Bohr and Schrödinger Models from Atomic Spectra

Suppose you pass white light through a prism. You get a smooth rainbow. Now do something similar with light from excited hydrogen gas. Instead of every colour, you see a few sharp lines.

That is strange. If an electron in an atom could have any energy it liked, you would expect it to release any amount of energy and therefore produce a continuous spread of wavelengths. Hydrogen does not do that.

Before reading on, make a prediction: what must be true about the electron’s allowed energies if the atom can emit only particular wavelengths?

The key idea is that the allowed energies must be discrete. There are gaps between them. Spectroscopy revealed those gaps before physicists had a satisfactory picture of what an electron inside an atom was actually doing.

01What the spectroscope actually measures

Picture the apparatus from left to right.

A low-pressure gas, such as hydrogen, sits inside a discharge tube. A high voltage supplies energy to the gas. Some atoms become excited, meaning their electrons gain energy.

Light from the tube then passes through a narrow slit. The slit produces a thin beam that enters a prism or diffraction grating. Different wavelengths leave at different angles, so they appear at different positions on a screen or detector.

If the source were ordinary white light, you would expect the wavelengths to merge into a continuous spectrum.

Excited hydrogen instead produces a line emission spectrum. Only particular wavelengths are present.

That tells us something important, but also something limited. A spectral line tells us about an energy difference inside the atom. It does not directly photograph an electron’s path.

For each emitted photon,

\[
E_{\text{photon}} = h\nu = \frac{hc}{\lambda}
\]

where:

• \(E_{\text{photon}}\) is the photon energy in joules (J)
• \(h\) is Planck’s constant, \(6.626 \times 10^{-34}\text{ J s}\)
• \(\nu\) is frequency in hertz (Hz)
• \(c\) is the speed of light, \(3.00 \times 10^8\text{ m s}^{-1}\)
• \(\lambda\) is wavelength in metres (m)

So measuring a wavelength gives you an energy difference.

Shorter wavelength means a larger photon energy. Longer wavelength means a smaller photon energy.

02The pattern that needed an explanation

By the late nineteenth century, measurements of hydrogen’s visible spectral lines showed an unexpectedly neat mathematical pattern.

In 1885, Johann Balmer found a relationship describing the visible hydrogen lines. Johannes Rydberg later expressed the pattern in a more general form:

\[
\frac{1}{\lambda}
=
R_{\text H}
\left(
\frac{1}{n_f^2}
–
\frac{1}{n_i^2}
\right)
\]

For an emission transition, \(n_i>n_f\).

Here:

• \(R_{\text H}\) is the Rydberg constant, approximately \(1.097 \times 10^7\text{ m}^{-1}\)
• \(n_i\) is an integer associated with the initial state
• \(n_f\) is an integer associated with the final state

The integers were the suspicious part.

The formula worked, but a formula that matches measurements is not automatically an explanation. Physicists still needed to answer: why should an atom care about whole numbers?

03Why the classical orbit picture was in trouble

After Rutherford’s nuclear model, it was tempting to picture an atom like a tiny solar system. A negatively charged electron travelled around a small positive nucleus.

There was a serious problem.

An orbiting electron is accelerating because its direction is constantly changing. According to classical electromagnetic theory, an accelerating charged particle should radiate energy. The electron should lose energy and spiral towards the nucleus.

Stable atoms do not behave that way.

The classical picture also gave no natural reason for hydrogen to emit only certain wavelengths.

Imagine a staircase and a ramp. On a ramp, you can stand at essentially any height. On a staircase, you can stand on the steps, but not halfway between them.

The classical expectation was closer to the ramp. Spectra were telling physicists that atomic energy behaved more like the staircase.

That analogy is useful only for energy. Atomic electrons are not literally standing on tiny physical stairs.

04Bohr’s 1913 repair: allow only certain energies

Niels Bohr kept part of the orbit idea but added a radical restriction.

An electron could occupy only certain allowed states. While in one of these states, it did not continuously lose energy. Light was absorbed or emitted only when the electron moved between allowed energies.

For hydrogen, Bohr’s allowed energies are:

\[
E_n
=
-\frac{2.18 \times 10^{-18}\text{ J}}{n^2}
\]

where \(E_n\) is the electron’s energy and \(n=1,2,3,\ldots\) is the principal quantum number.

The negative sign matters. The zero of energy is defined as a completely separated electron and nucleus.

So:

• \(n=1\) has the lowest energy and is the ground state
• larger \(n\) values have higher energies
• as \(n\) becomes very large, \(E_n\) approaches \(0\)

For example,

\[
E_1=-2.18\times10^{-18}\text{ J}
\]

while

\[
E_2=-5.45\times10^{-19}\text{ J}
\]

Notice that \(E_2\) is higher because it is less negative.

That detail causes plenty of mistakes. On an energy scale, \(-5\) is higher than \(-10\).

Emission and absorption

Suppose an electron begins in a higher energy level and moves down.

The atom loses energy, and a photon is emitted:

\[
E_{\text{photon}}=E_i-E_f
\]

where \(E_i\) is the initial electron energy and \(E_f\) is the lower final electron energy.

For absorption, the direction reverses. The electron gains exactly the energy supplied by the photon:

\[
E_{\text{photon}}=E_f-E_i
\]

A photon with the wrong energy is not enough. The energy has to match an allowed gap.

It is a bit like someone refusing every date except the people already on an extremely specific shortlist. Being “nearly right” does not get you into the next energy level. The analogy breaks because an atom is obeying quantum mechanics, not making questionable romantic decisions.

05Why Bohr’s success did not prove circular electron orbits

Here is a subtle point worth getting right.

Hydrogen’s line spectrum strongly supports quantised energy levels. It does not, by itself, prove that electrons travel around the nucleus in neat circular paths.

Those are different claims.

Bohr’s model combined them:

1. electron energies are restricted to particular values
2. electrons occupy particular orbits

The spectral evidence strongly supported the first idea. Later quantum mechanics kept quantised energies but abandoned the idea of definite classical orbits.

That is a common pattern in science. A model can contain one powerful idea and one part that later needs replacing.

06Where the Bohr model starts to fail

Bohr’s model works impressively for hydrogen because hydrogen contains only one electron.

It also works reasonably for other one-electron species, such as \(\mathrm{He}^{+}\), if the stronger nuclear charge is included.

But neutral helium already has two electrons. Now each electron is attracted to the nucleus and repelled by the other electron. A simple circular-orbit model cannot handle the full situation.

Real spectra also contain finer details than the original Bohr model predicts. External magnetic fields can split spectral lines, for example. Line intensities also require more information than just the sizes of the energy gaps.

So the next model had to preserve Bohr’s successful quantisation while replacing the unsupported picture of a little particle travelling on a fixed track.

07Schrödinger’s model: keep the energy levels, lose the tracks

By the 1920s, experiments had shown that microscopic particles could display wave-like behaviour.

Erwin Schrödinger developed a quantum mechanical model in 1926 in which an electron is described using a mathematical wavefunction, usually represented by \(\psi\).

You do not need to imagine the electron physically wobbling like a water wave.

Instead, the wavefunction contains information about the electron’s quantum state. The quantity

\[
|\psi|^2
\]

is related to the probability density of finding the electron in a particular region of space.

This is the point where an orbital replaces an orbit.

An orbit is a path.

An orbital is a quantum state with a particular spatial probability distribution.

Those two words sound annoyingly similar, but they describe very different ideas.

Why does Schrödinger’s model still give discrete energies?

A useful picture is a vibrating guitar string.

A string fixed at both ends cannot sustain every possible standing-wave shape. Only certain patterns fit the boundary conditions.

Schrödinger’s equation behaves differently in the mathematical details, but the broad idea is similar. Only certain wavefunctions satisfy the conditions required for a bound electron. Those allowed wavefunctions have particular energies.

So quantisation does not have to be added as an unexplained rule about permitted circular tracks. It appears naturally from the mathematics of the wave model.

The analogy breaks in an important place. An electron’s wavefunction is not a literal string vibrating through three-dimensional space. It is a quantum mechanical probability amplitude.

08The meaning of \(n\) changes

Bohr and Schrödinger both use the principal quantum number \(n\), but they do not give it exactly the same picture.

In the Bohr model, \(n\) labels an allowed orbit and its energy.

In the Schrödinger model, \(n\) labels a principal energy level, but each level can contain different subshells and orbitals.

For example, the \(n=2\) shell contains:

• a \(2s\) subshell
• a \(2p\) subshell

The \(2p\) subshell contains three orbitals with different spatial orientations.

That richer structure becomes essential when dealing with multi-electron atoms. It leads directly into spdf notation and electron configurations.

09What Schrödinger explains that Bohr cannot

The Schrödinger model keeps the central lesson from the hydrogen spectrum: electrons in atoms have quantised energies.

It then provides a much richer description of their possible states.

One qualification matters here. The Schrödinger equation is exactly solvable for hydrogen-like, one-electron atoms. Multi-electron atoms are more complicated because electrons interact with one another, so chemists use approximation methods.

That is still a huge improvement over pretending the electrons are travelling around fixed rings.

10What spectral evidence actually supports

It helps to separate three statements.

Observation: excited hydrogen produces light at particular wavelengths.

Inference: because \(E=hc/\lambda\), those wavelengths correspond to particular energy differences.

Model: Bohr and Schrödinger provide different descriptions of the states separated by those energy differences.

The observation does not force you to draw a circular orbit.

This is why saying “line spectra prove the Bohr model” is too strong. They provided powerful evidence for quantised atomic energies, which were a central feature of Bohr’s model. Later evidence showed that Bohr’s physical picture of electron motion was incomplete.

Schrödinger’s model preserved quantisation and gave a better account of the electron states themselves.

11Emission and absorption spectra tell the same energy-gap story

Suppose an atom can move between two states separated by

\[
\Delta E=3.0\times10^{-19}\text{ J}
\]

If the electron moves downward through that gap, a photon of that energy can be emitted.

If the electron begins in the lower state, a photon of the same energy can be absorbed to move it upward.

That is why an element’s emission and absorption lines are related. They are produced by the same set of atomic energy differences, although the appearance and intensity of a real spectrum also depend on which states are populated and which transitions are likely.

This same energy-level reasoning explains why different metal ions can produce characteristic colours in flame tests. You can follow that connection further in Flame Tests and Quantised Energy Levels for HSC Chemistry.

12A useful exam decision rule

When you are asked to compare Bohr and Schrödinger using spectral evidence, do not write only:

Bohr has shells, while Schrödinger has orbitals.

That is true, but it misses the evidence.

A stronger chain of reasoning is:

1. line spectra contain only particular wavelengths
2. particular wavelengths mean particular photon energies because \(E=hc/\lambda\)
3. therefore atoms have discrete energy differences
4. Bohr explained these differences using quantised energy levels and transitions between them
5. Bohr’s fixed-orbit description works mainly for one-electron atoms and cannot explain the full behaviour of multi-electron atoms
6. Schrödinger retained quantised energies but described electrons using wavefunctions and orbitals rather than definite paths

That connects the experiment to the model instead of listing features from memory.

13Common misconceptions

“An electron releases energy while it stays in one energy level”

Not in the Bohr or Schrödinger description of a stationary atomic state.

A spectral photon is associated with a transition between states. The photon’s energy corresponds to the energy difference.

“A larger \(n\) means a more negative energy”

It is the opposite for a bound hydrogen electron.

As \(n\) increases, the energy approaches \(0\). For example, \(-1\times10^{-19}\text{ J}\) is higher than \(-5\times10^{-19}\text{ J}\).

“An orbital is just a more complicated orbit”

No.

An orbit is a trajectory through space. An orbital describes a quantum state and probability distribution. It does not tell you that the electron travels repeatedly along the boundary of the orbital shape.

“The Schrödinger model removed energy levels”

It did not. Quantised energies remain central.

What changed was the description of the electron occupying those states.

“A line spectrum proves electrons are particles”

A line spectrum shows discrete energy differences. It does not, by itself, settle every question about the nature or motion of electrons.

14Questions and solutions

Question 1

An atom emits a photon of wavelength \(610\text{ nm}\). Calculate the energy carried by one photon.

Solution 1

The photon carries approximately \(3.26\times10^{-19}\text{ J}\) of energy.

Use

\[
E=\frac{hc}{\lambda}
\]

and convert the wavelength first:

\[
610\text{ nm}=6.10\times10^{-7}\text{ m}
\]

Then,

\[
\begin{aligned}
E
&=\frac{(6.626\times10^{-34}\text{ J s})(3.00\times10^8\text{ m s}^{-1})}
{6.10\times10^{-7}\text{ m}}\\
&=3.26\times10^{-19}\text{ J}
\end{aligned}
\]

The spectral line therefore reveals an atomic energy difference of \(3.26\times10^{-19}\text{ J}\).

The important idea is that the detector measures a wavelength, but the model interprets that wavelength as an energy gap.

Question 2

A hydrogen electron falls from \(n=4\) to \(n=2\). Calculate the energy of the emitted photon.

Use

\[
E_n=-\frac{2.18\times10^{-18}\text{ J}}{n^2}
\]

Solution 2

The emitted photon has an energy of approximately \(4.09\times10^{-19}\text{ J}\).

First calculate the two allowed electron energies:

\[
\begin{aligned}
E_4
&=-\frac{2.18\times10^{-18}}{16}\\
&=-1.36\times10^{-19}\text{ J}\\
E_2
&=-\frac{2.18\times10^{-18}}{4}\\
&=-5.45\times10^{-19}\text{ J}
\end{aligned}
\]

The electron moves from the higher \(n=4\) state to the lower \(n=2\) state, so the energy released is

\[
\begin{aligned}
E_{\text{photon}}
&=E_4-E_2\\
&=(-1.36\times10^{-19})-(-5.45\times10^{-19})\\
&=4.09\times10^{-19}\text{ J}
\end{aligned}
\]

The negative electron energies do not mean the emitted photon has negative energy. The photon carries away the positive magnitude of the energy lost by the atom.

Question 3

A student observes hydrogen’s line emission spectrum and says:

“Because the spectrum contains separate lines, the experiment proves that electrons move around the nucleus in fixed circular orbits.”

Evaluate the student’s conclusion.

Solution 3

The conclusion goes further than the evidence supports. The line spectrum supports discrete atomic energy levels, but it does not prove that electrons follow fixed circular paths.

Each spectral wavelength corresponds to a photon energy:

\[
E=\frac{hc}{\lambda}
\]

Because only particular wavelengths appear, only particular energy differences are involved.

Bohr successfully represented those energies using allowed electron orbits. However, the measured spectrum gives information about energy differences, not a direct measurement of the electron’s trajectory.

The Schrödinger model later retained quantised energies while replacing definite circular paths with wavefunctions and orbitals.

The tempting mistake is to treat one successful feature of a model as proof of every assumption inside the model.

Question 4

Helium produces a line spectrum rather than a continuous spectrum, showing that its electrons also occupy quantised states. However, calculations using the simple hydrogen Bohr equation do not correctly predict helium’s complete spectrum.

Explain why these two observations are not contradictory.

Solution 4

The observations are consistent because quantised energy states are real, while the simple hydrogen Bohr equation is only a limited model of those states.

Helium’s line spectrum still provides evidence for discrete energy differences. Each line corresponds to a photon emitted during a transition between allowed states.

However, neutral helium contains two electrons. Each electron is attracted to the nucleus and also repelled by the other electron. The simple hydrogen equation

\[
E_n=-\frac{2.18\times10^{-18}}{n^2}
\]

was developed for a one-electron hydrogen atom and does not include electron-electron repulsion.

Therefore, failure of that equation for helium does not mean quantisation disappears. It means Bohr’s particular description is too simple.

The Schrödinger framework can describe multi-electron atoms in principle, although approximation methods are needed because the interacting electrons make the mathematics much harder.

Question 5

A hydrogen atom begins in the \(n=4\) state. A student argues that every atom prepared in \(n=4\) must emit exactly one spectral line because all the atoms begin with the same energy.

Explain the flaw in this reasoning.

Solution 5

Beginning in one energy level does not guarantee one emitted wavelength because the electron can reach lower levels through different sequences of transitions.

For example, an electron in \(n=4\) could move directly to \(n=2\). That transition releases an energy

\[
E_4-E_2
\]

and produces one particular photon wavelength.

But an electron could instead undergo a sequence such as

\[
n=4\rightarrow n=3\rightarrow n=2
\]

The \(4\to3\) transition has one energy difference, while the \(3\to2\) transition has another. The emitted photons therefore have different wavelengths.

The student’s hidden assumption is that an excited atom has only one possible route to a lower energy state.

A spectrum records the allowed photon energies produced by the collection of transitions occurring in many atoms. This is why several spectral lines can arise even when atoms were initially excited to the same high level.

Question 6

Imagine two models both predict the same wavelengths for the main hydrogen spectral lines.

Model A says electrons travel on definite circular paths.

Model B says electrons occupy quantum states described by probability distributions and does not assign definite paths.

Can the agreement with those hydrogen wavelengths alone determine which description of electron motion is correct? Explain what further evidence would be needed.

Solution 6

No. Matching the same hydrogen wavelengths would not, by itself, distinguish the two descriptions of electron motion.

The wavelengths tell us the energy differences between states because

\[
\Delta E=\frac{hc}{\lambda}
\]

If both models contain the same relevant energy differences, both can reproduce those spectral positions.

To distinguish them, scientists need observations where the models make different predictions. These can include the behaviour of multi-electron atoms, finer structure in atomic spectra, the effects of magnetic fields, and evidence associated with the wave-like behaviour of electrons.

This question exposes an important scientific principle: evidence can strongly support one feature of a model without uniquely proving the entire model.

Hydrogen spectra established that atomic energies are quantised. The later quantum mechanical model was accepted not because quantisation disappeared, but because it preserved that successful prediction while explaining a much wider range of evidence.

15From spectra to electron configuration

The historical shift from Bohr to Schrödinger changes what an electron configuration actually means.

In the Bohr picture, it is tempting to imagine electrons simply filling circular shells. In the modern quantum model, electrons occupy orbitals grouped into subshells and principal energy levels. Their energies are affected by nuclear attraction, shielding, penetration, and electron-electron repulsion.

That is the next step: moving from the spectral evidence for discrete energies to the arrangement of electrons in real multi-electron atoms. The Electronic Configuration for HSC Chemistry resource develops that connection.