Quantum Measurement Theory on Haifei's Home

The collapsed state of position measurement

hfwang132@gmail.com (hfwang132) — Thu, 23 May 2024 01:30:11 +0800

Common sense tells us that an actual position measurement will not produce a delta function, as the delta function itself is pathological. So what should the collapsed state of an actual position measurement look like?

TL;DR:

Let the wave function of the system to be measured be $\varphi(x)$, then the collapsed wave function is:

$\varphi^{(q)}(x) = \mathcal{A}\psi(q-gx) \varphi(x)$

where $\psi(x)$ is the initial wave function of the instrument’s pointer. $q$ is the reading of the instrument’s pointer. $g$ is the coupling strength between the instrument and the system to be measured. $\begin{aligned} \mathcal{A} = \left[\int_\mathbb{R}\psi(q-gx) \varphi(x)\right]^{-1} \end{aligned}$ is the appropriate normalization constant.

The Difference Between Quantum Entanglement and Classical Correlations, and the Quantum Measurement Problem

hfwang132@gmail.com (hfwang132) — Tue, 30 Apr 2024 15:24:28 +0800

When many physicists popularize quantum entanglement to the general public, they often give an example:

Imagine that you have two boxes, one containing a pizza and the other containing a hamburger, and you cannot know what is inside before opening them. Alice and Bob each take one box and travel to places far apart. At this point, when Alice opens her box, she can know what is in Bob’s box far away.

Generalized Quantum Measurements: An Introduction to POVMs

hfwang132@gmail.com (hfwang132) — Sun, 21 Jan 2024 22:13:57 +0800

Projective Measurements

A measurement in the traditional sense (in the Von Neumann sense) is a collection of projection operators. By performing a spectral decomposition of the self-adjoint operator corresponding to an observable $O=\sum_i\lambda_i |\varphi_i\rangle\langle\varphi_i|$ , one obtains these projection operators $|\varphi_i\rangle\langle\varphi_i|$ . Anyone who has studied elementary quantum mechanics should be familiar with this part.

In addition to Von Neumann measurements, there is a more general type of measurement called a generalized measurement (Generalized Measurements), also known as a POVM (Positive Operator Valued Measure).