7D.3 Particle in 2D and 3D Boxes — Detailed Lecture Notes (English)

§1 Learning Objectives and Context

After 7D.1 (free particle) and 7D.2 (1D box) you already know how quantisation arises from boundary conditions in one dimension. But the real world is three-dimensional: electrons move on atomic orbitals, quantum dots confine carriers in all three directions, and semiconductor heterostructures confine electrons to thin layers. In this section we promote the 1D model to 2D and 3D geometries that are much closer to experiment.

By the end of this section you should be able to:

State the multi-dimensional infinite well potential $V=0$ inside, $V=\infty$ outside.
Explain why and when separation of variables $\psi(x,y)=X(x)Y(y)$ is valid.
Write down the eigenfunctions $\psi_{n_x,n_y}$ (and the 3D analogue) and the eigenenergies.
Identify and rationalise degeneracies, relating them to spatial symmetry.
Recognise physical systems that are modelled as 2D/3D boxes: quantum dots, 2D electron gases, free-electron gas in metals, $\pi$-electrons in planar aromatics.

Why it matters: Separation of variables and the concept of degeneracy appear everywhere in quantum mechanics — from atomic structure and molecular spectroscopy to solid-state physics. This lecture introduces both simultaneously.

§2 From 1D to Higher Dimensions: Why Extend?

2.1 Limitations of the 1D model

The 1D box teaches us that confinement forces the wavefunction to vanish at the walls, restricting $k$ and therefore $E$ to a discrete ladder. But 1D cannot describe:

$\pi$ electrons delocalised over a planar conjugated system (e.g. benzene ring);
semiconductor quantum wells (2D confinement), quantum wires (1D residual freedom), and quantum dots (full 3D confinement);
free electrons in a bulk metal (3D Fermi gas);
electrons bound to nuclei (which demand 3D, though in spherical coordinates).

2.2 The natural generalisation

Upgrade the position variable $x$ to a vector $\mathbf r=(x,y)$ or $(x,y,z)$, replace $d^2/dx^2$ with the Laplacian $\nabla^2$, and let the wavefunction depend on all coordinates.

1D: $\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi$
2D: $\displaystyle -\frac{\hbar^2}{2m}\!\left(\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial y^2}\right)+V(x,y)\psi=E\psi$
3D: $\displaystyle -\frac{\hbar^2}{2m}\nabla^2\psi+V(x,y,z)\psi=E\psi$, with $\nabla^2=\partial_x^2+\partial_y^2+\partial_z^2$.

§3 The Schrödinger Equation in a 2D Box

3.1 Model

Consider a rectangular box $x\in[0,L_x]$, $y\in[0,L_y]$ with

$$V(x,y)=\begin{cases}0,&0

Inside the box the particle is free; the walls are infinitely high, so the particle cannot escape.

3.2 Equation inside

$$-\frac{\hbar^2}{2m}\!\left(\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial y^2}\right)=E\,\psi(x,y).$$

3.3 Boundary conditions

Because $V$ is infinite outside, $\psi$ must vanish everywhere on the boundary:

$\psi(0,y)=\psi(L_x,y)=0$ for all $y\in[0,L_y]$
$\psi(x,0)=\psi(x,L_y)=0$ for all $x\in[0,L_x]$

Note that the entire edge must vanish — not merely the four corners. This is different from (and stronger than) the 1D condition.

§4 Separation of Variables

4.1 Why can we guess $\psi(x,y)=X(x)Y(y)$?

Three conditions must be met:

The equation must be linear (it is).
The potential must be separable: $V(x,y)=V_x(x)+V_y(y)$. Here $V=0+0$ inside the box.
The boundary must be rectangular: it factorises into independent $x$- and $y$-boundaries.

When all three hold, a theorem of partial differential equations guarantees that a complete set of solutions can be built as products. It is not a lucky guess.

4.2 Substitute and separate

Insert $\psi=X(x)Y(y)$:

$-\dfrac{\hbar^2}{2m}\bigl[Y\,X''+X\,Y''\bigr]=E\,X Y$
Divide both sides by $X Y$ (assumed non-zero):
$-\dfrac{\hbar^2}{2m}\dfrac{X''}{X}-\dfrac{\hbar^2}{2m}\dfrac{Y''}{Y}=E.$
The first term depends only on $x$ and the second only on $y$. For their sum to equal the constant $E$ for every $x$ and every $y$, each term must itself be constant:
$-\dfrac{\hbar^2}{2m}\dfrac{X''}{X}=E_x,\quad -\dfrac{\hbar^2}{2m}\dfrac{Y''}{Y}=E_y,\quad E=E_x+E_y.$

Result: the 2D problem reduces to two independent 1D problems: $$-\frac{\hbar^2}{2m}X''=E_xX,\qquad -\frac{\hbar^2}{2m}Y''=E_yY.$$ Each is the familiar 1D infinite-well equation.

4.3 Physical meaning

Mathematical separability corresponds physically to motion in the two directions being independent: the Hamiltonian $\hat H=\hat T_x+\hat T_y$ splits into commuting pieces, each carrying its own good quantum number.

§5 The 2D Wavefunctions and Energies

5.1 Each 1D factor

From 7D.2 we already know the 1D infinite-well solutions with boundary $X(0)=X(L_x)=0$:

$$X_{n_x}(x)=\sqrt{\tfrac{2}{L_x}}\sin\!\left(\tfrac{n_x\pi x}{L_x}\right),\quad E_x=\tfrac{n_x^2 h^2}{8mL_x^2},\ n_x=1,2,3,\dots$$

and similarly for $Y$.

5.2 Product eigenstates

2D rectangular box: $$\psi_{n_x,n_y}(x,y)=\frac{2}{\sqrt{L_xL_y}}\sin\!\left(\frac{n_x\pi x}{L_x}\right)\sin\!\left(\frac{n_y\pi y}{L_y}\right)$$ Energy eigenvalues: $$E_{n_x,n_y}=\frac{h^2}{8m}\!\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}\right).$$ Two independent quantum numbers $n_x,n_y\in\mathbb Z^+$.

5.3 Normalisation check

$\displaystyle\int_0^{L_x}\!\!\int_0^{L_y}|\psi|^2\,dx\,dy=\frac{4}{L_xL_y}\cdot\frac{L_x}{2}\cdot\frac{L_y}{2}=1.\ \checkmark$

5.4 The square box

If $L_x=L_y=L$:

$$\psi_{n_x,n_y}=\tfrac{2}{L}\sin\!\left(\tfrac{n_x\pi x}{L}\right)\sin\!\left(\tfrac{n_y\pi y}{L}\right),\quad E_{n_x,n_y}=\tfrac{h^2}{8mL^2}(n_x^2+n_y^2).$$

§6 Nodal Lines, Symmetry and Wavefunction Shape

6.1 Nodes become lines

In 1D a node is a point where $\psi=0$; in 2D it is a line. For the eigenstate $(n_x,n_y)$:

$n_x-1$ vertical nodal lines at $x=kL_x/n_x$ ($k=1,\dots,n_x-1$);
$n_y-1$ horizontal nodal lines at $y=kL_y/n_y$ ($k=1,\dots,n_y-1$).

Sketch: $(n_x,n_y)=(3,2)$ — the interior is divided by 2 vertical + 1 horizontal nodal lines into 6 rectangular cells of alternating sign.

6.2 Checkerboard sign pattern

Crossing a nodal line flips the sign of $\psi$. The whole box is therefore partitioned into a checkerboard of $\pm$ regions.

6.3 Antinodes

Antinodes — local maxima of $|\psi|$ — form an $n_x\times n_y$ grid of bright spots evenly spaced across the box.

§7 Probability Density Maps

The probability density is

$$|\psi_{n_x,n_y}(x,y)|^2=\tfrac{4}{L_xL_y}\sin^2\!\tfrac{n_x\pi x}{L_x}\sin^2\!\tfrac{n_y\pi y}{L_y}.$$

Zero on nodal lines (the particle is never found there);
Maximum $4/(L_xL_y)$ at antinodes;
Appears as an $n_x\times n_y$ grid of bright lobes.

7.1 Classical correspondence

At very large $n_x,n_y$ the rapid oscillations average to the uniform classical density $1/(L_xL_y)$ — Bohr's correspondence principle in 2D.

Use the companion interactive page (7D3_visualizations.html §3) to watch the bright-spot pattern morph as $(n_x,n_y)$ increases.

§8 Degeneracy

8.1 Definition

Degeneracy: a single energy eigenvalue is called g-fold degenerate if $g$ linearly independent eigenstates share that energy.

8.2 The square box ($L_x=L_y$)

Energy depends only on $n_x^2+n_y^2$; distinct integer pairs may sum to the same value.

$n_x^2+n_y^2$	$(n_x,n_y)$	$g$
2	(1,1)	1 (ground state)
5	(1,2), (2,1)	2
8	(2,2)	1
10	(1,3), (3,1)	2
13	(2,3), (3,2)	2
17	(1,4), (4,1)	2
18	(3,3)	1
20	(2,4), (4,2)	2
50	(1,7), (7,1), (5,5)	3 (accidental)

8.3 Three sources of degeneracy

Symmetry (systematic): if $L_x=L_y$, swapping $n_x\leftrightarrow n_y$ leaves the energy invariant, making every state with $n_x\ne n_y$ at least two-fold degenerate.
Accidental: the $(1,7),(7,1),(5,5)$ triple above is not forced by symmetry — it is an integer-theoretic coincidence.
Spin: (beyond this section) each spatial state accommodates one spin-up + one spin-down electron.

8.4 1D boxes never degenerate

In 1D the energies $n^2$ form a strictly increasing sequence, so no two states share an energy. Degeneracy requires two ingredients: higher dimension plus spatial symmetry.

§9 Extension to 3D Boxes

9.1 Setup

For a rectangular parallelepiped with sides $L_x,L_y,L_z$:

$$\psi_{n_x,n_y,n_z}=\sqrt{\tfrac{8}{L_xL_yL_z}}\sin\!\tfrac{n_x\pi x}{L_x}\sin\!\tfrac{n_y\pi y}{L_y}\sin\!\tfrac{n_z\pi z}{L_z},$$ $$E_{n_x,n_y,n_z}=\tfrac{h^2}{8m}\!\left(\tfrac{n_x^2}{L_x^2}+\tfrac{n_y^2}{L_y^2}+\tfrac{n_z^2}{L_z^2}\right).$$

The normalisation prefactor $\sqrt{8/V}$ is the product of three $\sqrt{2/L}$ factors.

9.2 The cubic box

With $L_x=L_y=L_z=L$, $E\propto n_x^2+n_y^2+n_z^2$. Degeneracies are richer:

$n_x^2+n_y^2+n_z^2$	states	$g$
3	(1,1,1)	1 (ground)
6	(1,1,2),(1,2,1),(2,1,1)	3
9	(1,2,2),(2,1,2),(2,2,1)	3
11	(1,1,3),(1,3,1),(3,1,1)	3
12	(2,2,2)	1
14	(1,2,3) and 5 more permutations	6
27	(1,1,5)×3, (3,3,3)×1	4

The multiplicity of an energy equals the number of ways its integer-squared-sum can be partitioned into three squares — a beautiful meeting point of number theory and physics.

§10 Symmetry Breaking and Lifting of Degeneracy

If we stretch the box so $L_y\ne L_x$, the swap $(n_x,n_y)\leftrightarrow(n_y,n_x)$ is no longer a symmetry and the levels split:

$E_{1,2}-E_{2,1}=\tfrac{h^2}{8m}\!\left(\tfrac{1}{L_x^2}+\tfrac{4}{L_y^2}\right)-\tfrac{h^2}{8m}\!\left(\tfrac{4}{L_x^2}+\tfrac{1}{L_y^2}\right)=\tfrac{3h^2}{8m}\!\left(\tfrac{1}{L_y^2}-\tfrac{1}{L_x^2}\right).$

The splitting vanishes at $L_x=L_y$ and grows with the asymmetry. The same logic underpins:

external fields breaking atomic spherical symmetry → Stark (electric) and Zeeman (magnetic) splittings;
low-symmetry point groups splitting degenerate vibrational modes;
crystal fields splitting the $d$-orbital manifold of transition-metal complexes, determining colour and magnetism.

Heuristic: Degeneracy is the shadow of symmetry. Observing a degeneracy suggests hidden symmetry; a lift tells you that something has broken that symmetry.

§11 Real-World Applications

11.1 Quantum dots

Semiconductor nanocrystals (a few nm across) are well modelled as 3D boxes. Changing the size $L$ tunes the optical gap $\Delta E\propto 1/L^2$ and therefore the emission colour.

CdSe quantum dots: ~2 nm emit blue, ~6 nm emit red.
Applications: QLED displays, biological fluorescent labels, qubits.

11.2 Two-dimensional electron gas (2DEG)

In GaAs/AlGaAs heterostructures, electrons are confined to a thin layer (quantised in $z$, free in $x,y$). The integer and fractional quantum Hall effects live in this world.

11.3 Free-electron gas in metals

Treat a bulk metal as a giant 3D box; electrons fill $(n_x,n_y,n_z)$ states up to the Fermi level. This is the starting point for the Sommerfeld theory of metals — Fermi energy, electronic heat capacity, Pauli paramagnetism.

11.4 $\pi$-electrons in planar aromatics

A rough 2D-box estimate reproduces the qualitative absorption spectra of planar conjugated molecules such as porphyrins or polycyclic aromatics.

§12 Worked Examples

Example 1 — Ratio of first two energies in a square box

For $L_x=L_y=L$, find $E_{1,1}$ and the first excited energy.

$E_{1,1}=\tfrac{h^2}{8mL^2}(1+1)=2\varepsilon$, where $\varepsilon\equiv h^2/(8mL^2)$.
First excited: $(1,2)$ or $(2,1)$, $E=(1+4)\varepsilon=5\varepsilon$.
Ratio $E_{2,1}/E_{1,1}=5/2=2.5$.

Example 2 — Transition wavelength

An electron in a 2D square box with $L=1.0$ nm. Find the wavelength for the $(1,1)\to(1,2)$ transition.

$\Delta E=(5-2)\varepsilon=3\varepsilon=\frac{3h^2}{8mL^2}$.
$\Delta E=\frac{3(6.626\times10^{-34})^2}{8(9.109\times10^{-31})(10^{-9})^2}\approx 1.81\times10^{-19}$ J $\approx 1.13$ eV.
$\lambda=hc/\Delta E\approx 1.10\ \mu$m (near-IR).

Example 3 — First six cubic-box levels

level	states	$E/\varepsilon$	$g$
1	(1,1,1)	3	1
2	(1,1,2) and perms	6	3
3	(1,2,2) and perms	9	3
4	(1,1,3) and perms	11	3
5	(2,2,2)	12	1
6	(1,2,3) and 5 perms	14	6

Example 4 — Level ordering in a rectangular box

Take $L_y=2L_x$. Find the energies of $(1,1),(2,1),(1,2),(2,2)$ in units of $\varepsilon_x\equiv h^2/(8mL_x^2)$:

$E_{1,1}=\varepsilon_x(1+1/4)=1.25\varepsilon_x$
$E_{2,1}=\varepsilon_x(4+1/4)=4.25\varepsilon_x$
$E_{1,2}=\varepsilon_x(1+1)=2\varepsilon_x$
$E_{2,2}=\varepsilon_x(4+1)=5\varepsilon_x$

Order: $E_{1,1}

Example 5 — CdSe quantum dot size for red emission

Using effective mass $m^*=0.13\,m_e$ and a 3D-box approximation, estimate the edge length $L$ that gives a HOMO-LUMO transition near $\lambda=600$ nm.

Crude estimate: $\Delta E\approx \tfrac{9h^2}{8m^*L^2}$. From $\Delta E=hc/\lambda=3.31\times 10^{-19}$ J we get $L\approx\sqrt{9h^2/(8m^*\Delta E)}\approx 3.5$ nm. (Quantitative work must add the bulk band gap and solve the actual problem, but the scaling is correct.)

§13 Key Takeaways

Separable potentials allow the multi-dimensional Schrödinger equation to split into independent 1D problems.
The full wavefunction is a product of 1D factors; the total energy is a sum of 1D energies.
Each direction gets its own quantum number; all quantum numbers are positive integers.
Nodes become lines in 2D and surfaces in 3D; crossing a node flips the sign of $\psi$.
Degeneracy arises from symmetry; breaking the symmetry lifts it.
The pattern "symmetry → degeneracy, perturbation → splitting" is the mathematical skeleton of Stark, Zeeman, and crystal-field splittings.
Applications: quantum dots (3D), 2DEG (2D), metallic free electrons (large 3D box), and $\pi$ systems (2D).

§14 Advanced Discussion Questions

If the 2D square box is replaced by a disc (radius $R$, infinite walls), is separation of variables still possible? What functions arise?
Why is the cubic-box ground state $(1,1,1)$ never degenerate while the first excited level $(1,1,2)$ is always 3-fold? Explain with the permutation group $S_3$.
Switch on a uniform magnetic field. Does the same separation still work? Which direction behaves differently?
Imagine adding a small repulsive $\delta$-like potential at the centre of a 2D square box. How does this perturbation split the $(1,2)/(2,1)$ degeneracy? (Hint: the perturbation vanishes for states with a node at the origin.)
How does the average degeneracy of the $N$-th level of a 3D box grow with $N$? Show it is consistent with the density of states $g(E)\propto\sqrt E$.

Question 5 is the gateway to the Fermi gas treatment of metals — these lecture formulas feed directly into solid-state physics.