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.
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"/>
Although one-dimensional problems are only an idealisation of naturally occurring physical situations, and hence may seem somewhat artificial, motion of a single particle in one dimension yields exact solutions of the Schrödinger equation which highlight most of the features of the three-dimensional quantum approach, avoiding, however, its complexity. Because the time-dependent Schrödinger equation in one dimension: 6.1 i ℏ ∂ Ψ ( x , t ) ∂ t = − ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + U ( x ) Ψ ( x , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_1_B.tif"/> has the general solution, Eq.(5.62) , which is expressed in terms of the eigenvalues and eigenfunctions of the Hamiltonian operator only: 6.2 Ψ ( x , t ) = ∑ n a n e − i E n t / ℏ ψ n ( x ) + ∫ a ( E ) e − i E t / ℏ ψ ( x , E ) d E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_2_B.tif"/> the problem is always reduced to solving the Schrödinger wave equation: 6.3 − ℏ 2 2 m ∂ 2 ψ ( x ) ∂ x 2 + U ( x ) ψ ( x ) = E ψ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_3_B.tif"/> This is the energy eigenvalue equation, which can also be written as: 6.4 d 2 ψ ( x ) d x 2 + 2 m ℏ 2 [ E − U ( x ) ] ψ ( x ) = d 2 ψ ( x ) d x 2 + k 2 ( x ) ψ ( x ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_4_B.tif"/> by setting: 6.5 k 2 ( x ) = 2 m ℏ 2 [ E − U ( x ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_5_B.tif"/> 203 It follows that the energy eigenvalue problem is similar to the calculation of monochromatic waves in a medium with non-uniform refractive index. If E < U ( x ), it is clear that k ( x ) becomes imaginary, namely k { x ) = iγ (x), where: 6.6 γ 2 ( x ) = 2 m ℏ 2 [ U ( x ) − E ] , E < U ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_6_B.tif"/> For any given eigenvalue E, the second-order differential equation (6.4) yields two linearly independent solutions ψ 1 ( x ) and ψ 2 ( x ), suchthat: 6.7 ψ ( x ) = a 1 ψ 1 ( x ) + a 2 ψ 2 ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_7_B.tif"/> It follows that any eigenvalue might be at most two-fold degenerate. The solutions of interest are those which satisfy certain regularity conditions which make them admissible eigenfunctions: i ψ ( x ) and its first derivative should be continuous functions of x, for all x. This implies that we allow for finite discontinuities of the potential energy U ( x ) at some points, where d 2 ψ/dx 2 should exhibit an opposite finite discontinuity, so preventing the departure from continuity of the right-hand side of Eq.(6.1) , whose left-hand side ∂ Ψ /∂t is a continuous function. A step-like behaviour of d 2 ψ/dx 2 requires the continuity of dψ/dx at the same points, otherwise the discontinuity becomes infinite. ii ψ ( x ) should be finite or single-valued for all x. This ensures the possibility of normalizing the wave functions, according to Eq.(1.37) , and having a uniquely defined probability density || ψ ( x ) | 2 . The eigenfunctions ψ ( x ) can always be chosen to be real, and this becomes apparent by conjugation of Eq.(6.4) , which gives: d 2 ψ * ( x ) d x 2 + k 2 ( x ) ψ * ( x ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_8_B.tif"/> where k 2 ( x ) is always real. Because ψ ( x ) and ψ* ( x ) are eigenfunctions with the same eigenvalue E, their real and imaginary parts, namely: 6.8 ψ r ( x ) = ψ ( x ) + ψ * ( x ) 2 , ψ i ( x ) = ψ ( x ) − ψ * ( x ) 2 i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_9_B.tif"/> are real eigenfunctions corresponding to E. Hence, we will limit ourselves to the real solutions ψ ( x ) to Eq.(6.4) .
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 .
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"/>