Q33(1) Calculate the expectation value of the two-particle observable represented by the operator in Eq. (33.9) in the state described by the vector Φ → ( c ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1602.tif"/> in Eq. (33.2).
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.
The solution of Schrödinger’s equation for systems composed of a large number of strongly interacting particles is an unsolved problem. Indeed, even the classical three-body problem cannot be solved in general form. Fortunately, the problem of obtaining the Ψ-function of such a system is not only insoluble but unnecessary; such a detailed description of the many-body problem is not required by any experiment that is possible in real life. Any real experimental set-up contains a comparatively small number of particle indicators, and hence, in cases where a large number of particles take part in a given process, can give only their averaged characteristics. Any attempt to determine the coordinates of every particle of such a system would lead (because of the uncertainty relation) to a complicated excited state and would indeed change the properties of the system. Thus, the description of a macroscopic system by means of a wave function is an inadequate approach * : we need 262 instead methods of incomplete description, which determine only the relations between averaged quantities. An example of such a description is hydrodynamics, the equations of which determine only the average velocity of the particles at each point (i. e. the velocity field). A second example, closer to our problem, is the kinetic equation, which allows us to find the distribution of particles with respect to velocity and position. A knowledge of the one-particle distribution function f ( r ˜ , p ˜ , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1304.tif"/> , which depends on the coordinate r ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1305.tif"/> and velocity p ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1306.tif"/> of the particles, allows us to calculate the mean values of additive quantities such as the density n ( r ˜ ) = ∑ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1307.tif"/> or the momentum per unit volume j ˜ ( r ˜ ) = ∑ i p ˜ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1308.tif"/> . The two-particle distribution function f ( r ˜ 1 , r ˜ 2 ; p ˜ 1 , p ˜ 2 ; t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1309.tif"/> allows us to find the correlations between the coordinates and the velocities of two particles and to determine the averages of quantities which depend on the coordinates of two particles, for instance the mean value of the pairwise interaction energy V = ∑ ik V ( r ˜ i - r ˜ k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1310.tif"/> . The periodic or weakly damped solutions of the equation for the distribution function give the eigenfrequencies of the system.
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?
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .
The postulates of quantum mechanics and their consequences, which have been developed to describe a single particle, are still valid when applied to a system of N particles. The system can be described by a wave function Ψ which depends on the time and on the 3 N coordinates of all the particles: 10.1 Ψ = Ψ ( r → 1 , … r → i , … r → N , t ) = Ψ ( x 1 , y 1 , z 1 , … , x i , y i , z i , … x N , y N , z N , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn10_1_B.tif"/>