How can Planck’s equation and the wave equation simultaneously coexist with contradicting each other? [closed]

Question

The equations:

E=hf (Planck’s equation, where E is the energy, h is the Planck’s constant and f is the frequency) and

V=fλ (where v is the speed (in a vacuum), f is frequency and λ is the wavelength) Can be combined to form the equation:

E = hv/λ Which is often used in calculations involving the absorption and emission spectrums of elements.

However, as atoms emit photons when electrons fall to lower levels, and all frequencies of light travel at the same speed, this equation is often shown as:

E=hc/λ

The emission spectrum of atoms is often shown like the image below:

From the diagram as energy is on the y axis, and energy and frequency are proportional as per Planck’s equation, which would indicate that the ‘longer arrows’ on the diagram are of higher frequency as they are of greater energy change.

HOWEVER, as the diagram also shows the quantum energy levels gradually getting closer together, would a ‘longer arrow’ not also indicate a greater wavelength as the electrons has dropped/moved up a higher distance, in which case wouldn’t a shorter arrow, with a shorter wavelength lead to a greater frequency.

Essentially, I am confused about whether an arrow of shorter distance means a greater frequency because wavelength and frequency vary inversely, or if an arrow of longer distance means a greater frequency because energy and frequency are directly proportional.

Answer 1

Essentially, I am confused about whether an arrow of shorter distance means a greater frequency because wavelength and frequency vary inversely, or if an arrow of longer distance means a greater frequency because energy and frequency are directly proportional.

The diagram you show is called an energy level diagram or sometimes a Grotrian diagram. As Poutnik pointed out, read the axes. The vertical-axis has E(nergy) written on it. The horizontal lines simply show the energy level number. So the arrow length shows the magnitude of energy change in going from one energy level to the other. You can give the arrow a pure geometric meaning, the length of an arrow show the "distance" between two lines in "energy units" from the starting point to the end point.

Longer arrow: Larger energy change and therefore corresponding higher frequency light emitted or absorbed (ultraviolet side)

Shorter arrow: Smaller energy change, therefore corresponding lower frequency light emitted or absorbed (red side)

This chart from NASA may help you visualize better Link: