# Can photons with same energies have different amplitude?

The word "photon" refers to an aspect of the nature of electromagnetic radiation that arises in the treatment of the fields by quantum theory. The best way to get an idea of what the word means is to think of any given field as composed of many modes. A mode is a pattern of oscillation; it refers to an oscillation all at one frequency, but the spatial distribution of the oscillation will vary from case to case. When a given mode of frequency $$\omega$$ is in a state of well-defined energy with energy $$(n + 1/2) \hbar \omega$$ then we say there are $$n$$ photons present in that mode.

Now I used the word "oscillation" there, and this relates to your question. But it is not always straightforward to say what is oscillating. When $$n$$ is large then it is possible to have a configuration with several modes of close frequency all having many photons, and when they combine in a certain way, called coherent state, the total electric field and the magnetic field are oscillating. This is the situation in the kind of light waves that are treated by classical electromagnetism. Light from a laser is like this, and light from the thermal source such as the Sun can be expressed as a sum of many coherent states.

The relationship between the amplitude of the electric field and the average number of photons in a coherent state is roughly $$\begin{eqnarray*} \mbox{energy} &\simeq& n \hbar \omega \\ \Rightarrow \epsilon_0 E^2 V &\simeq& n \hbar \omega \end{eqnarray*}$$ where $$E$$ is the electric field amplitude and $$V$$ is the volume occupied by the mode. The answer to your question is that each photon can be said to have a given energy, but it is a bit misleading to think of them as having amplitudes independently of one another. You could compare it to something more familiar, as follows. Suppose a classical wire were vibrating, such that the total energy in the vibration is $$1$$ joule. Asking about the oscillation amplitude for each photon would be like asking, what is the amplitude associated with each of the one thousand milli-joules which together form the total oscillation?

By the way, you can also have states where the idea of an oscillation does not apply in the ordinary sense of the word. When the state of a mode has a single precise value of $$n$$, as opposed to a superposition (for example, this often happens when $$n$$ is small), then the phase of the oscillation is spread out by quantum uncertainty, and then it is not correct to say the fields are oscillating in the ordinary sense of the word. There is a frequency but the electric field is not waving up and down. The situation is very similar to the energy eigenstates of a harmonic oscillator, if you know about those. The oscillator has a frequency $$\omega$$, but in a state of well-defined $$n$$ the modulus-squared wavefunction $$(|\psi|^2)$$ is completely static.

Could two photons with same frequency have different amplitudes

Photons do not have amplitude,i.e their energy is not distributed in space, let alone oscillate. They are elementary point particles in the standard model of physics. This is a quantum mechanical theory.

The particles in quantum mechanics can be represented by wave functions which are the solutions of the appropriate equations. For the photon the equation is quantized Maxwell's equation and you can read about the wave function $$Ψ$$ of a photon here.

This means there exists a probability for a photon to be at an (x,y,z,t) given by $$Ψ^*Ψ$$, the complex conjugate squared of the wavefunction. It is this probability that has a frequency given by the $$E=hν$$ where E is the energy of the photon and $$ν$$ the frequency of the classical light wave that zillions of such energy photons will build up. This can be shown with quantum field theory, for example here.

and so different peak velocities of oscillations perpendicular to their direction? (to make a larger distance in same interval you need to be quicker).

This has no meaning in the probabilistic quantum mechanical framework which models photons.

See this double slit experiment one photon at a time. The photons are all of the same energy and thus the light they build up is of one frequency.

Figure 1. 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 little dots are the footprints of single photons, no oscillations. At left they look random, but at the rightmost frame they have built up the classical interference pattern of light of that frequency. The plot on the right is the quantum mechanical probability distribution for the experiment "photons impinging on given double slit" with its boundary conditions.

Keep in mind, each photon appears at a definite (x,y) on the screen which is a distance z. No photon oscillations.