There is more information in a quantum state than in the classical initial values https://www.w3.org/1998/Math/MathML" display="inline"> ( x 0 , p 0 ) that determine the complete 1-dimensional motion of a point particle for a given force. The initial wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , 0 ) requires an infinite number of parameters, which can be expressed in observational terms as the two basic expectation values https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ^ ⟩ and https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ p ^ ⟩ together with the two variances https://www.w3.org/1998/Math/MathML" display="inline"> ( σ x 2 , σ p 2 ) and the covariance 5.1 https://www.w3.org/1998/Math/MathML" display="block"> C x p = 1 2 ⟨ x ^ p ^ + p ^ x ^ ⟩ − ⟨ x ^ ⟩ ⟨ p ^ ⟩ ,
The quantum revolution in the beginning of the 20th century changed our concepts of dynamics. The dynamic variables such as position 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-math1_1.tif"/> and momentum p $ {\mathbf{p}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_2.tif"/> are replaced by the corresponding position and momentum operators r ^ $ {\hat{\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-math1_3.tif"/> and p ^ $ {\hat{\mathbf{p}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_4.tif"/> , respectively When one would like to observe a dynamical variable, say ω , of a particle, one has to introduce a wavefunction ψ , which contains all the dynamical information of particles, and to apply the corresponding dynamical operator Ω ^ $ \hat{\Omega } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_5.tif"/> to the wavefunction ψ . Then one may obtain the desired dynamical value of the particle as the eigenvalue of the operator, 1.1 Ω ^ ψ = ω ψ . Ω^ψ=ωψ. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math1_1.tif"/>
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 .
If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .
This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p . The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation , i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .