Free Particle and Harmonic Oscillator

Eigenfunction and Eigenvalue

We find the eigenfunctions and eigenenergy values via the Feynman path integration with the help of the transition amplitude К for the free particle as well as the harmonic oscillator. This will be done without solving the Schrodinger equation and implies not solving the wave equation directly but everything determined from the transition amplitude К .

Free Particle

We examine an important step towards the path integral formulation of quantum mechanics by considering the time-dependent transition amplitude for a free particle of mass m and satisfies the Schrodinger equation with respect to the variables q and t:

Here, the eigenfunction of the momentum operator as well as the Hamiltonian is:

with a momentum eigenvalue p out of a continuous spectrum. Tire coefficient C_p(q',t') in 9.1 can be defined from the condition when the transition amplitude achieves the expression of the Dirac delta function S(q - q') when t = t':

This implies,

We show the validity of the result by recovering the transition amplitude via 9.1 and 9.4: or

or

From transition amplitude in 9.5 the energy of the free particle is:

The transition amplitude 9.7 can be rewritten in the 3D form:

It should be noted that the dependence of the transition amplitude on the difference of the coordinate (q - q') and time T is related to the homogeneity of space and time for the free particle.

m(r-r')²

From relation 9.9, the quantity — ^ at the exponent is the classical action as seen in 5.72 for the ID problem:

We examine some particular cases where the transition amplitude can again be expressed via the classical action. For brevity, we consider the ID wave function that is the solution of the time-dependent Schrddinger equation:

Substituting this in the time-dependent Schrddinger equation while separating the imaginary and real parts then we have:

that has the form of the Hamilton-Jacobi equation with S coinciding with the classical action and then

Here, the first summand from our condition is dependent only on t. It follows that the second summand is also only dependent on t. This can only be feasible when

So, expressions 8.11 as well as 8.21, where, S is the classical action are valid for the potential in 9.14 describing a time-dependent homogenous arbitrary electric and magnetic fields as well as a time-depen- dent driven harmonic oscillator with a variable frequency.

Transition Amplitude for a Particle in a Homogenous Field

We apply this to an example for the transition amplitude for a particle in a homogenous field. We find the transition amplitude for this case by first evaluating the action:

Here,

From the equation of motion then or

So,

or

From where, or

Here, q' is the initial coordinate and

the velocity at the initial time moment t'. From here, the action is obtained as follows:

We substitute this in 9.13 to obtain

For the case of a free particle then and we find F from the transition amplitude for a free particle:

So, the transition amplitude for a free particle subjected to a homogenous field is:

From the spectral representation of the transition amplitude in 7.45 considering 7.29, the eigenfunctions (expressed via Airy functions) and eigenvalues (roots of the Airy functions) are obtained respectively in 8.163 and 8.169.

From knowledge of the path integral of the harmonic oscillator we study another harmonic system which involves a particle of charge e in a homogenous electric field with electric field strength s . Hie transition amplitude for the 3D case: