In the Schroedinger picture the time development of a wave function subjected to some potential is entirely deterministic, provided that the system is left undisturbed. We may write

If the wave function at time t₁ is known everywhere, then the wave function at time t₂ can be obtained from this expression.

What is K(r₂,t₂;r₁,t₁)?

We may write We then can find the wave function by writing

or

We therefore have

is called the propagator for the Schroedinger equation.  Often we want to use the propagator only for t₂>t₁. We the write

where q(t₂-t₁) is the step function q(t₂-t₁) =1 if t₂>t₁, and q(t₂-t₁) =0 if t₂<t.

can be interpreted as the probability amplitude that a particle that at t₁ is located precisely at r₁ will be found at r₂ at time t₂.  If the Hamiltonian of a system does not explicitly depend on time then   We may also write

where {|f_n>} is an eigenbasis of the Hamiltonian, since   This yields

an expression for the propagator in terms of eigenstates of H.

The propagator is the Green’s function for the time-dependent Schroedinger equation.

since the are eigenfunctions of H and

Therefore

The propagator as a sum of partial amplitudes

We may write i.e. we divide the time interval t₂-t₁ into subintervals.  Then, by inserting the closure relation for each subinterval we obtain

Now consider the product We can interpret this term as the probability amplitude that particle that leaves r₁ at t₁ arrives at r₂ at t₂ after having passed successively through the points r_i at t_i.  To obtain K(2,1) we sum (integrate) over all possible positions r_i at t_i. If we let n approach infinity, then we sum the probability amplitudes for all possible paths from r₁ at t₁ to r₂ at t₂.  K(2,1) can be interpreted to be the coherent superposition of the probability amplitudes associated with all possible space-time paths starting at 1 and ending at 2.

This concept of the propagator as the coherent superposition of the probability amplitudes associated with all possible space-time paths has led to Feynman’s postulates, a new formulation of the postulate concerning the evolution of a physical system.  For a spinless particle (simplest case) we have the following postulates.

• i) K(2,1) is the probability amplitudes for the particle starting at r₁ at t₁ to arrive at r₂ at t₂.
• ii) K(2,1) is the sum of an infinite number of partial amplitudes, one for each space-time path leading from r₁ at t₁ to r₂ at t₂.
• iii) The partial amplitude K_G(2,1) associated with a particular path G is determined in the following way.  Let be the classical action calculated along G.  Then where N is a normalization constant.

The Schroedinger equation and the commutation relations between the components of the observables R and P follow as a consequence of these postulates.  The postulates therefore permit a different but equivalent formulation of quantum mechanics.

The classical limit

Consider a situation for which the classical action S_G is much larger than The variation of the action DS_G between different path is then usually also much larger than The phase of then varies rapidly and contributions from most of the paths G to K(2,1) cancel out.  If there, however, exists a path G₀ for which the action is stationary and does not vary to first order when one goes from G₀ to another infinitesimally close path, then the amplitudes of the paths in the neighborhood of G₀ will interfere constructively, since their phases are practically constant.  Therefore, to obtain K(2,1) in the classical limit we can ignore all paths except the ones infinitesimally close to the one for which the action is stationary.  This is the classical trajectory defined by Hamilton’s principle, the principle of least action.  We obtain Hamilton’s principle from Feynman’s postulates in the limit The wave packet associated with the particle explores the various paths and picks the one for which the action is the smallest.

Link:

• The Feynman page