Quantum Mechanics

Advanced physics enrichment path through wavefunctions, uncertainty, the Schrödinger equation, square wells, tunnelling, and the quantum oscillator.

Advanced Physics

Learning goals

Use state functions, boundary conditions, stationary states, and measurement to analyse introductory quantum systems.

Quantum mechanics replaces definite particle trajectories with a state that predicts measurement probabilities. This path develops that idea into solvable one-dimensional models.

Prerequisites: H2 Quantum Physics, Matter Waves, algebra, differentiation, and basic integration.

Scope: this is a MiniPhysics optional extension. Quantum mechanics is not one of the five additional topics named in the current 9814 H3 Physics syllabus.

Study the models

Follow the lesson sequence on this page because each model reuses notation and boundary conditions established earlier. When you reach the infinite well, compare its energy spectrum, wavefunctions and probability density before testing the classical limit.

Introduction to Quantum Mechanics
H3 Quantum Mechanics: what quantum mechanics is for, why classical physics fails at small scales, and how to interpret probability via |ψ|².
The Uncertainty Principle
H3 Quantum Mechanics: what Δ means in ΔxΔp ≥ ħ/2, why the uncertainty is intrinsic to the state, and how to use it in estimates.
Wavefunction and Schrödinger Equation
H3 Quantum Mechanics: what the wavefunction ψ means, why |ψ|² is a probability density, and how the Schrödinger equation constrains allowed energies.
Normalisation of the Wavefunction
H3 Quantum Mechanics: what it means to normalise a wavefunction so total probability is 1, and how to find the normalisation constant.
Time-Independent Schrödinger Equation
H3 Quantum Mechanics: the time-independent Schrödinger equation (TISE), how solutions behave for E>V vs E<V, and the boundary conditions used in 1D problems.
Particle Trapped in a Box of Length L
Set up the infinite square well and use its boundary conditions to obtain standing states, energy levels, and possible momentum magnitudes.
Allowed Energy States of a Trapped Particle
Derive infinite-well energy levels and distinguish fixed energy from the two possible momentum outcomes of a standing state.
Wavefunctions of a Particle Trapped Within a Box
H3 Quantum Mechanics: deriving ψn(x) in the infinite square well, applying boundary conditions, and interpreting |ψn|².
The Particle in a Box Revisited
H3 Quantum Mechanics: a compact recap of the infinite square well (particle in a box) — boundary conditions, ψn(x), En, and key interpretations.
A Particle in a Well of Finite Height
H3 Quantum Mechanics: the finite square well — exponential decay outside the well, boundary matching, and why tunnelling is possible.
Quantum Harmonic Oscillator
H3 Quantum Mechanics: the quantum harmonic oscillator — energy levels En=(n+1/2)ħω, equal spacing, and why the ground state energy cannot be zero.
Correspondence Principle
Use large-quantum-number, large-scale, and coarse-resolution limits to test how quantum predictions approach classical physics.

The four box lessons cover model setup, energy derivation, wavefunction derivation and synthesis. If you already know the derivations, use the revisited lesson to check your understanding.

Revision

Quick reference

Uncertainty: Δ x Δ p ≥ ħ/2.
Time-independent Schrödinger equation (1D): -(ħ²/2m)d²ψ/dx² + Vψ = Eψ.
Probability density: |ψ|² and normalisation ∫ |ψ|² dx = 1.
Particle-in-a-box energies: Eₙ = n²π²ħ²/2mL².
Finite forbidden region: ψ ∝ e^(-κ x) with κ = square root of (2m(V-E))/ħ.
Oscillator energies: Eₙ = (n + 1/2)ħω.

Problem workflow

Draw V(x) and state the zero of potential energy.
Compare E with V in each region; choose oscillatory or exponential solutions.
Reject divergent branches, then apply wall or matching conditions.
Normalise over the stated domain and interpret |ψ|², not ψ.
Check units, symmetry, nodes, and a sensible limiting case.

Common traps

Treating ψ as a probability rather than a generally complex amplitude.
Assigning one momentum to an infinite-well standing state; its outcomes are + pₙ and -pₙ.
Calling every exponential tail “transmission”; tunnelling through a barrier also needs an allowed region beyond it.
Assuming n = 0 means zero energy in every model: the box begins at n = 1, while the oscillator begins at n = 0 with E₀ > 0.

Practice

Normalise a piecewise wavefunction and calculate a probability over a subinterval.
Derive the infinite-well kₙ, ψₙ, and Eₙ results from the two wall conditions.
Compare infinite and finite walls without claiming that E < V forces ψ = 0.

These are extension exercises rather than claims about current 9814 examinable content.

Continue learning

Quantum Theory of Light applies energy and momentum quantisation to radiation.
H3 Physics returns to the current syllabus and paper map.

Check what I know

Check what I know: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 6 minutes

Check what I know

Answer 2 short questions. This starting check helps choose what to work on; it does not prove mastery.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

This check needs JavaScript to record answers. The reviewed questions are listed below for study.

A one-dimensional wave function $\psi(x)$ is normalised so that $\int_{-\infty}^{\infty}|\psi(x)|^2\,\mathrm{d}x=1$. What does this condition state physically?

The probability of finding the particle somewhere on the $x$-axis is exactly 1.
The maximum value of $\psi(x)$ is 1.
The particle's energy has been scaled to $1\,\mathrm{eV}$.
The wave function is real-valued everywhere.

A particle is confined in a one-dimensional infinite square well of width $L$, with allowed energies $E_n=\dfrac{n^2h^2}{8mL^2}$. What is the ratio $E_2/E_1$?

4
2
8
$\sqrt2$

Practise

Practise: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 10 minutes

Practise

Questions are selected when you start. Use the feedback to decide what to practise next; this does not prove mastery.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

This check needs JavaScript to record answers. The reviewed questions are listed below for study.

A wavefunction $\psi$ is normalised. If it is replaced by $2\psi$ without any further rescaling, what is the value of the integral of its probability density over the allowed domain?

$1/2$.
$1$.
$2$.
$4$.

For the same particle and quantum number in a one-dimensional infinite square well, the well width is doubled from $L$ to $2L$. How does the stationary-state energy change?

It doubles.
It halves.
It becomes one quarter as large.
It is unchanged.

For a particle in an infinite square well of width $L$ ($0\le x\le L$), the $n=2$ stationary state is $\psi_2(x)=\sqrt{2/L}\,\sin(2\pi x/L)$. At which position strictly inside the well is the probability density zero?

$x=L/2$
$x=L/4$
$x=L/3$
Nowhere inside the well; nodes occur only at the walls.

A particle is bound in a finite square well of width $L$ and depth $U_0$. Compared with an infinite well of the same width, how do the wave function and the ground-state energy differ?

The wave function penetrates the classically forbidden walls with exponential decay, and the ground-state energy is lower than in the infinite well.
The wave function is zero at the walls and the ground-state energy is higher.
Both the wave function and the energy are identical to the infinite-well case.
A finite well has no bound states because the particle can always tunnel out.

An electron of energy $E=1.5\,\mathrm{eV}$ meets a step where the potential energy rises from $V=0$ to $V_0=4.0\,\mathrm{eV}$. In each region the time-independent Schrödinger equation reads $\psi''=-\dfrac{2m(E-V)}{\hbar^2}\psi$. Use $h=6.63\times10^{-34}\,\mathrm{J\,s}$, $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$, $m_e=9.11\times10^{-31}\,\mathrm{kg}$, $e=1.60\times10^{-19}\,\mathrm{C}$ and $c=3.00\times10^{8}\,\mathrm{m\,s^{-1}}$.

Wavenumber $k=\sqrt{2mE}/\hbar$ of the oscillatory solution where $V=0$, in $\mathrm{nm^{-1}}$
Decay constant $\kappa=\sqrt{2m(V_0-E)}/\hbar$ of the evanescent solution beyond the step, in $\mathrm{nm^{-1}}$
Penetration depth $\delta=1/\kappa$ into the forbidden region, in nm

A student normalises $\psi(x)=A\sin(\pi x/L)$ for $0\le x\le L$ (zero elsewhere), taking $A$ to be real and positive, as follows. Line 1: $\int_0^L|\psi|^2\,\mathrm{d}x=1$ Line 2: $A\int_0^L\sin(\pi x/L)\,\mathrm{d}x=1$ Line 3: $A\cdot\dfrac{2L}{\pi}=1$ Line 4: $A=\dfrac{\pi}{2L}$ Use $\int_0^L\sin^2(\pi x/L)\,\mathrm{d}x=L/2$.

Number of the first line that is wrong
Correct value of $A\sqrt{L}$ (a pure number)

Practise

Practise after feedback: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 10 minutes

Practise

Questions are selected when you start. Use the feedback to decide what to practise next; this does not prove mastery.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

This check needs JavaScript to record answers. The reviewed questions are listed below for study.

An electron is localised to within $\Delta x=1.0\times10^{-10}\,\mathrm{m}$. Using $\Delta x\,\Delta p\geq\hbar/2$ with $\hbar=1.05\times10^{-34}\,\mathrm{J\,s}$, what is the minimum uncertainty in its momentum?

$5.3\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}$
$1.1\times10^{-24}\,\mathrm{kg\,m\,s^{-1}}$
$6.6\times10^{-24}\,\mathrm{kg\,m\,s^{-1}}$
$0$, because the position is known

For the infinite square well, $E_n\propto n^2$. How does the correspondence principle apply as $n$ becomes very large?

The fractional spacing $(E_{n+1}-E_n)/E_n\approx2/n$ tends to zero, so the discrete levels become indistinguishable from a classical continuum.
The absolute spacing $E_{n+1}-E_n$ tends to zero.
The energies become negative, as for bound states in atoms.
The wave function becomes constant across the well.

A particle is in the ground state of an infinite square well, $\psi_1(x)=\sqrt{2/L}\,\sin(\pi x/L)$ for $0\le x\le L$. Use $\int\sin^2(ax)\,\mathrm{d}x=\dfrac{x}{2}-\dfrac{\sin(2ax)}{4a}$.

Probability of finding the particle in the middle third, $L/3\le x\le 2L/3$
Probability of finding it in either outer third, combined

Check my progress

Check my progress: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 10 minutes

Check my progress

Answer 2 questions. If accepted, this result can contribute to your course progress.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

Check again

Check again: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 10 minutes

Check again

Answer 2 questions. If accepted, this result can contribute to your course progress.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

Review

Review: Quantum Mechanics

A text-first Quantum Mechanics assessment with labelled response controls.

About 10 minutes

Review

Answer 2 questions. A scheduled review can contribute to your course progress only when it is due and the result is accepted.

Recent attempts

History is stored only in this browser.

No completed attempts are saved yet.

How this activity affects progress

Course and syllabus information

Course: Advanced Physics
Edition: Advanced Physics

Quantum Mechanics

Study the models

Revision

Practice

Continue learning

Check what I know: Quantum Mechanics

Starting check complete

Practise: Quantum Mechanics

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Practise after feedback: Quantum Mechanics

Practice complete

Check my progress: Quantum Mechanics

Progress check complete

Check again: Quantum Mechanics

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Review: Quantum Mechanics

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