IB Physics E.1.4 Bohr Model and Quantized Energy Levels Notes

Energy Levels in Hydrogen

In the early 20th century, Niels Bohr proposed that the energy of an electron in a hydrogen atom is quantized. It can only take on specific, discrete values.

Note

This was a radical departure from classical physics, which suggested that electrons could have any energy.

The Energy Formula

Bohr discovered that the energy of an electron in the $n$-th energy level of a hydrogen atom is given by:

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

where:

$E$ is the energy of the electron,

$n$ is the principal quantum number ($n = 1, 2, 3, \dots$).

Tip

The negative sign indicates that the electron is bound to the nucleus, meaning energy must be added to free it.

Energy Levels and Transitions

Each value of $n$ corresponds to a specific energy level:

$n = 1$: Ground state ($E = -13.6 \, \text{eV}$).
$n = 2$: First excited state ($E = -3.40 \, \text{eV}$).
$n = 3$: Second excited state ($E = -1.51 \, \text{eV}$).
and so on...

When an electron transitions from a higher energy level ($n_{\text{high}}$) to a lower one ($n_{\text{low}}$), it emits a photon with energy equal to the difference between the two levels: $$E_{\text{photon}} = E_{n_{\text{low}}} - E_{n_{\text{high}}}$$

For instance, a transition from $n = 3$ to $n = 2$ releases a photon with energy: $$E_{\text{photon}} = -1.51 \, \text{eV} - (-3.40 \, \text{eV}) = 1.89 \, \text{eV}$$

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IB Physics E.1.4 Bohr Model and Quantized Energy Levels Notes

IB Physics E.1.4 Bohr Model and Quantized Energy Levels Notes

Energy Levels in Hydrogen

The Energy Formula

Energy Levels and Transitions

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Energy Levels in Hydrogen

The Energy Formula

Energy Levels and Transitions

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IB Physics E.1.4 Bohr Model and Quantized Energy Levels Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

IB Physics E.1.4 Bohr Model and Quantized Energy Levels Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

Energy Levels in Hydrogen

The Energy Formula

Energy Levels and Transitions

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Energy Levels in Hydrogen

The Energy Formula

Energy Levels and Transitions

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