Why move photons straight ahead and do not move wave like? A single photon can be seen as a wave. Normally, waves move in all direction. So one could expect a photon to form a spherical wave. But a photon moves straight on, only in one direction.
Is there any solution of the wave equation which explains this behavior?
 A: We actually don't know How Photon move? In straight line or jiggling a little bit or some other path. We don't talk of photon trajectory either.
QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.
The key components of Feynman's presentation of QED are three basic actions.

*

*A photon goes from one place and time to another place and time.

*An electron goes from one place and time to another place and time.

*An electron emits or absorbs a photon at a certain place and time.

As well as the visual shorthand for the actions Feynman introduces another kind of shorthand for the numerical quantities called probability amplitudes. The probability is the square of the absolute value of total probability amplitude
The basic rules of probability amplitudes that will be used are:
1.If an event can happen in a variety of different ways, then its probability amplitude is the sum of the probability amplitudes of the possible ways.
2.If a process involves a number of independent sub-processes, then its probability amplitude is the product of the component probability amplitudes.
A: (1) A model for a phenomenon is acceptable if it explains the behavior correctly.
(2) And it is better than other models if it is easy to understand.
(3) Moreover, such a model is correct as long as no new details of the phenomenon make the model wrong.
One of the possible models is the next

*

*A photon is a indivisible quant from its emission to its absorption.

*As long as undisturbed, it moves in the straightest possible line (the geodesic path) and between any two points it goes with the speed of light.

*The photon has an oscillating electric field component and an oscillating magnetic field component. Both fields are dipoles and develop symmetrically to the centre of the photon.

*In a vacuum (free of other fields), both components unfold perpendicular to the direction of propagation and at 90° to each other. This means that the development of the fields is symmetrical and there is no reason for the photon path to wobble.


Normally, waves move in all direction.

The dissipation of matter waves occurs in the directions in which the medium is elastic. The [water wave in a channel] (https://en.wikipedia.org/wiki/Soliton ) with the rigid channel walls does not dissipate over a longer time.
Without a medium, how does an EM wave dissipate? It doesn't. It is only "diluted". Light from distant stars arrives on earth as individual quanta.
What is known about electromagnetic waves?

*

*They are made of photons because there is no other way of radiation than from the emission of electrons or other subatomic particles.

*Usually the photons are emitted chaotically, with different energy content (wavelengths) and at different times. Furthermore with different directions directions of the electric / magnetic field (not polarized).

*From thermic sources it is impossible to measure any modulation. In this sense it is not a wave.

*The only way I know of to generate an EM wave is the synchronous and periodic acceleration of electrons on the surface of a conductor. The emitted photons are polarised and the amount of photons changes periodically. Such a wave is measurable.


One last point. Single photons propagating along an edge are deflected so that their distribution forms fringes behind the edge. The explanation of the dark areas by means of interference is wrong because destructive interference is impossible for low-energy photons.
The better explanation is the deflection due to the influence of the edge-surface electrons on the photons.
A: 
Why move photons straight ahead and do not move wave like?

First lets look at a clear experiment, single photons at a time:


Single-photon  camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

The points show the footprint of a particle, a single photon. The accumulation shows an interference pattern which, analyzed shows  a wave of frequency equal to the $hν$ of the photons in the beam.

Is there any solution of the wave equation which explains this behavior?

Sure, the quantum mechanical wave function for the solution of the scattering problem "photon + two slits a given width a given distance apart".  The wave equation is not the solution of classical Maxwell equations, but of a quantized form and it gives the probability distribution seen in the last frames.
