qitd331 - Stochastic Quantum Dynamics I. Born Rule Robert...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Stochastic Quantum Dynamics I. Born Rule Robert B. Grifths Version oF 25 January 2010 Contents 1 Introduction 1 2 Born Rule 1 2.1 Statement oF the Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . ....... 1 2.2 Incompatible sample spaces . . . . . . . . . . . . . . . . . . . . . . . ......... 3 2.3 Born rule using pre-probabilities . . . . . . . . . . . . . . . . . . ........... 5 2.4 Generalizations oF the Born rule . . . . . . . . . . . . . . . . . . . .......... 5 2.5 Limitations oF the Born rule . 7 3T w oq u b i t s 8 3.1 Pure states at t 1 ...................................... 8 3.2 Properties oF one qubit at t 2 ................................ 9 3.3 Correlations and conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . 10 ReFerences: CQT = Consistent Quantum Theory by Grifths (Cambridge, 2002), Chs. 8, 9, 10, 11. 1 Introduction ± Quantum dynamics, i.e., the time development oF quantum systems, is Fundamentally stochas- tic or probabilistic. It is unFortunate that this basic Fact is not clearly stated in textbooks, including QCQI, which present to the student a strange mishmash oF unitary dynamics Followed by probabilities introduced by means oF measurements . While the measurement talk is Fundamentally sound when it is properly interpreted it can be quite conFusing. Measurements: topic For separate set oF notes What we will do is to introduce probabilities by means oF some simple examples which show what one must be careFul about when dealing with quantum systems, and the Fundamental principles which allow the calculation oF certain key probabilities related to quantum dynamics. 2B o r n R u l e 2.1 Statement of the Born Rule ± The Born rule can be stated in the Following way. Let t 0 and t 1 be two times, and T ( t 1 ,t 0 ) the unitary time development operator For the quantum system oF interest. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This system must be isolated during the time period under discussion, or at least it must not decohere, as otherwise the unitary T ( t 1 ,t 0 ) cannot be defned. ± Assume a normalized state | ψ 0 ± at t 0 and an orthonormal basis {| φ k 1 ±} ,k =0 , 1 ,... at t 1 . Then Pr( φ k 1 )=Pr( φ k 1 | ψ 0 )= φ k 1 | T ( t 1 0 ) | ψ 0 ±| 2 = ² ψ 0 | T ( t 0 1 ) | φ k 1 ±² φ k 1 | T ( t 1 0 ) | ψ 0 ± (1) is the probability oF | φ k 1 ± at t 1 given the initial state | ψ 0 ± at t 0 . In the argument oF Pr() and sometimes in other places we omit the ket symbol; Pr( φ k 1 ) means the probability that the system is in the state | φ k 1 ± , or in the ray (one-dimensional subspace) generated by | φ k 1 ± , or has the property | φ k 1 ± corresponding to the projector | φ k 1 ±² φ k 1 | =[ φ k 1 ]. The condition ψ 0 to the right oF the vertical bar | is oFten omitted when it is evident From the context. (±or more on conditional probabilities, see separate notes.) The fnal equality in (1) is a consequence oF the general rule ² χ | A | ω ± * = ² ω | A | χ ± , together with the Fact that T ( t 1 0 ) = T ( t 0 1 ). Mnemonic. In (1) subscripts indicate the time. In the matrix elements the time t j as an argument oF T always lies next to the ket/bra corresponding to this time. ± It should be emphasized that the Born rule, (1), is on exactly the same level as the time- dependent Schr¨odinger equation in terms oF Fundamental quantum principles. That is, it is a postulate or an axiom that does not emerge From anything else.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 10

qitd331 - Stochastic Quantum Dynamics I. Born Rule Robert...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online