[Paper Review] Minimalisation of uncertainty relations in noncommutative quantum mechanics

Piotr Kosiński, Katarzyna Bolonek-Lasoń|ArXiv.org|Aug 29, 2002

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper constructs explicit quantum states that saturate the uncertainty relations in noncommutative quantum mechanics (NCQM), where position-position noncommutativity is governed by a parameter $\theta$. Using Fock space representations and Bogolubov transformations, the authors derive states that minimize uncertainty for each of the three fundamental uncertainty relations—$\Delta x_1\Delta x_2 \geq \theta/2$, $\Delta x_1\Delta p_1 \geq \hbar/2$, and $\Delta x_2\Delta p_2 \geq \hbar/2$—and prove that no single state can saturate more than one relation simultaneously.

ABSTRACT

The explicit constrtuction of states saturating uncertainty relations following from basic commutation rules of NCQM is given both in Fock space and coordinate representation

Motivation & Objective

• To explicitly construct quantum states that saturate the uncertainty relations arising from the noncommutative algebra $[\hat{x}_i, \hat{x}_j] = i\theta\epsilon_{ij}I$.
• To resolve the ambiguity in prior work by providing a complete and explicit characterization of minimal uncertainty states in NCQM.
• To demonstrate that, unlike in standard quantum mechanics, no state can simultaneously saturate more than one of the three uncertainty relations in NCQM.
• To provide a unified framework using Fock space and coherent state formalism to derive and classify minimal uncertainty states.

Proposed method

• Constructs a Fock space representation of the NCQM algebra via modified creation/annihilation operators $a_i, a_i^\dagger$ related to $\hat{x}_i, \hat{p}_i$ through a unitary transformation.
• Introduces new operators $b, b^\dagger$ and $c, c^\dagger$ to map the $\hat{x}_1, \hat{x}_2$ uncertainty relation to a standard Heisenberg-type form.
• Applies the standard coherent state formalism to the $b$-operators to generate states saturating $\Delta x_1 \Delta x_2 \geq \theta/2$, parameterized by complex $z$ and real $\gamma$.
• Uses Bogolubov transformations via the unitary operator $V(\gamma) = e^{-\frac{1}{4}\ln\gamma(a^2 - (a^\dagger)^2)}$ to generate minimal uncertainty states for $\Delta x_1\Delta p_1$ and $\Delta x_2\Delta p_2$ relations.
• Derives coordinate-space wave functions for the minimal uncertainty states and verifies analytically that no wave function can saturate more than one uncertainty relation.
• Employs the generalized Heisenberg inequality $(\Delta A)_\psi (\Delta B)_\psi \geq \frac{1}{2}|\langle C \rangle_\psi|$ to derive the saturation condition and confirms the structure of minimal states.

Experimental results

Research questions

• RQ1 What are the explicit forms of quantum states that saturate the uncertainty relation $\Delta x_1 \Delta x_2 \geq \theta/2$ in noncommutative quantum mechanics?
• RQ2 Can a single quantum state simultaneously saturate more than one of the three fundamental uncertainty relations in NCQM?
• RQ3 How do the dispersions $\Delta x_1$, $\Delta x_2$, $\Delta p_1$, $\Delta p_2$ behave in states that saturate individual uncertainty relations?
• RQ4 What is the role of Bogolubov transformations and coherent states in constructing minimal uncertainty states in Fock space for NCQM?
• RQ5 How do the coordinate-space wave functions of minimal uncertainty states reflect the noncommutative structure of the underlying algebra?

Key findings

• The paper constructs explicit minimal uncertainty states for each of the three uncertainty relations in NCQM: $\Delta x_1\Delta x_2 \geq \theta/2$, $\Delta x_1\Delta p_1 \geq \hbar/2$, and $\Delta x_2\Delta p_2 \geq \hbar/2$.
• The minimal uncertainty states for $\Delta x_1\Delta x_2 \geq \theta/2$ are given by $|z,\gamma\rangle_\phi = e^{-\frac{1}{2}|z|^2} e^{\frac{1}{4}\ln\gamma((b^\dagger)^2 - b^2)} e^{zb^\dagger}|\phi\rangle$, where $b$ is a combination of $a_+$ and $a_-$ operators.
• For the $\Delta x_1\Delta p_1$ and $\Delta x_2\Delta p_2$ relations, the minimal states are generated via a unitary transformation $V(\gamma)$ acting on standard coherent states, with $\gamma$ controlling the relative spread of $x$ and $p$.
• The dispersions in the minimal states are quantitatively related to $\gamma$: $(\Delta x)^2 = \frac{\gamma\hbar}{2}$ and $(\Delta p)^2 = \frac{\hbar}{2\gamma}$, confirming the saturation of the Heisenberg-type inequality.
• It is rigorously shown that no wave function can simultaneously saturate more than one of the three uncertainty relations, due to the noncommutative structure of $\hat{x}_i$.
• The coordinate-space wave functions of the minimal states are explicitly derived and shown to satisfy the uncertainty saturation conditions through direct computation.