Why an Electron Behaves Like a Wave and Refuses to Sit Still

Rutherford's nuclear model could not explain why atoms are stable or why hydrogen gives off light only at certain colours. Answering those questions meant treating energy as coming in fixed packets and electrons as behaving partly like waves.

This lesson covers Bohr's model, the dual nature of matter and radiation, and Heisenberg's uncertainty principle.

**Bohr's model states that the electron in a hydrogen atom moves only in certain circular orbits of fixed energy without radiating, and gains or loses energy only by jumping between orbits, with $E_n=-2.18\times 10^{-18}\frac{Z^2}{n^2}$ J and $r_n=52.9\frac{n^2}{Z}$ pm.

Postulates:

- The electron moves in fixed circular orbits, called stationary states**, without losing energy
- Each orbit has a definite energy, which increases with distance from the nucleus
- A jump between orbits absorbs or emits energy $\Delta E=h\nu$
- Angular momentum is quantised: $mvr=\frac{nh}{2\pi }$

Worked example 1 — energy. For hydrogen (Z = 1):

$$E_1=-2.18\times 10^{-18}\text{ J}{E}_2=\frac{-2.18\times 10^{-18}}{4}=-5.45\times 10^{-19}\text{ J}$$

Worked example 2 — a spectral line. For a jump from n = 3 to n = 2:

$$\Delta E=2.18\times 10^{-18}\left(\frac{1}{4}-\frac{1}{9}\right)=3.03\times 10^{-19}\text{ J}$$

$$\lambda =\frac{hc}{\Delta E}=\frac{6.626\times 10^{-34}\times 3.0\times 10^8}{3.03\times 10^{-19}}=6.56\times 10^{-7}\text{ m}=656\text{ nm}$$

This is the red line of the Balmer series.

Worked example 3 — radius. The second orbit of hydrogen has $r_2=52.9\times 2^2=211.6$ pm, and the first orbit of $\mathrm{H}{\mathrm{e}}^{+}$ (Z = 2) has $r_1=\frac{52.9}{2}=26.45$ pm.

An everyday example. The orange glow of sodium vapour street lamps comes from electrons jumping between fixed energy levels and emitting light of a particular wavelength.

The substance. Bohr's model works only for one-electron species such as H, $\mathrm{H}{\mathrm{e}}^{+}$ and $\mathrm{L}{\mathrm{i}}^{2+}$ — it fails for atoms with more electrons.

**Radiation behaves both as waves and as particles called photons, and moving matter also has wave properties, with a de Broglie wavelength $\lambda =\frac{h}{mv}=\frac{h}{p}$.

Particle nature of radiation:

- Energy comes in packets called photons**, with $E=h\nu =\frac{hc}{\lambda }$
- In the photoelectric effect, electrons leave a metal only when light is above a threshold frequency, however bright lower-frequency light may be
- Interference and diffraction show the wave nature of light

Wave nature of matter:

$$\lambda =\frac{h}{mv}$$

Worked example 1 — electron. An electron ($m=9.1\times 10^{-31}$ kg) moves at $1.0\times 10^6$ m s${}^{-1}$:

$$\lambda =\frac{6.626\times 10^{-34}}{9.1\times 10^{-31}\times 1.0\times 10^6}=7.3\times 10^{-10}\text{ m}=730\text{ pm}$$

Worked example 2 — cricket ball. A 0.15 kg cricket ball bowled at 40 m s${}^{-1}$:

$$\lambda =\frac{6.626\times 10^{-34}}{0.15\times 40}=1.1\times 10^{-34}\text{ m}$$

Worked example 3 — photon energy. Light of wavelength 500 nm carries

$$E=\frac{hc}{\lambda }=\frac{6.626\times 10^{-34}\times 3.0\times 10^8}{5.00\times 10^{-7}}=3.98\times 10^{-19}\text{ J per photon}$$

An everyday example. Electron microscopes in Indian research laboratories use the very short wavelengths of fast electrons to reveal details far smaller than ordinary light can show.

The substance. Every moving object has a wavelength, but only tiny particles show it — a cricket ball's wavelength is far too small to ever detect.

**Heisenberg's uncertainty principle states that the position and momentum of a tiny particle such as an electron cannot both be known exactly at the same time, $\Delta x\cdot \Delta p\ge \frac{h}{4\pi }$, so fixed orbits must give way to orbitals — regions where an electron is likely to be found.

The principle:**

$$\Delta x\cdot \Delta p\ge \frac{h}{4\pi }\text{or}\Delta x\cdot \Delta v\ge \frac{h}{4\pi m}$$

- Locating an electron needs light, which disturbs its momentum
- The smaller the uncertainty in position, the larger the uncertainty in velocity

Worked example. If an electron's position is known to within $1.0\times 10^{-10}$ m, the minimum uncertainty in its velocity is

$$\Delta v=\frac{6.626\times 10^{-34}}{4\times 3.14\times 9.1\times 10^{-31}\times 1.0\times 10^{-10}}=5.8\times 10^5{\text{ m s}}^{-1}$$

That uncertainty is enormous, so an exact path for the electron cannot be fixed.

Orbit versus orbital:

- Orbit — a definite circular path at a fixed distance from the nucleus, as in Bohr's model; it conflicts with the uncertainty principle
- Orbital — a three-dimensional region around the nucleus where the probability of finding an electron is high, described by a wave function
- Orbitals have shapes, such as the spherical s and dumb-bell-shaped p orbitals, while orbits are flat circles

An everyday example. Feeling for a coin on a dark floor by sweeping your hand nudges the coin the moment you touch it — in the same way, the light used to locate an electron disturbs its motion.

The substance. The uncertainty principle matters only for very small particles — for a cricket ball, the uncertainties are far too tiny to notice.