20 Basic formalism then Tr(Pa)=1. (1.129) We consider now a mixed ensemble that contains the states ),B)...etc.with probabilities ,w...,respectively.We define a density operator p as p=∑waPa=∑wala)al. (1.130) Since wa,being the probability,is real and P is Hermitian,p is,therefore,Hermitian, p'=p. (1.131) From(1.117)and(1.128)the ensemble average is e=∑wTr(4P.). (1.132) We note that since is a number and not a matrix,and,at the same time,since the operator A is independent of a,and thus does not participate in the summation over a,we can reshuffle the terms in(1.133)to obtain (4)av Tr(4>WaPa=Tr(Ap)=Tr(PA). (1.133) The last equality follows from the property that the trace of a product of two matrices is invariant under interchange of the matrices From(1.117)we find,by taking 4=1,that m=∑aa川a)=∑u=1 (1.134) Therefore,from(1.134),we get (1)av Tr(p) (1.135) Finally,(1.135)and(1.136)imply Tr(p)=1. (1.136 We will discuss the properties of p in Chapter 14 for the specific case of the spinparticles, and again in the chapter on two-level problems. 1.13 Measurement When a measurement of a dynamical variable is made on a system and a specific,real, alue for a physical observable is found,then that value is an eigenvalue of the operator
20 Basic formalism then Tr(Pα) = 1. (1.129) We consider now a mixed ensemble that contains the states |α, |β, ... etc. with probabilities wα, wβ ... , respectively. We define a density operator ρ as ρ = α wαPα = α wα|αα|. (1.130) Since wα, being the probability, is real and Pα is Hermitian, ρ is, therefore, Hermitian, ρ† = ρ. (1.131) From (1.117) and (1.128) the ensemble average is Aav = α wαTr(APα). (1.132) We note that since wα is a number and not a matrix, and, at the same time, since the operator A is independent of α, and thus does not participate in the summation over α, we can reshuffle the terms in (1.133) to obtain Aav = Tr A α wαPα = Tr(Aρ) = Tr(ρA). (1.133) The last equality follows from the property that the trace of a product of two matrices is invariant under interchange of the matrices. From (1.117) we find, by taking A = 1, that 1av = α ωα α |1| α = α ωα = 1. (1.134) Therefore, from (1.134), we get 1av = Tr(ρ). (1.135) Finally, (1.135) and (1.136) imply Tr(ρ) = 1. (1.136) We will discussthe properties of ρ in Chapter 14 for the specific case of the spin ½ particles, and again in the chapter on two-level problems. 1.13 Measurement When a measurement of a dynamical variable is made on a system and a specific, real, value for a physical observable is found, then that value is an eigenvalue of the operator
21 1.14 Problems representing the observable.In other words,irrespective ofthe state,the act ofmeasurement kicks the state of the operator From the superposition principle,the state of a system,say )can always be expressed as a superposition ofbasis states,which we take to be normalized kets.One can choose these states to be the eigenstates of the operator,e.g.,).Thus one can write the superposition as )-∑snla). (1.137刀 When a single measurement is made,one of the eigenstates of this operator in this superposition will be observed and the corresponding eigenvalue will be measured.In other words,the state )will "collapse"to a state )The probability that a particular eigenvalue,will be measured in asingle measurement is given byBu t a second measurement ofan identical system may yield another eigenstate with a different eigenvalue and a different probability.Repeated measurements on identically prepared systems,will then give the probability distribution of the eigenvalues and yield information on the nature ofthe syst In practice,one prefers tomake mea rements on a large number ofidentical systems.which gives the same probability distribution. Similar arguments follow if a measurement is made to determine whether the system is in state).In this case the original state)will "jump"into the state)with the probability)2 The concept of measurement and the probability interpretation in quantum mechanics is a complex issue that we will come back to when we discuss Stern-Gerlach experiments in Chapter 7 and entangled states in Chapter 30. 1.14 Problems 1.Define the following two state vectors as column matrices: la)0 and la2)0 with their Hermitian conjugates given by al=[10]and(a2l=[01] respectively.Show the following for i,j=1,2: (i)The )'s are orthonormal (ii)Any column matrix [ can be written as a linear combination of the la)'s
21 1.14 Problems representing the observable. In other words, irrespective of the state, the act of measurement itself kicks the state into an eigenstate of the operator in question. From the superposition principle, the state of a system, say |φ, can always be expressed as a superposition of basis states, which we take to be normalized kets. One can choose these states to be the eigenstates of the operator, e.g., |an. Thus one can write the superposition as |φ = n cn |an. (1.137) When a single measurement is made, one of the eigenstates of this operator in this superposition will be observed and the corresponding eigenvalue will be measured. In other words, the state |φ will “collapse” to a state |an. The probability that a particular eigenvalue, e.g., an will be measured in a single measurement is given by |cn| 2. But a second measurement of an identical system may yield another eigenstate with a different eigenvalue and a different probability. Repeated measurements on identically prepared systems, will then give the probability distribution of the eigenvalues and yield information on the nature of the system. In practice, one prefers to make measurements on a large number of identical systems, which gives the same probability distribution. Similar arguments follow if a measurement is made to determine whether the system is in state |ψ. In this case the original state |φ will “jump” into the state |ψ with the probability |ψ |φ|2. The concept of measurement and the probability interpretation in quantum mechanics is a complex issue that we will come back to when we discuss Stern–Gerlach experiments in Chapter 7 and entangled states in Chapter 30. 1.14 Problems 1. Define the following two state vectors as column matrices: |α1 = 1 0 and |α2 = 0 1 with their Hermitian conjugates given by α1| = 1 0 and α2| = 0 1 respectively. Show the following for i, j = 1, 2: (i) The |αi’s are orthonormal. (ii) Any column matrix a b can be written as a linear combination of the |αi’s.
22 Basic formalism (iii)The outer products la)(ail form 2 x 2 matrices which can serve as operators. (iv)The a)'s satisfy completeness relation ∑la〉(al=1 where 1 represents a unit 2x 2 matrix (v)Write 4-e剑 as a linear combination of the four matrices formed by la)(jl. (vi)Determine the matrix elements of 4 such that la)and la2)are simultaneously the eigenstates of 4 satisfying the relations Ala1）=+la1）and Alo2〉=-la2) (The above properties signify that the )'s span a Hilbert space.These abstract ons of the state significance.They Chapter 5.) 2.Show that if an operator A is a function of then dA- d 3.Show that a unitary operator Ucan be written as U= 1+i迟 1- where K is a Hermitian operator.Show that one can also write U=ec where C is a Hermitian operator.If U=A+iB Show that A and B commute.Express these matrices in terms of C.You can assume that =1+M+ 2+ where M is an arbitrary matrix. 4.Show that det(e）=eo
22 Basic formalism (iii) The outer products |αiαj| form 2 × 2 matrices which can serve as operators. (iv) The |αi’s satisfy completeness relation i |αi αi| = 1 where 1 represents a unit 2 × 2 matrix. (v) Write A = a b c d as a linear combination of the four matrices formed by |αiαj|. (vi) Determine the matrix elements of A such that |α1 and |α2 are simultaneously the eigenstates of A satisfying the relations A |α1 = + |α1 and A |α2 = − |α2. (The above properties signify that the |αi’s span a Hilbert space. These abstract representations of the state vectors actually have a profound significance. They represent the states of particles with spin ½. We will discuss this in detail in Chapter 5.) 2. Show that if an operator A is a function of λ then dA−1 dλ = −A−1 dA dλ A−1. 3. Show that a unitary operator U can be written as U = 1 + iK 1 − iK where K is a Hermitian operator. Show that one can also write U = eiC where C is a Hermitian operator. If U = A + iB. Show that A and B commute. Express these matrices in terms of C. You can assume that eM = 1 + M + M2 2! +··· where M is an arbitrary matrix. 4. Show that det eA = eTr(A)
23 1.14 Problems 5.For two arbitrary state vectors la)and |B)show that Tr [la)(BI]=(Bl a) 6.Consider a two-dimensional space spanned by two orthonormal state vectors la)and IB).An operator is expressed in terms of these vectors as A la)(a+B)(al+*la)(B1+ulB)(B1. Determine the eigenstates of4 for the case where(①x=l,u=±l,(iiλ=i,u=±l. Do this problem also by expressing as a2x2 matrix with eigenstates as the column matrices
23 1.14 Problems 5. For two arbitrary state vectors |α and |β show that Tr [|α β|] = β| α. 6. Consider a two-dimensional space spanned by two orthonormal state vectors |α and |β. An operator is expressed in terms of these vectors as A = |α α| + λ |β α| + λ∗ |α β| + μ|β β| . Determine the eigenstates of A for the case where (i) λ = 1, μ = ±1, (ii) λ = i, μ = ±1. Do this problem also by expressing A as a 2 × 2 matrix with eigenstates as the column matrices
Fundamental commutator and time evolution of state vectors and operators In the previous chapter weoutlined the basic mathematical structure essential for our studies. We are now ready to make contact with physics.This means introducing the fundamental constant hi,the Planck constant,which controls the quantum phenomena.Our first step will be to discuss the so-called fundamental commutator,also known as the canonica commutator,which is proportional to hand which essentially dictates how the quantum processes are described.We then examine how time enters the formalism and thus set the stage for writing equations of motion for a physical system. 2.1 Continuous variables:X and P operators Eigenvalues need not always be discrete as we stated earlier.For example,consider a one- dimensional,continuous(indenumerable)infinite-dimensional position space,the x-space. One could have an eigenstateof a continuous operator xk)=xk) (2.1) where xcorresponds to the value of the x-variable. The ket has all the properties of the kets la)and of the eigenstates lan)that were outlined in the previous chapter.The exceptions are those cases where the fact that x is a continuous variable makes an essential difference 8-function (x-x),which has the following properties: 8x-x=0forx≠x (2.2) dk8x-x0=1. 2.3) From these two definitions it follows that dxf(x)8(x-x)=f(x'). (2.4) -0 The properties of the delta function are discussed in considerable detail in Appendix 2.9
2 Fundamental commutator and time evolution of state vectors and operators In the previous chapter we outlined the basic mathematical structure essential for our studies. We are now ready to make contact with physics. This means introducing the fundamental constant , the Planck constant, which controls the quantum phenomena. Our first step will be to discuss the so-called fundamental commutator, also known as the canonical commutator, which is proportional to and which essentially dictates how the quantum processes are described. We then examine how time enters the formalism and thus set the stage for writing equations of motion for a physical system. 2.1 Continuous variables: X and P operators Eigenvalues need not always be discrete as we stated earlier. For example, consider a onedimensional, continuous (indenumerable) infinite-dimensional position space, the x-space. One could have an eigenstate x of a continuous operator X , X x = x x (2.1) where x corresponds to the value of the x-variable. The ket x has all the properties of the kets |α and of the eigenstates |an that were outlined in the previous chapter. The exceptions are those cases where the fact that x is a continuous variable makes an essential difference. δ-function δ(x − x ), which has the following properties: δ(x − x ) = 0 for x = x , (2.2) ! ∞ −∞ dx δ(x − x ) = 1. (2.3) From these two definitions it follows that ! ∞ −∞ dx f (x) δ(x − x ) = f (x ). (2.4) The properties of the delta function are discussed in considerable detail in Appendix 2.9.