The Schrödinger Equation’s Green’s Function
The transition amplitude
satisfies the wave function:
and the time-dependent Schrodinger equation:
from where
In the above relation, the factor in the last term:
For tb > t„,
Also, in the limit tb = ta from equation 7.20 then
So, if
then
Hence, the retarded propagator satisfies the equation whose right-hand side is proportional to a four-dimensional delta function:
From here, 6(tb - ta)K(qb, tb qa, f„) the retarded propagator is a Green's function for the Schrodinger equation. Physically, in equation 7.20 , the wave function y¥(qa>ta) acts as a source term, and the amplitude K(qby tb q„, t„) propagates the effect of that source to the space-time point (qb, tb), where it generates its contribution to (qb, tb). Integration over all such sources gives the total wave function at (qb, tb).
Transition Amplitude for a Time-Independent Hamiltonian
The propagator contains the complete information about the eigenenergies £„ and the corresponding eigenstates or eigenvectors |Ф„). So, we will write the propagator in the basis formed from the eigenfunctions of the time-independent Schrodinger. We assume a system with a time-independent Hamiltonian H corresponding to any situation where the action S does not depend explicitly on time. Here, the transition amplitude К (qb>tbqa,ta) is a function only of the time interval T=tb- ta and maps the state vector |'P(f)) described by the time-dependent Schrodinger equation:
This has the solution
with the eigenstate |Ф) satisfying the stationary eigenvalue problem:
This permits the obtaining of the energy spectrum £„ and the eigenvectors |Ф„), or, equivalently, the eigenfunctions represent a complete orthonormal set of eigenfunctions of the Hamiltonian H :
in the coordinate representation. The general solution of 7.31 can be expressed using a linear combination of the complete basis eigenvectors |Ф„):
where
We find the transition amplitude K(q(>, tb; qa, ta) in terms of the states in 7.36 and suppose the system starts at time ta in the position eigenstate q„). We can find C„ via a scalar product with (Ф,„| from the left:
and so
From here, the state vector at a later time tb is found from 4.8:
So, we have the following identity connecting the Schrodinger eigenfunctions and eigenenergies to Feynman path integral in the form of the transition amplitude:
or
or
or summarily
Here, the Heaviside step function is:
The expression might be viewed as if the particle moves around in all possible locations in the energy space (acquiring different phase factors) as it propagates from the selected initial position to the final position. The representation in 7.43 can be used as the definition of the transition amplitude K, provided the spectrum is known and the n -sum converges.
Suppose in 7.43 we set tb to an arbitrary time t then the spectral representation of the transition amplitude:
where £„ are the eigenvalues and |Ф,,(
We apply the transition amplitude 7.45 to the case of a free particle in 5.94 then this permits us to read off the normalized eigenfunctions
and eigenenergies
