# Time in Quantum Measurement

## Abstract

**:**

**94A17**;

**81P15**

## 1 Introduction

## 2 Time

#### 2.1 Dissipation

_{i}, i = 1, 2 the states of the apparatus and ε

_{i}, i = 1, 2 the states of the environment. We may assume that $\langle 1|2\rangle =\langle {A}_{0}|{A}_{1}\rangle =\langle {\epsilon}_{0}|{\epsilon}_{1}\rangle =0$. Let $\mathcal{E}$ be in thermal equilibrium of temperature T. For the simplicity of notation we will just write down the coefficients ${\alpha}_{i}=\mathbb{C}$ and their conjugates ${\alpha}_{i}^{*}\in \mathbb{C},i=1,2$ of the density matrix and omit the basis vectors in each step of sequence (1). This leaves us with

_{i}, i = 1, 2.

#### 2.2 External Time

#### 2.2.1 Scope

_{O}. Following [1] , the intrinsic energy uncertainty on the basis of quantities given in our model is

#### 2.2.2 Application

^{2}for some fixed $c\in \mathbb{R}$ and all x ≥ 0, 2) L

^{4}(${\mathbb{R}}^{3}$) ⊂ L

^{2}(${\mathbb{R}}^{3}$), 3) lim

_{x→0}x

^{s}ln x → 0, s, x > 0.

_{R}needed to measure our particle at a distance of at least R would by (4) be

_{R}is defined as

_{φ}> 0 denotes the constant depending upon the set up (3).

_{R}is the minimal time it takes for the particle to "reach" a point at distance R or further.

_{R}→ 0 as R → ∞, the intuitive fact that it takes longer to reach more distant regions, is assured.

_{R}from R. Since φ ∈ L

^{2}(ℝ

^{3}) we have for some R

_{0}> 0, ε > 0

_{0}big enough

**Interpretation**We try to reconstruct as closely as possible an analogue of classical velocity from quantum mechanics. Thereby we focus on the reconstruction of the most simple situation where there is a free particle in space of temperature T. Due to the probabilistic nature of quantum mechanics it is clear that the reconstruction can only be done by using quantities which are at most analogous to the classical notions of "distance passed on a straight line" and "time to pass through that distance" which form the definition of classical velocity.

_{ϕ}, (6) would give a general bound on the velocity of a free particle.

## 3 Comments

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**MDPI and ACS Style**

Schlatter, A.E.
Time in Quantum Measurement. *Entropy* **2006**, *8*, 182-187.
https://doi.org/10.3390/e8030182

**AMA Style**

Schlatter AE.
Time in Quantum Measurement. *Entropy*. 2006; 8(3):182-187.
https://doi.org/10.3390/e8030182

**Chicago/Turabian Style**

Schlatter, Andreas E.
2006. "Time in Quantum Measurement" *Entropy* 8, no. 3: 182-187.
https://doi.org/10.3390/e8030182