces? (b) Let us denote the reduced density matrix in momentum space by $\phi\left(\mathbf{p}_1, \mathbf{p}_2\right)$. Show that if $\phi$ is diagonal, that is, $$ \phi\left(\mathbf{p}_1, \mathbf{p}_2\right)=f\left(\mathbf{p}_1\right) \delta_{\mathbf{p}_1, \mathbf{p}_2}, $$ then the diagonal elements of the position density matrix are constant.

(a) What are the reduced density matrices in position and momentum spaces?
(b) Let us denote the reduced density matrix in momentum space by $\phi\left(\mathbf{p}_1, \mathbf{p}_2\right)$. Show that if $\phi$ is diagonal, that is,
$$
\phi\left(\mathbf{p}_1, \mathbf{p}_2\right)=f\left(\mathbf{p}_1\right) \delta_{\mathbf{p}_1, \mathbf{p}_2},
$$
then the diagonal elements of the position density matrix are constant.

Key Concepts

Reduced Density Matrix

A reduced density matrix is obtained by tracing out part of a larger quantum system’s degrees of freedom, effectively describing the state of a subsystem. It retains complete statistical information about the subsystem itself, including quantum coherences and correlations, while discarding information about the rest of the system.

Position and Momentum Space Representations

Quantum states can be represented in different bases, with position and momentum spaces being the two most commonly used. The density matrix expressed in the position basis provides probabilities and coherences with respect to spatial coordinates, while in the momentum basis it describes the distribution over momentum eigenstates. The two representations are connected via Fourier transforms.

Diagonal Density Matrices

When a density matrix is diagonal in a given basis, it indicates that in that representation, the state lacks coherences between different eigenstates and is essentially a statistical mixture of the corresponding eigenstates. In the momentum basis, a diagonal density matrix implies that the momentum distribution is incoherent, leading to specific properties when transformed to the position basis.

Fourier Transform and Constant Diagonal in Position Space

The Fourier transform connects the momentum and position representations of quantum states. In the case where the momentum-space density matrix is diagonal, the transform yields a position-space density matrix whose diagonal elements (i.e., the local probabilities) are found to be constant, reflecting a uniform distribution in position due to the structure of the momentum-space function.

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