Extended reading: the electron in a box

Where does the quantisation come from?

In the main course, we met the energy levels of hydrogen. Bohr put them in by hand. The quantum model gets them without any extra rule. The reason is simple: an electron is a wave, and a wave held in a small space can have only certain shapes.

A guitar string is a good picture. It is fixed at both ends, so it can vibrate as one half-wave, or as two, or as three. It cannot vibrate as one and a half. The electron in an atom is held by the pull of the nucleus, and it behaves in the same way.

The simplest case: a box

Take an electron that can move along a line of length $L$, and cannot leave it. The walls are so strong that the wave must be zero at both walls.

Only whole half-waves fit between the walls:

From de Broglie, $p = \dfrac{h}{\lambda} = \dfrac{nh}{2L}$. Inside the box there is no force, so all the energy is kinetic: $E = \dfrac{p^2}{2m}$. Putting the two together,

What the formula tells us

• The energy is quantised. Only $E_1$, $4E_1$, $9E_1$, and so on, are allowed. The gaps grow with $n$ here. In an atom the gaps shrink, because the pull of the nucleus changes the shape of the problem. What is the same is that only certain energies exist.
• The lowest energy is not zero. For $n = 1$, $E_1 = \dfrac{h^2}{8mL^2}$ is above zero. The electron can never be at rest. This is the uncertainty principle in action: if the electron is held within a distance $L$, its momentum cannot be exactly zero. The energy that it must have is called the zero-point energy.
• A bigger box means lower energy. $E \propto \dfrac{1}{L^2}$. If we double the length, every level falls to one quarter, and the levels come closer together.

Numbers for an electron

An atom is about $0.5\ \text{nm}$ across, so take $L = 0.5\ \text{nm} = 5 \times 10^{-10}\ \text{m}$.

Then $E_2 = 4E_1 = 6.02\ \text{eV}$. A jump from $n = 2$ to $n = 1$ gives a photon of $6.02 - 1.50 = 4.51\ \text{eV}$, and its wavelength is $\dfrac{1240}{4.51} = 275\ \text{nm}$. This is in the ultraviolet, and the size of the energies is right for an electron in an atom. The model is rough, but the order of magnitude is correct.

You should find that each step up in $n$ adds one more half-wave, and that halving $L$ makes every energy four times larger.

Colour from a box

Many dyes and pigments are long molecules in which some electrons can move along the whole chain. We can treat such an electron as an electron in a box, with $L$ the length of the chain. A longer chain is a bigger box, so the gap between the levels is smaller, and the molecule absorbs light of a longer wavelength. This is why the pigment of a carrot, with a long chain, absorbs blue light and looks orange, and a short chain absorbs only ultraviolet and looks colourless.

Practise with new numbers

Try it in Python

This program finds the first four energies of an electron in a box of $0.5\ \text{nm}$, and the photon of the jump from $n = 2$ to $n = 1$.

Remember these

• A wave in a box fits whole half-waves: $\lambda = \dfrac{2L}{n}$, and $E_n = \dfrac{n^2 h^2}{8mL^2}$.
• The energy is quantised, it is never zero, and it falls as $\dfrac{1}{L^2}$ when the box grows.
• A longer conjugated chain is a bigger box, so it absorbs light of a longer wavelength.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many half-waves fit in the box when $n = 3$?

2. The box is made half as long. By what factor does $E_1$ change?

3. What is $E_3$ in units of $E_1$?

4. Why can a long chain molecule be coloured while a short one is not?