In this chapter, we develop results that will prove to be very important later. First, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0001.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0002.tif"/> are compatible , that is, if their Hermitian operator representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0003.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0004.tif"/> commute, then these representatives have a complete orthonormal set of common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0005.tif"/> which are the solutions of the simultaneous eigenvalue problem A ^ Ψ a n , b m ( r ) = a n Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0006.tif"/> and B ^ Ψ a n , b m ( r ) = b n Ψ a n , b m ( r ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0007.tif"/> Consequently, from the discussion at the end of Section 4.5 , when the system is in a quantum state specified by one of the common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0008.tif"/> , both A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0009.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0010.tif"/> have definite values, a n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0011.tif"/> for A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0012.tif"/> and b m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0013.tif"/> for B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0014.tif"/> In contrast, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0015.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0016.tif"/> are incompatible , that is, if their representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0279.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0017.tif"/> do not commute, then these representatives have few if any common eigenfunctions. As a consequence rarely if ever can one find quantum states for which two incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0018.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0019.tif"/> both have definite values rather than spreads of values. Therefore, two incompatible observables cannot (with rare exceptions) be simultaneously precisely measured. Rather, in analogy to the Heisenberg uncertainty relation Δ x Δ p x ≥ ℏ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0020.tif"/> which exists between the incompatible position and momentum observables, a Heisenberg uncertainty relation exists between any pair of incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0021.tif"/> and B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0022.tif"/>
The quantised angular momentum operators are obtained in §27.4 . These operators do not commute. They satisfy the commutation relations in Eqs. (27.111) to (27.113) . It follows that L ^ c x , L ^ c y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7135.tif"/> and L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7136.tif"/> are essentially incompatible. They do not possess a complete orthonormal set of common eigenvectors. However, there is a state, the state of zero orbital angular momentum values, in which they do possess a definite value, i.e., the value 0. 1 L ^ c z , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7137.tif"/> are compatible and they possess common eigenvectors and are simultaneously measurable. This is also true for L ^ c x , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7138.tif"/> and for L ^ c y , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7139.tif"/> . However, the eigenvectors of L ^ c z , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7140.tif"/> are different from that of L ^ c x , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7141.tif"/> or that of L ^ c y , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7142.tif"/> .
In view of the Heisenberg uncertainty principle, introduced in Chapter 1 , the application of Postulate 3 to a dynamical system faces one problem never encountered in classical mechanics, namely to define the conditions which make possible the simultaneous measurement of different observables. Because each physical observable takes definite values in the eigenstates of its associated operator, the simultaneous measurements of different observable will only be possible if their associated operators have a common eigenbasis. If such a complete orthonormal set of vectors ψ i are simultaneous eigenfunctions of two operators  and B̂: 4.1  ψ i = λ i ψ i , B ^ ψ i = µ i ψ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_1_B.tif"/> we have for each basis vector:  B ^ ψ i =  µ i ψ i = λ i µ i ψ i = µ i λ i ψ i = B ^  ψ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_2_B.tif"/> It follows that for any state vector Ψ, which can be expanded in the form ( 3.55) , we have: 4.2 (  B ^ − B ^  ) Ψ = 0 or  B ^ − B ^  = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_3_B.tif"/> This means, in view of Eq.(2.80) , that simultaneous measurements on a dynamical system are restricted to those observables for which the associated operators commute.
The representation of physical observables by Hermitian linear operators was postulated assuming that an expectation value, defined by Eqs.(2.11) and (2.12) , is associated with any observable A measured in a quantum state Ψ of a dynamical system. In view of measurement uncertainties, defined by the dispersion: ( Δ A ) 2 = 〈 A 2 〉 − 〈 A 〉 2 = 〈 ( A − 〈 A 〉 ) 2 〉 = 〈 ( δ A ) 2 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_1_B.tif"/> the expectation value 〈 A 〉 is taken in a state Ψ if and only if (Δ Α ) 2 vanishes. It is convenient to introduce the deviation operator: 3.1 δ  =  − 〈 A 〉 Î https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_2_B.tif"/> such that the condition of a vanishing dispersion for the observable A in a given state Ψ reads: 〈 Ψ |( δ  ) 2 Ψ 〉 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_3_B.tif"/> Because δ is a Hermitian operator, we can also write: 〈 δ Â Ψ | δ Â Ψ 〉 = | δ Â Ψ | 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_4_B.tif"/> which means that the norm of δ Â Ψ is zero and this implies that we have a null function δ Â Ψ = 0. In other words: 3.2 Â Ψ = 〈 A 〉 Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_5_B.tif"/> It follows that the state function Ψ where a definite expectation value 〈 A 〉 of the observable A is observed, should be a solution of Eq.(3.2) rather than an arbitrary 112 function. The action of the operator  on Ψ has to be simply that of rescaling. All the state functions which are solutions of Eq.(3.2) are called eigenfunctions of  , and the constant factor 〈 A 〉 is said to be the corresponding eigenvalue. Hence we have shown that if ((δ  ) 2 ) = 0, the state function should be an eigenfunction of  , which means that a physical observable only takes definite values in states represented by eigenfunctions of the associated quantum operator. Conversely, given a quantum operator  one can systematically determine all its eigenvalues and eigenfunctions, which are solutions of the so-called eigenvalue equation: 3.3 Â Ψ = λ Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_6_B.tif"/> It is visible that the eigenvalue equation fixes the eigenfunctions only up to an overall scale factor, which may always be chosen such that the eigenfunctions be considered normalized. Then, in view of Eq.(2.11) , Eq.(3.3) gives: 〈  〉 = ∫ − ∞ ∞ Ψ * Â Ψ d x = λ ∫ − ∞ ∞ Ψ * Ψ d x = λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_7_B.tif"/> If we further apply the operator  to Eq.(3.3) , we obtain  2 Ψ = λ 2 Ψ and this leads to: 〈  2 〉 = ∫ − ∞ ∞ Ψ *  2 Ψ d x = λ 2 ∫ − ∞ ∞ Ψ * Ψ d x = λ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_8_B.tif"/> It follows that: ( Δ A ) 2 = 〈  2 〉 − 〈  〉 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_9_B.tif"/> and this shows that the dispersion vanishes if a physical observable is measured in states represented by eigenfunctions of the associated operator.
Q22(1) The commutation relation between the selfadjoint position and momentum operators x ^ = x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1151.tif"/> and p ^ = − i ℏ d / d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1152.tif"/> in the Hilbert space L → 2 ( I R ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1153.tif"/> is often written as [ x ^ , p ^ ] = i ℏ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0444.tif"/> As pointed out in the discussion in §17.7 this relation should be expressed as an inequality, i.e., [ x ^ , p ^ ] ⊂ i ℏ I I ^ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0445.tif"/> or ∗ [ x ^ , p ^ ] ϕ → = i ℏ ϕ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0446.tif"/> for an appropriate set of vectors ϕ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1154.tif"/> in L → 2 ( I R ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1155.tif"/> . What are the conditions ϕ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1156.tif"/> must satisfy in order for the equality to hold?
In Section 1.3 , we developed the Bohr theory of the hydrogen-like atoms: H , H e + , L i 2 + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0001.tif"/> , and so on. Here, we develop the modern quantum theory of these atoms. Again, we deal with a two-particle system comprised of an electron of mass 10.1 m e = 9.10938972 × 10 − 31 k g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0002.tif"/>