Quantized Energy Levels

At first glance, it might seem reasonable to imagine that an electron in an atom can have any amount of energy. In reality, electrons are restricted to specific allowed energies.

These allowed energies are called quantized energy levels.

The word quantized means that something can exist only in certain discrete amounts rather than taking every possible value.

An electron can occupy one allowed energy level or another, but it cannot have an energy between the allowed levels.

This is one of the fundamental ideas of quantum physics and helps explain why atoms absorb and emit only particular wavelengths of light.

What Does Quantized Mean?

Imagine climbing a staircase.

You can stand on:

• the first step
• the second step
• the third step
• the fourth step

But you cannot stand normally between two steps.

Atomic energy levels behave somewhat like this.

An electron can have:

E₁, E₂, E₃, E₄...

but it cannot simply choose any energy between these allowed values.

For example, if an atom had allowed energies of:

−10 eV

−5 eV

−2 eV

then an electron could occupy one of those states, but it could not occupy a state at:

−7 eV

if −7 eV is not an allowed energy.

This is what we mean by quantization.

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Continuous vs Quantized Energy

A continuous quantity can take any value within a range.

For example, classical models might allow energies such as:

2.0 J

2.1 J

2.15 J

2.151 J

and so on.

There is no required gap between the possible values.

A quantized quantity has only specific allowed values.

For example:

E₁

E₂

E₃

E₄

with no permitted energies between them.

Atomic electrons have quantized energies.

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Why Are Atomic Energies Quantized?

Electrons in atoms are described by quantum mechanics.

An electron does not behave simply like a tiny planet orbiting a nucleus. It also has wave-like properties.

Only certain electron wave patterns are allowed around a nucleus. These permitted quantum states correspond to particular energies.

As a result:

only certain electron energies are allowed.

This is similar to a vibrating string.

A guitar string cannot form just any stable standing-wave pattern. Only particular patterns fit the string correctly.

Likewise, only particular electron states satisfy the quantum conditions of an atom.

Energy Levels in an Atom

Energy levels are often represented using horizontal lines.

For example:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

Each line represents an allowed electron energy.

The vertical separation between lines represents the difference in energy.

Higher lines correspond to higher-energy states.

Lower lines correspond to lower-energy states.

Importantly, the electron does not occupy the empty spaces between the lines.

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The Ground State

The ground state is the lowest possible energy state of an atom or electron.

If an electron occupies the lowest available energy level, it is in its ground state.

For example:

E₃ ─────────────

E₂ ─────────────

E₁ ─────●───────

Here, the electron is at E₁.

It has the lowest energy permitted by the atom.

The ground state is generally the atom's most stable state.

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Excited States

An excited state occurs when an electron has more energy than it has in the ground state.

For example:

E₃ ─────●───────

E₂ ─────────────

E₁ ─────────────

The electron is now in a higher energy level.

It may have reached this level by absorbing energy.

Possible energy sources include:

• light
• heat
• electrical energy
• collisions with other particles

Ground State vs Excited State

An atom can have many possible excited states but only one lowest-energy ground state for a given configuration.

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Moving Between Energy Levels

Electrons can move between energy levels by gaining or losing energy.

A transition upward requires the electron to absorb energy.

A transition downward releases energy.

Suppose:

E₂ − E₁ = 3.0 eV

To move from E₁ to E₂, the electron must absorb exactly:

3.0 eV

It cannot absorb 2.5 eV and move partway upward.

It cannot occupy the gap between E₁ and E₂.

This is one of the most important consequences of quantization.

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Energy Differences

The energy involved in a transition is determined by the difference between the two levels:

ΔE = E_final − E_initial

where:

ΔE = change in energy

E_final = final energy level

E_initial = initial energy level

For an upward transition, ΔE is positive because energy must be absorbed.

For a downward transition, the atom releases energy.

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Worked Example: Excitation

Suppose:

E₁ = −10 eV

E₂ = −6 eV

An electron moves from E₁ to E₂.

Calculate the energy required.

ΔE = E₂ − E₁

ΔE = −6 − (−10)

ΔE = 4 eV

The electron must therefore absorb:

4 eV

of energy.

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Worked Example: Returning to a Lower Level

Suppose an electron moves from:

E₃ = −2 eV

to:

E₁ = −10 eV

The magnitude of the energy released is:

ΔE = |−10 − (−2)|

ΔE = 8 eV

Therefore, the atom releases 8 eV of energy.

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Absorption of Photons

Light comes in packets of energy called photons.

The energy of a photon is:

E = hf

where:

E = photon energy

h = Planck's constant

f = frequency

For an electron to move upward between two energy levels, the incoming photon must have the correct energy.

If:

ΔE = 4 eV

then the atom needs a photon with an energy corresponding to that transition.

This is why atoms absorb only certain frequencies of light.

What If the Photon Has the Wrong Energy?

Suppose an electron needs:

4 eV

to move to the next allowed level.

A photon arrives with:

3 eV

The electron cannot simply move 3/4 of the way to the next energy level.

There is no allowed state there.

For that particular transition, the photon does not have the required energy.

Similarly, a photon with 3.8 eV would not cause a transition requiring exactly 4.0 eV in the idealized model.

This is a direct consequence of quantization.

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Emission of Photons

Excited states are often unstable.

An excited electron can move to a lower energy level.

When this happens, energy is released as a photon.

The photon carries an energy equal to the difference between the two energy levels:

E_photon = ΔE

Therefore:

E_photon = hf

and:

ΔE = hf

Different transitions have different energy differences, so they produce photons with different frequencies.

Frequency and Wavelength

The speed of light is related to frequency and wavelength by:

c = fλ

where:

c = speed of light

f = frequency

λ = wavelength

Since:

E = hf

higher-frequency photons have higher energy.

Because:

c = fλ

higher frequencies correspond to shorter wavelengths.

Therefore:

larger energy transition → higher frequency → shorter wavelength

and:

smaller energy transition → lower frequency → longer wavelength

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Worked Example: Comparing Transitions

Suppose an atom has two possible downward transitions.

Transition A releases:

5 eV

Transition B releases:

2 eV

Transition A produces the higher-energy photon.

Therefore, Transition A also produces:

• higher frequency
• shorter wavelength

Transition B produces:

• lower frequency
• longer wavelength

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Energy Level Diagrams

An energy level diagram represents the allowed energies of electrons.

Consider:

E₄ ─────────────

E₃ ─────────────

E₂ ─────────────

E₁ ─────────────

An upward arrow represents:

energy absorption

A downward arrow represents:

energy emission

The length of the transition on the energy scale corresponds to the energy difference.

A larger vertical change means a larger ΔE.

Interpreting an Energy Level Diagram

When examining an energy-level diagram, ask:

Where does the electron begin?

This is the initial state.

Where does it finish?

This is the final state.

Is the arrow upward?

Energy is being absorbed.

Is the arrow downward?

Energy is being released.

How large is the gap?

A larger energy gap means a larger photon energy.

Which level is lowest?

That represents the ground state.

Higher allowed levels represent excited states.

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Example Energy Levels

Suppose an atom has:

E₄ = −1 eV

E₃ = −3 eV

E₂ = −6 eV

E₁ = −10 eV

Consider the transition:

E₁ → E₃

Energy required:

ΔE = −3 − (−10)

= 7 eV

Now consider:

E₃ → E₂

Energy released:

|−6 − (−3)|

= 3 eV

The first transition involves a larger amount of energy.

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Why Are Some Energy Values Negative?

Atomic energy levels are often shown using negative values.

A common convention defines:

E = 0

as the energy of a free electron completely separated from the atom.

An electron bound to the nucleus has less energy than this reference level, so its energy is negative.

For example:

E = −13.6 eV

indicates that energy must be supplied to remove the electron completely from the atom.

The more negative the value, the more strongly bound the electron is in this simplified picture.

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Ionization

If enough energy is supplied, an electron can leave the atom completely.

This process is called ionization.

For a simplified energy diagram:

E = 0 ─────── ionization limit

E₃ ─────────

E₂ ─────────

E₁ ─────────

An electron reaching or exceeding the ionization limit is no longer bound in one of the atom's discrete bound energy levels.

The atom becomes an ion because it has lost an electron.

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Ionization Energy

The ionization energy is the minimum energy required to remove an electron from an atom in a specified state.

Suppose the ground-state electron has:

E₁ = −13.6 eV

and the ionization limit is:

E = 0 eV

Then:

energy required

= 0 − (−13.6)

= 13.6 eV

This is the ionization energy for that electron from the ground state in this model.

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Why Energy Levels Get Closer Together

In atoms such as hydrogen, higher energy levels become increasingly close together.

Near the nucleus, the differences between successive energy levels are relatively large.

At high energies, the levels become closer together.

Eventually they approach the ionization limit.

This means that the energy required to move between neighboring high-energy states is generally smaller than the energy required between neighboring low-energy states in hydrogen.

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Quantization and Atomic Spectra

One of the strongest pieces of evidence for quantized atomic energy levels comes from atomic spectra.

When excited atoms release energy, they do not produce every possible wavelength.

Instead, they produce specific wavelengths.

When the light is separated using a spectroscope, we see individual bright lines.

This is called a line emission spectrum.

Why Do We See Spectral Lines?

Each spectral line corresponds to a particular electron transition.

For example:

E₃ → E₂

produces one photon energy.

E₄ → E₂

produces another.

E₅ → E₂

produces another.

Since:

ΔE = hf

each transition produces a particular frequency.

Therefore, only certain frequencies are emitted.

This creates discrete spectral lines rather than a continuous rainbow.

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Emission Spectrum vs Continuous Spectrum

A continuous spectrum contains essentially all wavelengths across a range.

An atomic emission spectrum contains only specific wavelengths.

This difference provides evidence that atomic energies are quantized.

If electrons could have any energy, we would expect a continuous range of possible photon energies.

Instead, atoms produce characteristic lines.

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Absorption Spectra

Atoms can also absorb particular photon energies.

When continuous light passes through a gas, atoms may absorb photons corresponding to allowed upward transitions.

The result can be an absorption spectrum containing dark lines.

Those dark lines occur at wavelengths absorbed by the atoms.

The same basic energy-level structure determines both emission and absorption.

Every Element Has Its Own Energy Levels

Different elements have different arrangements of electrons and different interactions within their atoms.

Therefore, different elements have different allowed energy levels.

As a result, each element produces its own characteristic spectrum.

This is sometimes compared to an atomic fingerprint.

Scientists can identify elements by examining the wavelengths of light they emit or absorb.

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Spectroscopy

The study of how matter interacts with electromagnetic radiation is called spectroscopy.

Spectroscopy allows scientists to determine the composition of substances without physically collecting them.

Applications include:

• identifying elements in stars
• analysing gases
• studying distant galaxies
• detecting pollutants
• analysing chemical samples
• examining plasmas
• studying planetary atmospheres

The connection between spectroscopy and quantized energy levels is fundamental.

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Example: Identifying Elements in Stars

Scientists cannot travel to most stars and collect samples.

Instead, they analyse starlight.

If particular absorption lines appear in a star's spectrum, those lines can be compared with known laboratory spectra.

Because elements have characteristic energy-level structures, their spectral lines reveal which elements are present.

This is how scientists can determine that stars contain elements such as:

• hydrogen
• helium
• sodium
• calcium
• iron

Quantized energy levels therefore allow us to investigate matter across enormous astronomical distances.

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The Electron Does Not Travel Through Intermediate Energy States

A common misunderstanding is to imagine an electron gradually climbing through the empty space between energy levels.

In an energy-level diagram, the vertical position represents energy, not a literal physical height.

When an electron makes a transition, its quantum state changes from one allowed state to another.

It does not occupy all the intermediate energies.

Therefore, an energy-level diagram should not be interpreted as a picture of an electron physically jumping upward through space.

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Energy Levels Are Not Literal Orbits

Simple atomic models sometimes show electrons moving around the nucleus in circular shells.

These diagrams can be useful introductions, but modern quantum mechanics describes electrons using orbitals and probability distributions rather than simple planetary orbits.

An energy level describes an allowed quantum state and energy.

It should not automatically be interpreted as a fixed circular path around the nucleus.

Quantization Beyond Atoms

Quantization is not limited to atomic electrons.

Quantum physics shows that many physical quantities can occur in discrete amounts under particular conditions.

Examples include:

• photon energies
• molecular rotational energies
• molecular vibrational energies
• angular momentum
• electron spin

The discovery of quantization was one of the major changes from classical physics to quantum physics.

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Planck's Constant

The relationship:

E = hf

contains Planck's constant, h.

Its approximate value is:

h = 6.63 × 10⁻³⁴ J·s

The extremely small value of Planck's constant helps explain why quantum effects are especially noticeable at atomic and subatomic scales.

For everyday objects, energy differences associated with quantum states are usually far too small and numerous for us to notice individual levels directly.

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Worked Example Using Photon Energy

Suppose an electron transition releases:

3.31 × 10⁻¹⁹ J

Use:

E = hf

Rearrange:

f = E/h

Substitute:

f = (3.31 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 4.99 × 10¹⁴ Hz

So the emitted photon has a frequency of approximately:

5.0 × 10¹⁴ Hz

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Worked Example Using Wavelength

Using:

f = 5.0 × 10¹⁴ Hz

and:

c = 3.0 × 10⁸ m/s

Use:

c = fλ

Rearrange:

λ = c/f

Substitute:

λ = (3.0 × 10⁸)/(5.0 × 10¹⁴)

λ = 6.0 × 10⁻⁷ m

or:

600 nm

This wavelength lies in the visible region of the electromagnetic spectrum.

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Connecting the Ideas

The entire process can be summarized as:

Electron absorbs energy

↓

Electron moves to a higher allowed energy level

↓

Atom is in an excited state

↓

Electron eventually moves to a lower allowed energy level

↓

Energy is emitted as a photon

↓

Photon energy equals the difference between the levels

↓

The photon has a specific frequency and wavelength

↓

Specific wavelengths produce atomic spectral lines

This chain of reasoning connects quantum energy levels directly to observable light.

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Common Mistakes

Thinking electrons can have any energy

Bound electrons in atoms occupy specific allowed energy states.

Thinking the gaps are additional energy levels

The spaces between lines on an energy-level diagram represent energies that are not allowed bound states.

Confusing ground state with zero energy

The ground state is the lowest allowed state, not necessarily E = 0.

Thinking an excited electron has left the atom

An excited electron is still bound to the atom unless it has been ionized.

Confusing excitation and ionization

Excitation: electron moves to a higher bound energy level.

Ionization: electron leaves the atom.

Thinking upward transitions release energy

Upward transitions require absorption.

Thinking downward transitions absorb energy

Downward transitions generally involve emission.

Forgetting that the photon must match the energy gap

An allowed transition requires the appropriate energy difference.

Assuming every energy gap is the same

Atomic energy levels are generally not equally spaced.

Treating energy diagrams as physical pictures

Vertical distance represents energy difference, not physical distance from the nucleus.

Thinking spectral lines are random

Each spectral line corresponds to a specific allowed energy transition.

Confusing frequency and wavelength

Higher photon energy means:

higher frequency but shorter wavelength.

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Key Terms

Quantized — Restricted to specific discrete values rather than any value in a continuous range.

Quantum — A discrete amount associated with a physical quantity in quantum theory.

Energy level — An allowed energy state of an electron in an atom.

Discrete — Existing as separate individual values rather than continuously.

Continuous — Able to take any value within a range.

Ground state — The lowest-energy state available to an atom or electron.

Excited state — An allowed state with energy greater than the ground state.

Excitation — The process of moving an electron to a higher allowed energy state.

Transition — A change from one allowed quantum state to another.

Energy gap — The difference in energy between two allowed levels.

Absorption — The process in which an atom gains energy, potentially causing an electron to move upward.

Emission — The release of energy when an electron moves to a lower state.

Photon — A discrete packet of electromagnetic energy.

Planck's constant — The constant relating photon energy to frequency.

Frequency (f) — The number of wave cycles passing a point each second.

Wavelength (λ) — The distance between corresponding points on successive waves.

Ionization — The removal of an electron from an atom.

Ionization energy — The minimum energy required to remove an electron from a specified atomic state.

Ionization limit — The energy at which an electron is no longer bound to the atom.

Bound state — A state in which an electron remains associated with the atom.

Atomic spectrum — The pattern of wavelengths emitted or absorbed by atoms.

Emission spectrum — A spectrum containing wavelengths emitted during downward transitions.

Absorption spectrum — A spectrum showing wavelengths removed when photons are absorbed.

Spectral line — A particular wavelength associated with an allowed atomic transition.

Spectroscopy — The study of interactions between matter and electromagnetic radiation.

Orbital — A quantum-mechanical description of where an electron is likely to be found.

Electron volt (eV) — A unit of energy commonly used in atomic and particle physics.

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Key Takeaways

• Atomic electrons have quantized energies.
• Quantized means only particular discrete values are allowed.
• Electrons cannot occupy energies between allowed bound energy levels.
• The ground state is the lowest allowed energy state.
• An excited state has more energy than the ground state.
• Electrons move to higher energy states by absorbing energy.
• Electrons move to lower states by releasing energy.
• The energy involved in a transition is determined by the difference between the energy levels.
• The basic relationship is:

ΔE = E_final − E_initial

• Light consists of packets of energy called photons.
• Photon energy is related to frequency by:

E = hf

• Frequency and wavelength are related by:

c = fλ

• Larger energy gaps correspond to higher-frequency photons.
• Higher-frequency photons have shorter wavelengths.
• An electron cannot simply absorb part of the required energy and occupy an intermediate state.
• Energy-level diagrams show energy, not literal physical height.
• Atomic energy levels are generally not equally spaced.
• Excitation and ionization are different processes.
• Excitation keeps the electron bound to the atom.
• Ionization removes the electron from the atom.
• Downward transitions produce characteristic photon energies.
• These photons create line emission spectra.
• Upward transitions can produce characteristic absorption lines.
• Each element has its own characteristic set of energy levels.
• Each element therefore has a distinctive atomic spectrum.
• Atomic spectra provide strong experimental evidence for quantized energy levels.
• Spectroscopy allows scientists to identify elements in laboratory samples, gases, stars, and distant astronomical objects.
• Quantization is one of the central ideas distinguishing quantum physics from classical physics.

A useful way to remember the process is:

Absorb energy → move up

Emit energy → move down

Only allowed levels → only specific photon energies → specific spectral lines

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Check Your Understanding

1. What does the word quantized mean?

2. How does a quantized quantity differ from a continuous quantity?

3. What is an atomic energy level?

4. Why can an electron not occupy an energy between two allowed levels?

5. Define the ground state.

6. Define an excited state.

7. What must happen for an electron to move from a lower energy level to a higher one?

8. What generally happens when an electron moves from a higher level to a lower one?

9. Suppose:

E₁ = −12 eV

E₂ = −7 eV

Calculate the energy required for the transition:

E₁ → E₂

10. Would the transition in Question 9 involve absorption or emission?

11. Suppose:

E₃ = −2 eV

E₁ = −12 eV

Calculate the magnitude of the energy released during:

E₃ → E₁

12. Explain why an electron requiring 5 eV for a particular excitation cannot simply absorb 3 eV and move partway to the next level.

13. What is a photon?

14. State the relationship between photon energy and frequency.

15. What does Planck's constant represent in the equation E = hf?

16. Which photon has more energy: one with a frequency of 4 × 10¹⁴ Hz or one with a frequency of 8 × 10¹⁴ Hz?

17. Which of those photons has the shorter wavelength?

18. Explain the difference between excitation and ionization.

19. What is the ionization limit?

20. Why are bound atomic energy levels often shown using negative energies?

21. Explain what an upward arrow means on an energy-level diagram.

22. Explain what a downward arrow means.

23. Why do different downward transitions produce photons with different wavelengths?

24. What is an atomic emission spectrum?

25. Why does an atomic emission spectrum contain separate lines rather than every possible wavelength?

26. Explain how atomic spectra provide evidence for quantized energy levels.

27. Why can different elements be identified using their spectra?

28. Explain why an energy-level diagram should not be interpreted as showing the literal physical height of an electron.

29. Describe one practical application of spectroscopy.

30. Challenge: An atom has the following allowed energy levels:

E₄ = −1.0 eV
E₃ = −3.0 eV
E₂ = −6.0 eV
E₁ = −10.0 eV

a. Identify the ground state.
b. Identify the possible excited states shown.
c. Calculate the energy needed for E₁ → E₂.
d. Calculate the energy needed for E₁ → E₃.
e. Calculate the energy needed for E₂ → E₄.
f. Calculate the energy released for E₄ → E₁.
g. Which possible downward transition shown would produce the highest-energy photon?
h. Which would produce the highest-frequency photon?
i. Which would produce the shortest-wavelength photon?
j. Explain why the answers to g, h, and i are related.
k. Could an electron in E₁ absorb 2 eV and move to an allowed level shown? Explain.
l. Could it absorb 4 eV? Explain.
m. Could it absorb 7 eV? Explain.
n. Describe what happens when an electron in E₄ returns to E₁.
o. Explain how these transitions could produce an observable line spectrum.