{
  "id": "1407.5519",
  "version": "v3",
  "published": "2014-07-21T15:07:05.000Z",
  "updated": "2014-10-15T15:25:45.000Z",
  "title": "A mathematical model for measurements in Quantum Mechanics",
  "authors": [
    "Tuyen Trung Truong"
  ],
  "comment": "7 pages. Largely revised and extended. Main change: Add that the model can be extended to be compatible with quantum entanglement",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "Let $V=\\mathbb{C}^N$, and $H$ (an observable) a Hermitian linear operator on $V$. Let $v_1,..., v_n$ be an orthonormal basis for $V$. Let $\\mathcal{M}$ be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the $v_j$'s. Here we propose a simple model which explains the Born rule and is compatible with entanglement.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-23T14:20:46.000Z",
      "abstract": "Let $V=\\mathbb{C}^N$, and $H$ (an observable) a Hermitian linear operator on $V$. Let $v_1,\\ldots ,v_n$ be an orthonormal basis for $V$. Let $\\mathcal{M}$ be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the $v_j$'s. The states of $\\mathcal{M}$ are represented by $W=\\mathbb{C}^m$. Let the time interval to perform a measurement to be $[0,1]$. Let $\\mathcal{S}$, with state $\\xi$, be an observed system to be measured by $\\mathcal{M}$. Then, as well-known, the combined system $\\mathcal{S}+\\mathcal{M}$ can be regarded to have a Schrodinger's unitary evolution $\\widehat{U}(t)=e^{-i\\widehat{H}t/\\hbar}:V\\otimes W\\rightarrow V\\otimes W$, $t\\in [0,1]$. By the set up of $\\mathcal{M}$, there are $N$ \"gates\" $W_j:=\\widehat{U}(1)^{-1}(v_j\\otimes W)$ from which $\\xi \\otimes W$ must choose one to go through. To explain the randomness of the results, we define a notion of capacity for the gates, and a complete ordering on capacities of different gates. For a linear subspace $Z\\subset V\\otimes W$, we let $P_Z$ be the projection onto $Z$. The evolutions $$\\widehat{U}_j(t):=\\widehat{U}(t)\\widehat{U}(1)^{-1}P_{v_j\\otimes W}\\widehat{U}(1)$$ are characteristic for $\\mathcal{M}$, satisfy the same Schrodinger's equation as $\\widehat{U}(t)$, and $\\sum _{j=1}^N\\widehat{U}_j(t)=\\widehat{U}(t)$. To obtain the picture of what happens from the view of $V$ alone, we take the traces of $\\widehat{U}_j(t)$, that is $$U _j(t)\\xi =Tr_W(\\widehat{U}(t)\\widehat{U}(1)^{-1}P_{v_j\\otimes W}\\widehat{U}(1))\\xi.$$ Here, $w_1,\\ldots ,w_m$ are an orthonormal basis for $W$. If $Tr_W(\\widehat{H}P_{v_j\\otimes W}\\widehat{U}(1))= HTr_W(P_{v_j\\otimes W}\\widehat{U}(1))$ for all $j=1,\\ldots ,N$, then $\\mathcal{S}$ and $\\mathcal{M}$ may be regarded as independent after the measurement.",
      "comment": "6 pages. Proof for the claim on probabilities in Section 0.6 added. Definitions of $\\widehat{U}_j$ and $U_j$ slightly modified. Generalizations to the case where the states collapses to one of orthogonal linear subspaces added. Comments are welcome!",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-15T15:25:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum mechanics",
      "mathematical model",
      "orthonormal basis",
      "hermitian linear operator",
      "schrodingers unitary evolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5519T"
    }
  }
}