Hydrogen Energy Levels

In one sentence: A hydrogen electron can only sit on certain energy steps, and every jump between steps gives off or takes in light of one exact color.

$E_n = -\frac{13.6\ \text{eV}}{n^2}$

Read it out loud: “E sub n equals negative 13.6 electron volts divided by n squared. E sub n is the energy of level n, and n is a whole number: 1, 2, 3 and so on.”

What each symbol means

Symbol	Meaning
$E_n$	The energy of the electron in level $n$ , in electron volts (eV). The negative sign means the electron is bound to the proton.
$n$	The principal quantum number: a whole number 1, 2, 3 and so on. A larger $n$ means a larger orbit and a level closer to zero energy.
$13.6\ \text{eV}$	The rounded ionization energy of hydrogen from the ground state, 13.6 electron volts.

What the formula is doing

Hydrogen is one electron bound to one proton. Solving the Schrödinger equation for this system gives a set of allowed energies, labeled by the whole number $n$ . The Bohr model reaches the same formula with a simpler picture, and the formula above gives those allowed energies.

The number $n$ sets the level. For $n = 1$ the electron is in its lowest level, at $-13.6$ eV. For $n = 2$ the energy is $-3.4$ eV, and for $n = 3$ it is about $-1.51$ eV. Higher levels are closer to zero, and the gaps between them shrink as $n$ grows.

The negative sign means the electron is bound. Raising the energy from the ground level to zero takes 13.6 eV, which is the ionization energy. At zero energy the electron is free and no longer tied to the proton.

When an electron falls from a higher level to a lower one, it emits one photon whose energy equals the gap between the two levels, following $E = hf$ . When it climbs, it absorbs a photon of exactly that energy. Each gap gives one specific wavelength, so hydrogen light is made of sharp lines rather than a smooth rainbow.

The first Balmer lines, which end on $n = 2$ , are visible light. The red H-alpha line at about 656 nm comes from the jump from $n = 3$ to $n = 2$ . Jumps that end on $n = 1$ (the Lyman series) are in the ultraviolet.

An everyday picture

Think of a staircase where each step is closer to the next as you go up, and the top is the free electron at zero energy. The electron can stand only on the steps, never between them. Dropping down a few steps releases a fixed amount of energy, and that amount sets the color of the light that comes out.

Worked example: The red H-alpha line, from n = 3 to n = 2

An electron falls from level $n = 3$ to level $n = 2$ . Use $E_n = -13.6\ \text{eV} / n^2$ and $hc = 1239.84$ eV nm.

Energy of level 3: $E_3 = -\frac{13.6}{3^2} \approx -1.51$ eV.
Energy of level 2: $E_2 = -\frac{13.6}{2^2} = -3.40$ eV.
Energy of the photon, the upper level minus the lower level: $E_{\text{photon}} = E_3 - E_2 = (-1.51) - (-3.40) \approx 1.89$ eV.
Wavelength: $\lambda = \frac{hc}{E_{\text{photon}}} = \frac{1239.84\ \text{eV nm}}{1.89\ \text{eV}} \approx 656$ nm.

Answer: The electron emits a photon of 1.89 eV, which has a wavelength of about 656 nm. That is the red H-alpha line, one of the reasons hydrogen gas glows red-pink in discharge tubes. Ionizing a hydrogen atom from its lowest level would take 13.6 eV instead.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Dropping the minus sign. The energies are negative, so $n = 1$ is the lowest level: $-13.6$ eV is less than $-3.4$ eV.
Subtracting in the wrong order. The photon energy is the upper-level energy minus the lower-level energy. Doing it the other way gives a negative photon energy.
Assuming every jump gives visible light. Only jumps that land on $n = 2$ are in the visible range. The jumps that land on $n = 1$ are in the ultraviolet.
Getting the ionization sign wrong. Ionizing from $n = 1$ takes the electron from $-13.6$ eV up to zero, which is $+13.6$ eV of energy added.

Where you will meet it

Astronomy, where the hydrogen lines of the Balmer series identify hydrogen in stars and nebulae.
Spectroscopy labs, where students measure the red H-alpha line to check the formula against an experiment.
Gas-discharge lamps, where excited hydrogen atoms emit their characteristic lines.
Introductory quantum mechanics, where the hydrogen atom is the standard worked example for energy levels.

Questions people ask

Why is the energy negative?

The zero point is set at a free electron, far away from the proton. A bound electron has less energy than that, so its energy is negative.

Why do the levels get closer together as n grows?

The gap between level $n$ and level $n+1$ shrinks, because $1/n^2$ changes less and less for large $n$ . Levels near zero are packed tightly, and very high levels blur into the free state.

Is 13.6 eV exact?

No, 13.6 eV is a rounded value. For a hydrogen-like atom with an infinitely heavy nucleus the precise value is 13.6057 eV, and using the real proton mass changes it only slightly.

Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.
Schrödinger Equation: The Schrödinger equation says how a quantum particle’s wave changes over time, the way Newton’s laws say how a moving ball changes its position.
Photoelectric Effect: Light can knock electrons out of a metal, but only if each packet of light carries enough energy, and the leftover energy becomes the electron’s speed.