Quantum Mechanical Model

The quantum mechanical model of the atom is the modern scientific model used to describe electrons in atoms.

Unlike the Bohr model, it does not picture electrons as tiny particles travelling around the nucleus in fixed circular paths.

Instead, quantum mechanics describes electrons using:

• wavefunctions
• quantized energy states
• atomic orbitals
• probability distributions

The central idea is:

We generally describe where an electron is likely to be detected rather than assigning it a fixed classical orbit around the nucleus.

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From Bohr to Quantum Mechanics

Bohr's model was a major improvement over Rutherford's model.

It introduced:

quantized energy levels

and successfully explained much of the hydrogen spectrum.

However, the Bohr model pictured electrons travelling along specific:

orbits

This idea eventually proved inadequate.

Experiments involving:

• electron diffraction
• atomic spectra
• multi-electron atoms
• magnetic fields
• electron wave behaviour

required a more sophisticated theory.

This became:

quantum mechanics

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Wave Nature of Electrons

Louis de Broglie proposed that particles such as electrons have an associated wavelength:

λ = h/p

where:

• λ = de Broglie wavelength
• h = Planck's constant
• p = momentum

Experiments confirmed that electrons can produce:

• diffraction
• interference

Therefore, electrons cannot be described completely as simple classical particles.

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Schrödinger's Contribution

In 1926, Austrian physicist Erwin Schrödinger developed an equation that describes the wave-like behaviour of quantum systems.

It became known as the:

Schrödinger equation

Its solutions provide information about the allowed quantum states of electrons.

For atoms, these solutions lead to the concept of:

atomic orbitals

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The Wavefunction

A quantum state can be represented mathematically by a:

wavefunction

usually written:

ψ

The wavefunction contains information about the quantum state.

However, ψ itself is not simply interpreted as the physical position of the electron.

Instead:

|ψ|²

is related to the:

probability density

for detecting the electron.

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Probability Density

Probability density tells us how likely an electron is to be detected in a particular region of space.

A region with:

large |ψ|²

has a greater probability density.

A region with:

small |ψ|²

has a lower probability density.

This does not mean the electron is physically smeared through the atom like a cloud of material.

The "cloud" represents:

probability

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Electron Probability Clouds

Electron probability distributions are often represented visually using:

electron clouds

In such a diagram:

• denser regions represent greater probability
• less dense regions represent lower probability
• the nucleus is located near the center

The cloud does not represent a fuzzy physical substance.

It represents where repeated measurements are more or less likely to detect the electron.

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Probability and Measurement

Suppose scientists prepared many identical hydrogen atoms in the same quantum state.

If they measured the electron's position repeatedly, they would not necessarily obtain exactly the same position every time.

Instead, the results would form a distribution.

After many measurements:

measurement locations → probability distribution

The electron cloud represents this distribution.

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Atomic Orbitals

An atomic orbital is a quantum state associated with a particular electron probability distribution around the nucleus.

Orbitals have characteristic:

• energies
• shapes
• orientations

Common orbital types include:

• s orbitals
• p orbitals
• d orbitals
• f orbitals

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Orbit vs Orbital

These two words sound similar but have very different meanings.

Orbit

Used in the simplified Bohr model.

An orbit is pictured as:

a definite path followed by an electron

Orbital

Used in quantum mechanics.

An orbital describes:

a quantum state with a characteristic probability distribution

Therefore:

orbit ≠ orbital

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The 1s Orbital

The lowest-energy state of hydrogen is the:

1s orbital

It has a:

spherical probability distribution

around the nucleus.

The probability density is greatest near the nucleus and generally decreases with increasing distance.

There is no thin circular track showing where the electron travels.

Instead, there is a three-dimensional probability distribution.

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s Orbitals

s orbitals are spherically symmetric.

Examples include:

• 1s
• 2s
• 3s

As the principal quantum number increases, the orbital generally becomes larger and more complex.

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p Orbitals

p orbitals have a different shape.

They are commonly represented as two-lobed structures.

For a p subshell there are three orientations:

• pₓ
• pᵧ
• p_z

These point along different spatial axes.

The probability distribution therefore has both:

shape and orientation

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d and f Orbitals

Higher-energy atoms also involve:

d orbitals

and:

f orbitals

These have more complicated probability distributions.

The increasing complexity of these orbitals helps quantum mechanics explain:

• electron configurations
• periodic table structure
• chemical bonding
• magnetic behaviour
• atomic spectra

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Orbitals Are Three-Dimensional

Atomic orbitals are not flat diagrams.

They describe probability distributions in:

three-dimensional space

A textbook drawing is only a visual representation.

For example, the two lobes of a p orbital represent regions associated with the electron's quantum state, not two physical containers holding pieces of an electron.

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Quantized Energy Still Exists

Quantum mechanics did not discard Bohr's idea that atomic energies are quantized.

Electrons still occupy:

discrete allowed energy states

However, modern quantum mechanics provides a much more complete description of those states.

Bohr:

energy levels + fixed circular orbits

Quantum mechanics:

energy levels + wavefunctions + orbitals + probabilities

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Principal Energy Levels

Electron states can be organized partly using the:

principal quantum number, n

Possible values are:

n = 1, 2, 3, 4, ...

Larger values of n generally correspond to:

• higher energy
• larger characteristic distance from the nucleus

But unlike the Bohr model, each principal level can contain different:

sublevels and orbitals

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Sublevels

Principal energy levels contain sublevels commonly labelled:

s, p, d, f

For example:

n = 1 contains:

1s

n = 2 contains:

2s and 2p

n = 3 contains:

3s, 3p and 3d

n = 4 contains:

4s, 4p, 4d and 4f

This structure is much richer than Bohr's simple circular shells.

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Number of Orbitals

Each type of sublevel contains a particular number of orbitals.

Each individual orbital can contain a maximum of:

2 electrons

with opposite spin states.

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Quantum Numbers

Electron states can be described using a set of:

quantum numbers

These provide information about an electron's quantum state.

The four commonly introduced quantum numbers are:

• principal quantum number, n
• orbital angular momentum quantum number, l
• magnetic quantum number, mₗ
• spin quantum number, mₛ

Together, they provide a much more detailed description than simply stating an electron's Bohr orbit.

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Principal Quantum Number

The principal quantum number:

n

is associated with the main energy level.

Values are:

1, 2, 3, 4, ...

It is related to the characteristic size and energy of the orbital.

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Orbital Angular Momentum Quantum Number

The quantum number:

l

is associated with orbital type.

Common correspondence:

l = 0 → s

l = 1 → p

l = 2 → d

l = 3 → f

This helps determine the shape of the orbital.

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Magnetic Quantum Number

The magnetic quantum number:

mₗ

describes the possible orientations associated with an orbital.

For example, the p sublevel has three orbitals.

These correspond to different spatial orientations commonly represented as:

• pₓ
• pᵧ
• p_z

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Electron Spin

Electrons also possess an intrinsic quantum property called:

spin

The spin quantum number can take values:

mₛ = +½

or:

mₛ = −½

These are often described informally as:

spin up

and:

spin down

Electron spin is a quantum property and should not be imagined literally as a tiny ball physically rotating around its axis.

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Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in the same atom can have exactly the same set of four quantum numbers.

One important consequence is:

each orbital can contain a maximum of two electrons

and those electrons have opposite spin states.

This principle is essential for understanding electron configurations and the periodic table.

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Nodes

Some orbitals contain regions where the probability density is:

zero

These regions are called:

nodes

An electron in that quantum state will not be detected at a node.

Nodes are a natural consequence of the wave nature of electrons.

They are similar in concept to nodes in standing waves.

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Standing Waves

The wave nature of electrons provides a useful analogy.

A vibrating string can form only certain stable standing-wave patterns.

Similarly, only certain electron wavefunctions satisfy the conditions required for stable atomic states.

This helps explain why atomic energies are:

quantized

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Quantization from Wave Behaviour

In the Bohr model, quantized orbits were largely introduced as a rule.

Quantum mechanics provides a deeper explanation.

Only certain wavefunctions satisfy the physical conditions around the nucleus.

Therefore:

only certain quantum states are allowed

and those states have particular:

energies

This makes quantization a natural result of the mathematical theory rather than simply an added assumption.

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The Uncertainty Principle

Quantum mechanics also introduces the:

Heisenberg uncertainty principle

It states that position and momentum cannot both be specified with unlimited precision.

A simplified form is:

ΔxΔp ≥ ħ/2

where:

• Δx = uncertainty in position
• Δp = uncertainty in momentum
• ħ = h/(2π)

This is a fundamental quantum relationship, not simply a limitation of poor measuring instruments.

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Why Fixed Electron Orbits Are a Problem

A classical orbit assumes that an electron has a precisely defined:

• position
• momentum
• trajectory

at every moment.

Quantum mechanics does not generally allow that classical description.

Instead, electron states are represented using:

wavefunctions and probabilities

This is one reason the fixed-orbit picture is replaced by orbitals.

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Electron Cloud vs Fixed Orbit

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Interpreting an Electron Cloud

Suppose an electron cloud diagram contains many dots.

A common interpretation is:

more dots in a region → greater probability of detecting the electron there

It does not mean:

the electron spends exactly that percentage of time as a classical particle moving through that region

The quantum interpretation concerns the probabilities of measurement outcomes.

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Probability Does Not Mean "We Just Don't Know"

In classical physics, probability can sometimes reflect missing information.

For example, if we do not know exactly how a coin was thrown, we might describe its outcome probabilistically.

Quantum probability is more fundamental.

The quantum state itself determines probabilities for possible measurement outcomes.

Quantum mechanics does not simply say:

"the electron has a definite classical orbit but we don't know which one."

That hidden classical-orbit picture does not reproduce quantum behaviour.

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Electron Transitions

Electrons can still transition between quantum energy states.

If an atom changes from a higher energy state to a lower one:

photon emitted

with:

E_photon = ΔE

and:

E = hf

Therefore:

ΔE = hf

This explains discrete atomic emission spectra.

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Absorption

An atom can absorb a photon if its energy matches an allowed energy difference.

Therefore:

photon absorbed → atom moves to higher-energy state

Because only certain transitions are possible, atoms absorb specific wavelengths.

This produces:

absorption spectra

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Atomic Spectra

Quantum mechanics explains why each element has a characteristic spectrum.

Different atoms have different:

• nuclear charges
• numbers of electrons
• electron-electron interactions
• allowed quantum states

Therefore, they have different energy differences.

Different energy differences produce different:

photon wavelengths

This gives each element its characteristic spectral pattern.

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Why Quantum Mechanics Improves on Bohr

Bohr's model works very well for hydrogen.

However, atoms such as:

• helium
• carbon
• oxygen
• sodium
• iron

contain several electrons.

These electrons interact with:

• the nucleus
• one another

Quantum mechanics can incorporate these interactions far more successfully than the simple Bohr model.

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Multi-Electron Atoms

In a multi-electron atom, electron-electron repulsion changes the energies of electron states.

This produces:

• sublevels
• shielding effects
• complex spectra
• electron configurations

The simple formula:

Eₙ = −13.6/n² eV

works for hydrogen but cannot describe ordinary multi-electron atoms in that simple form.

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Explaining the Periodic Table

The quantum mechanical model helps explain the structure of the:

periodic table

Electron configurations determine many chemical properties.

For example:

• elements in the same group often have similar outer-electron configurations
• orbital filling helps explain periods
• d orbitals help explain transition metals
• f orbitals help explain lanthanides and actinides

The quantum model therefore connects:

physics → atomic structure → chemistry

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Explaining Chemical Bonds

Chemical bonding depends strongly on electron wavefunctions and orbitals.

Quantum mechanics helps explain:

• covalent bonds
• molecular orbitals
• bond energies
• molecular shapes
• electron sharing

The Bohr model cannot provide a complete explanation of these phenomena.

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Explaining Spectral Details

The quantum mechanical model can explain features that the original Bohr model could not adequately describe, including:

• multi-electron spectra
• fine structure
• effects of magnetic fields
• effects of electric fields
• electron spin

These provide important evidence supporting the modern model.

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Magnetic Effects

When atoms are placed in magnetic fields, spectral lines can split.

This is associated with the:

Zeeman effect

Quantum mechanics explains this using properties such as:

• angular momentum
• magnetic quantum numbers
• electron spin

The simple Bohr model cannot fully account for these observations.

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The Shape of Orbitals

Different orbitals arise because different mathematical solutions of the Schrödinger equation have different spatial forms.

For example:

s → spherical

p → two-lobed

d → more complex

The shapes are therefore not arbitrary artistic diagrams.

They represent mathematical properties of quantum states.

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Probability Surfaces

Orbital diagrams often draw a boundary around a region containing a high percentage of the probability of finding the electron.

This boundary is sometimes called a:

probability surface

The electron is not absolutely confined inside that surface.

There can still be some probability of detecting it outside.

This is important when interpreting orbital diagrams.

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Most Probable Does Not Mean Certain

Suppose an orbital diagram shows a region where the electron is highly likely to be detected.

That does not mean:

the electron must always be inside that region

Quantum mechanics predicts probabilities.

A high-probability region is:

likely

not:

guaranteed

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Radial Probability

For spherical states such as hydrogen's 1s orbital, scientists can calculate the probability of finding the electron at different distances from the nucleus.

Interestingly, the point of greatest probability density and the most probable radial distance are not exactly the same concept.

This illustrates why quantum probability distributions are more sophisticated than simply drawing a fuzzy version of a Bohr orbit.

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Quantum Mechanics and Chemistry

The quantum model explains why electrons fill atoms in structured ways.

For example:

Hydrogen:

1s¹

Helium:

1s²

Lithium:

1s² 2s¹

Carbon:

1s² 2s² 2p²

Oxygen:

1s² 2s² 2p⁴

These electron configurations help explain the chemical behaviour of the elements.

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Worked Example: Identifying an Orbital

An electron is described as being in a:

2p orbital

What can we infer?

The number:

2

indicates the principal energy level:

n = 2

The letter:

p

indicates the orbital type.

A p orbital has a characteristic two-lobed probability distribution.

Therefore, 2p describes a:

quantum state

not a circular path at a fixed distance from the nucleus.

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Worked Example: Probability

Suppose two regions of an electron cloud are compared.

Region A has a high probability density.

Region B has a low probability density.

This means:

the electron is more likely to be detected in Region A than Region B

It does not mean that part of the electron is permanently stored in each region.

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Worked Example: Electron Transition

An electron changes between two atomic states with an energy difference:

ΔE = 4.0 × 10⁻¹⁹ J

Find the frequency of the photon.

Use:

ΔE = hf

Rearrange:

f = ΔE/h

f = (4.0 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 6.03 × 10¹⁴ Hz

Therefore, the photon frequency is approximately:

6.0 × 10¹⁴ Hz

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Worked Example: Finding Wavelength

Using:

c = fλ

Rearrange:

λ = c/f

λ = (3.00 × 10⁸)/(6.03 × 10¹⁴)

λ ≈ 4.98 × 10⁻⁷ m

or:

498 nm

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

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Comparing Rutherford, Bohr, and Quantum Models

Each model represented an important stage in the development of atomic theory.

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Why Earlier Models Were Still Important

Rutherford's model established the existence of the:

nucleus

Bohr introduced:

quantized atomic energy

de Broglie introduced:

matter waves

Schrödinger developed:

wave mechanics

Heisenberg helped establish:

uncertainty and quantum mechanics

Modern atomic theory developed from all of these contributions.

Scientific models evolve as new experimental evidence becomes available.

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Why the Quantum Model Is More Accurate

The quantum mechanical model successfully explains a much wider range of observations.

These include:

• atomic stability
• quantized energy levels
• hydrogen spectra
• multi-electron spectra
• orbital shapes
• electron configurations
• chemical bonding
• periodic trends
• magnetic properties
• electron diffraction
• fine spectral structure

It also makes extremely precise quantitative predictions that agree with experiments.

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Accuracy Does Not Mean Easy to Visualize

The Bohr model is easy to picture:

tiny electrons orbiting a nucleus

The quantum model is harder to visualize because nature at atomic scales does not behave like familiar everyday objects.

A model being easy to visualize does not necessarily make it more accurate.

Quantum mechanics relies heavily on:

mathematics + probability + experimental evidence

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Classical Thinking vs Quantum Thinking

Classically, we might ask:

"Where exactly is the electron, and what exact path is it following?"

Quantum mechanically, a more appropriate question is often:

"What is the probability of detecting the electron in a particular region?"

This represents one of the biggest conceptual changes from classical to quantum physics.

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Common Mistakes

Mistake: Saying an orbital is the same as an orbit.
An orbit is a fixed path in the Bohr model. An orbital is a quantum state associated with a probability distribution.

Mistake: Saying an electron cloud is a cloud of electron material.
It represents probability density.

Mistake: Saying electrons move around inside orbitals like planets inside containers.
Orbitals describe quantum states, not physical containers.

Mistake: Saying quantum mechanics abandoned quantized energy.
Quantization remains fundamental and is explained more deeply by quantum mechanics.

Mistake: Saying the electron is definitely located where the cloud is darkest.
A darker or denser region represents greater probability, not certainty.

Mistake: Saying an electron has no position.
An electron can be detected at a position. Quantum mechanics predicts probabilities for possible measurement outcomes.

Mistake: Saying uncertainty results only from poor equipment.
The uncertainty principle is a fundamental feature of quantum mechanics.

Mistake: Drawing p orbitals as two electrons.
The two lobes are parts of one orbital's probability distribution.

Mistake: Saying Bohr's model is completely useless.
It remains a useful simplified model for introducing quantized energies and hydrogen spectra.

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Key Terms

Quantum mechanical model — modern model describing atomic electrons using quantum states, wavefunctions, orbitals, and probabilities.

Wavefunction (ψ) — mathematical function describing a quantum state.

Probability density — quantity related to the likelihood of detecting a particle in a particular region.

Atomic orbital — quantum state associated with a characteristic electron probability distribution.

Electron cloud — visual representation of electron probability density.

Orbit — fixed electron path used in the Bohr model.

Ground state — lowest-energy quantum state of an atom.

Excited state — atomic state with energy above the ground state.

Principal quantum number — quantum number associated mainly with principal energy level and orbital size.

Sublevel — subdivision of a principal energy level, labelled s, p, d, or f.

Quantum numbers — values used to describe quantum states of electrons.

Spin — intrinsic quantum property of particles such as electrons.

Pauli exclusion principle — principle stating that no two electrons in an atom can have identical sets of four quantum numbers.

Node — region where a particular wavefunction has zero probability density.

Uncertainty principle — fundamental relationship limiting simultaneous precision in quantities such as position and momentum.

Electron configuration — arrangement of electrons among atomic orbitals.

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Key Takeaways

• The quantum mechanical model is the modern model of atomic structure.
• It replaces Bohr's fixed electron paths with quantum states and atomic orbitals.
• Electrons have wave-like as well as particle-like properties.
• A quantum state is represented mathematically by a wavefunction, ψ.
• |ψ|² is related to electron probability density.
• Electron clouds represent probability distributions, not physical clouds of electron material.
• An orbital is not the same as an orbit.
• Orbitals have characteristic energies, shapes, and orientations.
• Common orbital types are s, p, d, and f.
• s orbitals are spherically symmetric.
• p orbitals have characteristic two-lobed probability distributions.
• Orbitals can contain regions of zero probability called nodes.
• Atomic energies remain quantized in the quantum model.
• Quantization arises naturally from the allowed solutions of quantum mechanics.
• Electron states can be described using quantum numbers.
• Each orbital can contain a maximum of two electrons with opposite spin states.
• The uncertainty principle prevents a fully classical description involving precise electron trajectories.
• Quantum mechanics explains multi-electron atoms much more successfully than the Bohr model.
• It helps explain atomic spectra, electron configurations, periodic trends, chemical bonding, and magnetic behaviour.
• The Bohr model remains useful as a simplified model, but the quantum mechanical model agrees with a much wider range of experimental evidence.
• The progression from Rutherford to Bohr to quantum mechanics demonstrates how scientific models become more sophisticated as new evidence is discovered.