Part 5 · Chapter 39

More About Matter Waves

We left the last chapter with a single radical idea: a particle is also a wave, with a de Broglie wavelength \(\lambda = h/p\). A free electron, like an endless stretched string, can carry any wavelength and so any energy. But confine that wave — pen it between walls — and something familiar to anyone who has plucked a guitar string happens: only certain standing patterns fit, so only certain energies are allowed. This is the heart of quantum mechanics. From the simplest "electron in a box" we extract quantized energy levels, a wave function whose square tells us where the electron is likely to be, and a surprising zero-point energy that can never be removed. We then loosen the walls (the finite well), build real traps atom by atom (quantum dots and corrals), step up to two and three dimensions where levels can coincide, and finally meet the most important trap of all — the hydrogen atom, where this same wave logic reproduces the Bohr radius, the energy ladder \(E_n = -13.6\,\mathrm{eV}/n^2\), and the spectral series that first cracked the atom open.

Fundamentals of Physics Prof. Mithun Mondal Reading time ≈ 95 min
i What you'll learn
  • Confinement quantizes a wave. Just as a string fixed at both ends supports only standing waves with \(L = n\,\dfrac{\lambda}{2}\), a trapped electron is allowed only discrete states labelled by a quantum number \(n = 1, 2, 3, \dots\)
  • For a one-dimensional infinite well of width \(L\), the allowed energies are \(E_n = \dfrac{h^2}{8mL^2}\,n^2\), and a photon is absorbed or emitted only when its energy matches a gap, \(hf = E_{\text{high}} - E_{\text{low}}\).
  • The state is described by a wave function \(\psi_n(x) = \sqrt{\dfrac{2}{L}}\,\sin\!\dfrac{n\pi x}{L}\); the probability density \(|\psi_n|^2\) integrates to \(1\) (normalization), and the lowest state has a nonzero zero-point energy.
  • A finite well lets the wave function leak into the walls and holds only a limited number of bound states; nanocrystallites, quantum dots, and quantum corrals are real engineered traps.
  • Two- and three-dimensional traps give \(E = \dfrac{h^2}{8m}\!\left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)\), where different quantum numbers can share one energy — degeneracy.
  • The hydrogen atom is a natural trap: the Bohr radius \(a \approx 52.9\,\mathrm{pm}\), quantized radii \(r = a n^2\), energies \(E_n = -\dfrac{13.6\,\mathrm{eV}}{n^2}\), the Rydberg formula, the quantum numbers \(n, \ell, m_\ell\), and a ground-state probability cloud peaking at \(r = a\).
Section 39-1

What Is Physics?

One of physics' oldest ambitions is to understand the atom — how its electrons are arranged, how they move, how an atom emits and absorbs light, and why atoms are stable at all. Until the 1920s none of this was known, and without it there was no way to explain how atoms bond into molecules or assemble into solids. The previous chapter gave us the decisive clue: an electron is a matter wave. The task of this chapter is to take that clue seriously and follow where it leads when an electron is not free to roam but is confined — first in idealized boxes we can solve exactly, then in real engineered traps, and finally in the natural trap that is the hydrogen atom. The single thread running through all of it is a rule of startling simplicity and reach: confine a wave and you quantize its energy.

Section 39-2

String Waves and Matter Waves

Picture a free electron moving along the \(x\) axis with no net force on it. Its matter wave is like a wave on an infinitely long string: it can have any wavelength, hence any reasonable energy. Now picture an electron bound inside an atom by the pull of the nucleus. It is no longer free — and, like a wave on a string of finite length clamped at both ends, it can settle only into certain standing patterns. A clamped string has fixed nodes at its two ends, so it fits only when its length equals a whole number of half-wavelengths.

Standing-wave (confinement) condition
\[ L = n\,\frac{\lambda}{2}, \qquad n = 1, 2, 3, \dots \]
Each integer \(n\) picks out one allowed standing pattern; in quantum language \(n\) is a quantum number. The same arithmetic that limits a guitar string to a discrete set of harmonics will limit a trapped electron to a discrete set of energies.
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The confinement principle
Confinement of a wave leads to quantization — to a set of discrete states with discrete energies. The wave can possess only those energies and no others.

This one sentence is the engine of the whole chapter. It applies to waves on strings, to sound in pipes, and — the new and astonishing case — to the matter wave of a confined electron. A free electron has a continuous range of energies; the instant we pen it in, that continuum collapses into rungs on a ladder.

Section 39-3

Energies of a Trapped Electron

Take the simplest possible trap: a one-dimensional infinite potential well. Imagine an electron free to move along a segment of length \(L\) where its potential energy is zero, but walled in by an infinitely high potential on either side so it can never escape. Its matter wave must vanish at both walls, exactly like the clamped string. Combine the standing-wave condition \(\lambda = 2L/n\) with de Broglie's \(p = h/\lambda\) and the non-relativistic kinetic energy \(E = p^2/2m\) (the potential energy inside is zero, so all the energy is kinetic), and the allowed energies fall out at once.

Energy levels of a one-dimensional infinite well
\[ E_n = \left(\frac{h^2}{8mL^2}\right) n^2, \qquad n = 1, 2, 3, \dots \]
The electron may have only these energies — nothing in between. Because \(E_n \propto n^2\), the rungs spread apart as we climb: the gaps widen with \(n\). And because \(E_n \propto 1/L^2\), squeezing the box (smaller \(L\)) drives every level up — tighter confinement costs more energy, a fact that engineers exploit to tune the color of quantum dots.

A diagram of these allowed values against energy is an energy-level diagram. The lowest level, \(n = 1\), is the ground state; the higher levels are excited states. An electron jumps between levels only by absorbing or emitting energy equal to the gap. When that energy arrives as light, the photon condition is just energy conservation.

Photon absorbed or emitted on a quantum jump
\[ hf = E_{\text{high}} - E_{\text{low}} = \frac{h^2}{8mL^2}\left(n_{\text{high}}^2 - n_{\text{low}}^2\right) \]
A trapped electron can absorb only photons whose energy matches a gap exactly, and it emits the same discrete set when it falls back down. This is why confined electrons interact with light only at sharp, characteristic wavelengths — the origin of line spectra.
Section 39-4

Wave Functions of a Trapped Electron

The energies tell us what states exist; the wave function \(\psi_n(x)\) tells us what each state looks like. Solving Schrödinger's equation for the infinite well gives a sine that fits a whole number of half-waves between the walls and is zero outside them.

Wave functions of the infinite well
\[ \psi_n(x) = A\,\sin\!\left(\frac{n\pi x}{L}\right), \qquad 0 \le x \le L \]
Outside the range the wave function is zero — the electron is never found in the walls. Each \(n\) has one more half-wave (and one more interior node) than the last, exactly mirroring the harmonics of a clamped string.

What we actually observe is not \(\psi\) but its square. The probability density \(|\psi_n(x)|^2\) gives the probability per unit length of detecting the electron near \(x\). To find the chance of catching it in a stretch from \(x_1\) to \(x_2\), integrate the density over that stretch.

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Normalization fixes the amplitude
The electron must be somewhere in the well, so the total probability is exactly one: \(\displaystyle\int_0^{L} |\psi_n|^2\,dx = 1\). Carrying out the integral forces \(A = \sqrt{2/L}\) for every \(n\).

So the fully normalized probability density is \(|\psi_n(x)|^2 = \dfrac{2}{L}\sin^2\!\left(\dfrac{n\pi x}{L}\right)\). For the ground state it bulges in the middle of the well; for higher \(n\) it develops \(n\) humps separated by points of zero probability. As \(n\) grows very large the humps blur together into the flat, classical "equally likely anywhere" distribution — the correspondence principle in action.

One feature has no classical counterpart at all. The lowest allowed state is \(n = 1\), not \(n = 0\): setting \(n = 0\) would make \(\psi = 0\) everywhere, meaning no electron at all. So a confined electron can never be brought fully to rest.

Zero-point energy (ground state, n = 1)
\[ E_1 = \frac{h^2}{8mL^2} \ne 0 \]
The minimum energy of a trapped particle is strictly positive. A confined electron is always in motion; "frozen and motionless" is forbidden. This zero-point energy is a direct consequence of confinement and is consistent with Heisenberg's principle — pinning a particle into a region \(L\) forces a spread in momentum, hence a minimum energy.
Section 39-5

An Electron in a Finite Well

Real traps do not have infinitely high walls. In a finite well the potential energy outside is some finite \(U_0\) rather than infinity. The energies are still quantized, and the wave functions still look sinusoidal inside, but two things change — and both echo the tunneling of the previous chapter.

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The wave leaks, and the levels are fewer
In a finite well the wave function does not snap to zero at the walls; it decays exponentially into the forbidden region, so there is a real probability of finding the electron slightly outside the well even though classically it lacks the energy to be there.

Because the wave spreads a little beyond the walls, the electron behaves as if confined to a slightly wider box, so every energy level sits lower than in an infinite well of the same width. More striking, a finite well holds only a limited number of bound states: levels with energy above \(U_0\) are not trapped at all. Supply enough energy — say, a photon whose energy lifts the electron above \(U_0\) — and the electron escapes the well entirely, the bound-state analog of ionization.

Section 39-6

More Electron Traps

These wells are not just blackboard exercises. Three kinds of real, engineered traps put the physics to work, and in each the controlling idea is the same: smaller trap, wider energy gaps, bluer light.

Nanocrystallites
A semiconductor ground into a powder of nanometer-scale granules makes each granule a tiny three-dimensional trap. Larger granules behave like wider wells with smaller gaps and absorb across more of the visible spectrum (appearing dark red or black); shrink the granules and the gaps widen, so they absorb only the bluer, higher-energy light and transmit the rest — the same powder changes color with grain size alone.
Quantum dots — "artificial atoms"
Using the lithography that builds computer chips, one can fabricate an individual potential-energy well — a quantum dot — that traps a controllable handful of electrons and behaves in many ways like a designer atom. Their tunable, confinement-set energy levels make them valuable in electro-optics, displays, and emerging quantum-computing schemes.
Quantum corrals
With a scanning tunneling microscope, individual atoms can be nudged into a closed ring on a copper surface, penning the surface electrons inside. The standing matter waves of the trapped electrons appear as concentric ripples whose spacing matches quantum predictions precisely — confinement made directly visible.
Section 39-7

Two- and Three-Dimensional Electron Traps

Quantum dots and corrals confine an electron in more than one direction, so we need more than one quantum number. For a rectangular infinite trap, the motion along each axis is independent, and the total energy is simply the sum of the one-dimensional energies for each direction.

Energies of a rectangular 2-D / 3-D infinite trap
\[ E_{n_x,n_y,n_z} = \frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right) \]
Each direction gets its own quantum number (\(n_x, n_y, n_z = 1, 2, 3, \dots\)) and its own width. Drop the last term for a two-dimensional trap. The wave function is correspondingly a product of sines, one per direction.
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Degeneracy: different states, same energy
In a symmetric trap (say a square, \(L_x = L_y = L\)), distinct quantum-number sets can give identical energies. The states \((n_x, n_y) = (2,1)\) and \((1,2)\) both yield \(n_x^2 + n_y^2 = 5\), so they share one energy level.

Energy levels that correspond to more than one independent state are called degenerate. Degeneracy is a fingerprint of symmetry: lift the symmetry — stretch the square into a rectangle — and the once-equal levels split apart. This same idea, applied in three dimensions to the hydrogen atom, is what organizes the periodic table.

Section 39-8

The Hydrogen Atom

Now to the natural trap. A hydrogen atom is one electron held near a proton by the Coulomb attraction. In 1913, well before Schrödinger, Niels Bohr built a model by boldly assuming the electron's angular momentum is quantized in steps of \(\hbar = h/2\pi\). The assumption seems arbitrary until you recognize it as a standing-wave condition in disguise — a whole number of de Broglie wavelengths must fit around each orbit. From it, both the orbital radii and the energies come out quantized.

Bohr radius and quantized orbits
\[ r = a\,n^2, \qquad a = \frac{\varepsilon_0 h^2}{\pi m e^2} \approx 52.9\,\mathrm{pm}, \qquad n = 1, 2, 3, \dots \]
The smallest orbit (\(n = 1\)) has radius equal to the Bohr radius \(a \approx 52.9\,\mathrm{pm}\), which sets the natural size of an atom. Higher states swell as \(n^2\).
Energy levels of hydrogen
\[ E_n = -\frac{m e^4}{8\varepsilon_0^2 h^2}\,\frac{1}{n^2} = -\frac{13.60\,\mathrm{eV}}{n^2}, \qquad n = 1, 2, 3, \dots \]
The energies are negative because the electron is bound — energy must be added to free it. The ground state is \(E_1 = -13.6\,\mathrm{eV}\), so the ionization energy of hydrogen is \(13.6\,\mathrm{eV}\). As \(n \to \infty\) the levels crowd toward \(E = 0\), the threshold of freedom.

Transitions between these levels produce light at sharp wavelengths. Writing \(hf = hc/\lambda = E_{\text{high}} - E_{\text{low}}\) and substituting the energy ladder reproduces the empirical formula that 19th-century spectroscopists had wrung from hydrogen's spectrum.

The Rydberg formula for hydrogen lines
\[ \frac{1}{\lambda} = R\left(\frac{1}{n_{\text{low}}^2} - \frac{1}{n_{\text{high}}^2}\right), \qquad R \approx 1.097\times10^{7}\,\mathrm{m^{-1}} \]
With \(n_{\text{low}} = 1\) this gives the ultraviolet Lyman series; \(n_{\text{low}} = 2\) gives the visible Balmer series (whose red \(n=3\to2\) line is the familiar H-α); \(n_{\text{low}} = 3\) gives the infrared Paschen series. Bohr's recovery of this formula, and of the Rydberg constant \(R\) from fundamental constants, was the triumph that launched quantum physics.

Bohr's model gets hydrogen's energies right but pictures the electron as a tiny planet on a sharp orbit — which the uncertainty principle forbids. Schrödinger's equation replaces the orbit with a probability cloud. Solving it in three dimensions yields the same energies but now labels each state with three quantum numbers: the principal \(n\) (energy and size), the orbital \(\ell\) (magnitude of angular momentum, \(0 \le \ell \le n-1\)), and the magnetic \(m_\ell\) (its orientation, \(-\ell \le m_\ell \le +\ell\)).

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The ground state: a cloud, not an orbit
For the hydrogen ground state the wave function is spherically symmetric and falls off as \(\psi(r) \propto e^{-r/a}\), so the probability density \(|\psi|^2\) is largest at the nucleus and fades smoothly outward — the electron has no definite orbit at all.

To ask "how far is the electron from the proton?" we use the radial probability density \(P(r) = \dfrac{4}{a^3}\,r^2 e^{-2r/a}\), the probability of finding it in a thin shell of radius \(r\). Two competing factors — the shrinking density \(e^{-2r/a}\) and the growing shell volume \(\propto r^2\) — peak at \(r = a\). The most probable distance is exactly the Bohr radius, so Bohr's orbit survives, reinterpreted as where the electron is most likely to be rather than where it must be.

Worked Examples

Putting It to Work

1 Energy levels of an electron in a box

Problem. An electron is trapped in a one-dimensional infinite well of width \(L = 100\,\mathrm{pm}\) (about an atomic diameter). Find (a) the ground-state energy and (b) the wavelength of light it must absorb to jump from \(n = 1\) to \(n = 3\).

Solution. Use \(E_n = (h^2/8mL^2)n^2\) for part (a); for (b) the photon energy equals the level gap, and \(\lambda = hc/\Delta E\).

E₁ = h² / (8mL²)
\[ E_1 = \frac{(6.63\times10^{-34})^2}{8(9.11\times10^{-31})(100\times10^{-12})^2} \approx 6.03\times10^{-18}\,\mathrm{J} \approx 37.7\,\mathrm{eV} \]
ΔE = E₃ − E₁ = E₁(9 − 1), then λ = hc/ΔE
\[ \Delta E = 37.7\,(3^2 - 1^2)\,\mathrm{eV} = 301\,\mathrm{eV}, \qquad \lambda = \frac{1240\,\mathrm{eV\cdot nm}}{301\,\mathrm{eV}} \approx 4.12\,\mathrm{nm} \]

The gaps are tens to hundreds of electron-volts because the box is so small — confine an electron to atomic dimensions and its energy ladder is steep. The required light is soft X-ray / extreme-UV, not visible, a direct measure of how tight the trap is.

2 Where is the electron likely to be?

Problem. For the ground state (\(n = 1\)) of an infinite well, what is the probability of detecting the electron in the left third of the well, between \(x = 0\) and \(x = L/3\)?

Solution. Integrate the normalized probability density \(|\psi_1|^2 = (2/L)\sin^2(\pi x/L)\) over the region, using \(\sin^2\theta = \tfrac12(1 - \cos 2\theta)\).

P = ∫₀^{L/3} (2/L) sin²(πx/L) dx
\[ P = \frac{1}{3} - \frac{1}{2\pi}\sin\!\left(\frac{2\pi}{3}\right) = 0.333 - \frac{0.866}{2\pi} \approx 0.20 \]

Only about \(20\%\) — noticeably less than the \(33\%\) a classical particle bouncing uniformly would give, because the ground-state cloud is concentrated in the middle of the well. By symmetry the right third is also \(\approx 0.20\), so the central third holds the remaining \(\approx 0.61\) of the probability.

3 Degeneracy in a square corral

Problem. An electron is confined to a two-dimensional square trap of side \(L\). List the three lowest energies in units of \(h^2/8mL^2\) and state which are degenerate.

Solution. With \(E = (h^2/8mL^2)(n_x^2 + n_y^2)\), tabulate the smallest values of \(n_x^2 + n_y^2\).

Lowest 2-D square-trap levels
State(s) \((n_x,n_y)\)\(n_x^2+n_y^2\)Degeneracy
\((1,1)\)21 (non-degenerate)
\((2,1),\ (1,2)\)52 (degenerate)
\((2,2)\)81 (non-degenerate)

The first excited level is doubly degenerate purely because the trap is square. Stretch it into a rectangle (\(L_x \ne L_y\)) and the \((2,1)\) and \((1,2)\) energies separate — symmetry broken, degeneracy lifted.

4 The red Balmer (H-α) line

Problem. A hydrogen electron falls from \(n = 3\) to \(n = 2\). What is the wavelength of the emitted light?

Solution. Evaluate the two energy levels with \(E_n = -13.6/n^2\,\mathrm{eV}\), take the difference, and convert to wavelength with \(\lambda = hc/\Delta E\).

ΔE = E₃ − E₂
\[ E_2 = -\frac{13.6}{4} = -3.40\,\mathrm{eV}, \quad E_3 = -\frac{13.6}{9} = -1.51\,\mathrm{eV}, \quad \Delta E = 1.89\,\mathrm{eV} \]
λ = hc/ΔE
\[ \lambda = \frac{1240\,\mathrm{eV\cdot nm}}{1.89\,\mathrm{eV}} \approx 656\,\mathrm{nm} \]

That \(656\,\mathrm{nm}\) is the deep-red H-α line — the brightest visible feature of hydrogen and the glow that tints emission nebulae across the night sky. The same arithmetic with \(n_{\text{low}} = 1\) would land in the ultraviolet Lyman series.

5 Most probable radius of the ground state

Problem. Using the hydrogen ground-state radial probability density \(P(r) = \dfrac{4}{a^3}r^2 e^{-2r/a}\), show that the electron is most likely found at \(r = a\).

Solution. The most probable radius maximizes \(P(r)\), so set \(dP/dr = 0\).

dP/dr = 0
\[ \frac{dP}{dr} \propto \frac{d}{dr}\!\left(r^2 e^{-2r/a}\right) = 2r\,e^{-2r/a}\left(1 - \frac{r}{a}\right) = 0 \;\Rightarrow\; r = a \]

The maximum sits exactly at the Bohr radius, \(a \approx 52.9\,\mathrm{pm}\). The cloud picture and Bohr's orbit agree on where the electron most likely is — they disagree only on whether that location is a sharp track (Bohr) or the peak of a smeared-out probability (Schrödinger). Quantum mechanics keeps the number and discards the orbit.

Review

Chapter Summary

✓ Confinement principle

Penning a wave between walls quantizes it: only standing patterns \(L = n\lambda/2\) fit, giving discrete states labelled by \(n\).

✓ Energy levels

Infinite 1-D well: \(E_n = \dfrac{h^2}{8mL^2}n^2\). Gaps widen with \(n\) and with smaller \(L\).

✓ Quantum jumps

Light is absorbed/emitted only when \(hf = E_{\text{high}} - E_{\text{low}}\) — the origin of line spectra.

✓ Wave function

\(\psi_n = \sqrt{2/L}\,\sin(n\pi x/L)\); \(|\psi_n|^2\) is the probability density, normalized to \(1\).

✓ Zero-point energy

\(n=0\) is forbidden, so \(E_1 = h^2/8mL^2 \ne 0\): a trapped electron is never at rest.

✓ Finite wells & traps

Finite walls let the wave leak out and hold few bound states; nanocrystallites, quantum dots, and corrals are real traps.

✓ 2-D / 3-D & degeneracy

\(E = \dfrac{h^2}{8m}\!\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)\); symmetry makes different states share an energy.

✓ Hydrogen atom

\(a \approx 52.9\,\mathrm{pm}\), \(E_n = -13.6\,\mathrm{eV}/n^2\), Rydberg series, quantum numbers \(n,\ell,m_\ell\); ground-state cloud peaks at \(r = a\).

Practice

Problems

Take \(h = 6.63\times10^{-34}\,\mathrm{J\cdot s} = 4.14\times10^{-15}\,\mathrm{eV\cdot s}\), electron mass \(m = 9.11\times10^{-31}\,\mathrm{kg}\), \(hc = 1240\,\mathrm{eV\cdot nm}\), the Bohr radius \(a = 52.9\,\mathrm{pm}\), and the hydrogen ground-state energy \(-13.6\,\mathrm{eV}\). For a one-dimensional infinite well use \(E_n = (h^2/8mL^2)n^2\) and \(|\psi_n|^2 = (2/L)\sin^2(n\pi x/L)\); for hydrogen use \(E_n = -13.6/n^2\,\mathrm{eV}\).

  1. An electron is confined to an infinite well of width \(L = 250\,\mathrm{pm}\). Find the energies of the \(n = 1, 2,\) and \(3\) states.
  2. For the well of Problem 1, what is the energy of the photon emitted when the electron drops from \(n = 3\) to \(n = 1\)? What is its wavelength?
  3. An electron in an infinite well has a ground-state energy of \(2.6\,\mathrm{eV}\). How wide is the well?
  4. List, in units of \(h^2/8mL^2\), the energies of the four lowest levels of a 1-D infinite well, and give the energy of every photon that can be emitted by an electron starting in the \(n = 4\) state.
  5. Show by direct integration that \(\displaystyle\int_0^{L} (2/L)\sin^2(n\pi x/L)\,dx = 1\) for any positive integer \(n\), confirming the normalization constant \(A = \sqrt{2/L}\).
  6. For the \(n = 2\) state of an infinite well, where along \(x\) is the electron least likely to be found, and where is it most likely to be found?
  7. For the ground state of an infinite well, find the probability of detecting the electron in the central third of the well, between \(x = L/3\) and \(x = 2L/3\).
  8. Explain why an electron trapped in a finite well of depth \(U_0\) always has energy levels lower than those of an infinite well of the same width.
  9. An electron is trapped in a two-dimensional square corral of side \(L\). Find, in units of \(h^2/8mL^2\), the energies and degeneracies of the four lowest energy levels.
  10. A three-dimensional cubical box has side \(L\). What is the degeneracy of the level with \(n_x^2 + n_y^2 + n_z^2 = 6\)? List the contributing states.
  11. Using \(E_n = -13.6/n^2\,\mathrm{eV}\), compute the energies of the \(n = 1\) through \(n = 4\) levels of hydrogen, and state the atom's ionization energy.
  12. Find the wavelength of the series limit (\(n_{\text{high}} \to \infty\)) of (a) the Lyman series and (b) the Balmer series of hydrogen.
  13. What is the longest-wavelength line of the Lyman series of hydrogen, and in what part of the spectrum does it lie?
  14. A hydrogen atom absorbs a photon and the electron jumps from \(n = 1\) to \(n = 4\). (a) What photon energy is required? (b) When the electron returns to the ground state, list all the photon energies it could emit.
  15. Using the radial probability density \(P(r) = (4/a^3)r^2 e^{-2r/a}\), estimate the probability of finding the ground-state hydrogen electron within one Bohr radius of the nucleus (you may set up the integral and evaluate it numerically).
  16. Compare the zero-point energy of an electron confined to \(L = 0.10\,\mathrm{nm}\) (atomic scale) with that of a \(1\,\mathrm{g}\) bead confined to \(L = 1\,\mathrm{cm}\). Comment on why the bead's quantization is unobservable.
Tip: three habits carry this whole chapter. First, treat every "trapped particle" problem as a standing-wave bookkeeping exercise: identify the trap's width \(L\) (and its dimensionality), write \(E_n = (h^2/8mL^2)n^2\) for 1-D or the sum-of-squares form for 2-D/3-D, and remember that gaps grow with \(n^2\) and shrink with \(L^2\) — tighter boxes mean bigger, bluer jumps. Second, when light is involved, the only equation you ever need for a transition is energy conservation, \(hf = hc/\lambda = E_{\text{high}} - E_{\text{low}}\); absorption climbs the ladder, emission descends it, and nothing happens unless a photon's energy matches a gap exactly. Third, keep \(\psi\) and \(|\psi|^2\) straight: \(\psi\) is the abstract wave, but the physical, measurable thing is the probability density \(|\psi|^2\), which must integrate to \(1\) (normalization) and which — for hydrogen — you weight by the shell volume \(\propto r^2\) to get the radial probability that peaks, beautifully, right at the Bohr radius.