Chapter 1Basic Concepts of the QuantumTheory (I): HeisenbergUncertainty Principle1.1 Vectors and operatorsIn a classical system, there exists the direct correspondence between the state of the systemand the dynamical variables. Such direct correspondence does not exist in a quantumsystem. In Dirac formulation of quantum mechanics [1], we can deal with two strangecreatures, vectors and operators, in a Hilbert space to describe the state and the dynamicalvariable, respectively. In order to predict an experimental result, we have to project theoperator onto the vector.1.1.1 State vectorsThe state of a quantum object is described by a state vector, |ϕi (ket vector)(or equiva-lently by hϕ| (bra vector)), both of which describe the identical physical state. If the stateis a linear superposition state, expressed by|ϕi =XnCn|ϕni , (1.1)the corresponding bra vector is given by its hermitian adjointhϕ| =XnC∗nhϕn| =ÃXnCn|ϕni!+. (1.2)With each pair of ket vectors |ψi and |ϕi, we can define a scalar product, hϕ|ψi =hψ|ϕi∗, which is a c-number. The Schr¨odinger wavefunction in q-representation, ψ(q) ≡hq|ψi is just the projected coordinate of a state vector |ψi as shown in Fig. 1.1. hq| is aneigen-bra vector of coordinate. In contrast to a real vector in an ordinary space, thoseprojected coordinates are complex numbers rather than real numbers. The c-numbercoordinate is the probability amplitude, by which a quantum system is found in a positioneigen-state |qii. An important departure of the quantum theory from the classical theory1originates from (1.3) the fact that those probability amplitudes are c-numbers which carrythe amplitude and phase information simultaneously.ψ:qψ:Schrodinger wavefunctionstate vectorq2q1q3q3ψq1ψ..Figure 1.1: The state vector |ψi and the Schr¨odinger wavefunction ψ(q).The Schr¨odinger wavefunction in p-representation is given by [2]ϕ(p) ≡ hp|ψi =1√2π~Zψ(q) expµ−i~pq¶dq . (1.3)ϕ(p) and ψ(q) are a Fourier transform pair and includes exactly same information of aquantum object.If we know ψ(q) for all q values, it is said we have a complete information about thesystem and this situation is called a “pure state”. If we do not know ψ (q) completelybut have only partial information, that situation is called a “mixed state”. A state vectoris insufficient to describe such a situation. We need a new mathematical tool (densityoperator) for describing a mixed state, which will be introduced in Sec.2.1.1.1.2 Linear operatorsIf we associate each ket |ai in the space to another ket |bi by an operatorˆD,|bi =ˆD|ai , (1.4)andˆD satisfies the relationˆD(|a1i + |a2i) =ˆD|a1i +ˆD|a2i , (1.5)ˆD(c|ai) = cˆD|ai , (1.6)ˆD is called a linear operator.A linear operator which associates a ket vector |ψi to another ket vector |ϕi is alsocalled a projection operator, and is expressed byˆA ≡ |ϕihψ| . (1.7)A linear operatorˆA+which associates a bra vector hψ| to another bra vector hϕ| iscalled an adjoint operator toˆA:2lcl|ψiˆA=|ϕihψ|−−−−−−−−−−→ |ϕi =ˆA|ψi (1.8)hψ|ˆA+=|ψihϕ|−−−−−−−−−−→ hϕ| = hψ|ˆA+.IfˆA =ˆA+, the projection operatorˆA is called an Hermitian operator (or self-adjointoperator). IfˆA|ai = a|ai , (1.9)is satisfied, we have the following relations:ha|ˆA|ai = aha|aiha|ˆA|ai∗= ha|ˆA+|ai = a∗ha|ai (1.10)ButˆA =ˆA+→ a = a∗.The eigenvalue of an Hermitian operator is a real number.If we measure a dynamical variable of a physical system, such as p osition, momentum,angular momentum, energy, etc., the obtained values are always “real numbers”. Since theeigenvalues of Hermitian operators are “real numbers”, we can let an Hermitian operatorrepresent a dynamical variable. An Hermitian operator is in this sense called an observable.1.1.3 Probability interpretationIf an observableˆA is measured for a quantum object in a state |ψi, a measurement re-sult is one of the eigenvalues ofˆA. Which specific eigenvalue aiis obtained for a singlemeasurement is totally unknown. However, if we rep eat the preparation of a quantumobject in the same state and the measurement of the same observable, the probability ofobtaining a specific result aiis equal to |hai|ψi|2, which is the square of the Schr¨odingerwavefunction. This correspondence between the Schr¨odinger wavefunction and the “en-semble” measurement is only connection between the quantum theory and experimentalresult.The sum of the probabilities for all possible measurement results is unity :Xq|hq|ψi|2=Xqhψ|qihq|ψi = 1 , (1.11)Xq|qihq| =ˆI . (1.12)This relation(1.12) is called “completeness”. The eigen–states of an Hermitian operator(observable) form a complete set. If an Hermitian operator has continuous eigenvaluerather than discrete eigenvalues, the completeness relation is replaced byR|qihq|dq =ˆI.3Figure 1.2: Connection between the squared Schr¨odinger wavefunction and the probabilityof measurement results.1.1.4 Single quantum systemThe standard quantum theory describes a following ensemble measurement:1. Preparation of an ensemble of identical systems2. Noise-free measurement of a specific observableThe uncertainty (probability distribution) of the measurement results is attributed to thecharacteristics of the initial state.However, a following experimental situation is often encountered and becomes moreand more important recently:1. Prepare one and only one quantum system2. This is single quantum system couples to an unknown force (information source).3. To extract the information of the unknown force, a second quantum system (calledprobe) couples to the quantum system and the observable of the probe is measured.4. Repeat the process 2 and process 3 to monitor a time dependent unknown force, asshown in Fig. 1.3.In order to analyze the above situation, we must know not only the influence of the un-known force on the quantum system but also the influence of the coupling of the quantumprobe and the measurement of the probe observable on the quantum system. We mustgo beyond the standard probability interpretation of the Schr¨odinger wavefunction. Diracformulation of quantum mechanics is particularly useful for this goal, as will be discussedin Chapter 2.4ψF t( )systeminitial statemeas.probe#1meas.probe#2meas.probe#3Figure 1.3: A continuous monitoring unknown force F (t) by a single quantum system.1.2 Heisenberg uncertainty principleToday, the Heisenberg uncertainty