DeBroglie's particle waves and the particle-in-a-box

The allowed wavelengths of the standing waves of a giant "classical" spring are found to be "quantized."

Ingredients: giant spring

Procedure: A complete recipe follows.

1. Attach one end of the spring to a fixed point.

2. Extend the spring to straighten it, developing substantial tension.

3. Add vibrational energy to the spring by moving the opposite end.

4. Observe the allowed wavelengths of the standing waves that are created in the spring.

Understanding: The oscillation of the giant spring reminds us of our many experiences with waves formed on vibrating strings, including jump ropes and the strings of a guitar. Quickly moving one end of the spring up and down sets a wave pulse moving along the length of the spring. The propagating wave reflects off the far end of the spring, returning to our hands, reflecting again, and so on, until it dissipates. We find that we can create waves of practically any length, by varying how quickly we "pulse" the spring.

If we add energy to the spring, but keeping the ends fixed, we create standing waves. For the lowest energy wave, the two ends of the spring are held fixed and the amplitude of the wave has one maximum that oscillates up and down with a fixed frequency. The wavelength of the wave is twice the length of the spring. Adding energy to the spring induces more oscillations in the wave. The points along the wave where the amplitude is zero are called nodes. A wave with one node in the center has a wavelength equal to the length of the spring. A wave with three nodes has a wavelength equal to one-half the length of the spring. The ends of the spring don't count as nodes.

It is not hard to determine all of the possible wavelengths for the standing waves of a spring of length L. The longest wavelength for a standing wave is λ=2L. Anything longer and the wave will not have fixed points at the beginning and end of the spring. We can also have a wavelength of λ=L, where one full oscillation of the wave fits on the spring. Or we can have a wavelength of λ=2L/3, where 3 half-wavelengths fit along the spring. And so on. In general, there are an infinite number of allowed wavelengths

λn = 2L/n         n = 1,2,3...

We can call n/2 the wavenumber because it is the number of wavelengths λn that fit in a box of length L. But what does that have to do with the discrete nature of the emission and absorption line spectra of the hydrogen atom?

A general unhappiness with the Bohr model and its orbits

Many people were unhappy with the basic physical assumptions that led to the "quantized" spectra predicted by Bohr's model of the hydrogen atom. In the Bohr model, the electron is imagined to be a point particle in a circular orbit around the proton nucleus, allowed to assume only certain values of the angular momentum

mvr = nh/2π

That assumption led to good agreement with experiment. But why should the angular momentum be restricted in such a peculiar way?

And there were more problems. Vigorous attempts to extend Bohr's model to multielectron atoms and to molecules had been frustrating and largely unproductive. There was a general feeling that there must be a different and better way to explain the absorption spectra of the hydrogen atom, that could be extended to multielectron atoms and molecules.

Louis de Broglie was the first see the answer. He recalled the in classical physics, "quantization" arises as a natural property of waves. Think of waves that can move along a string that has ends that are free to move. Waves of any wavelength are possible, depending on how the ends of the string are moved. However, if we fix the position of the ends of the string, we find that only certain waves are possible. Why? If the ends are fixed, then the ends must correspond to nodes of the wave where the wave's amplitude is zero. That means that only certain wavelengths are possible - those wavelengths that can fit along the string just right, with fixed points at the end of the string.

That's very interesting! If we assume that the electron behaves like a wave, and it is confined to "oscillate" in a limited region of space, only certain quantized wavelengths would be possible. And since the energy of the wave depends on the wave's wavelength, each allowed wavelength would correspond to a particular "quantized" allowed energy. That is just what we need to make sense of the quantized line-spectra of the hydrogen atom!

An electron can behave as a particle and as a wave

So how can we "connect" the wave-like properties of an electron and the particle-like properties of the electron? For a sinusoidal wave, of the sort that we created using our vibrating spring, the wave is characterized by the frequency of oscillation, ν, and a characteristic wavelength, λ. For simplicity, we only consider waves that move with a speed that is independent of the wavelength of the wave. That wave is called dispersionless. The velocity of a dispersionless wave is simply

v = ν λ

That looks just like the familiar relation for light waves, c = λ ν, where c is the speed of light. Here v is the speed of our matter wave.

The special theory of relativity tells us that the energy of the photon is given in terms of the photon's momentum and the speed of light

E = pc

From Einstein's theory of the photoelectric effect, we know that the energy of the photon is

E = hν

Equating the two expressions of the photon energy leads to

λ = h/p

which relates the photon's wavelength to its momentum. That simple equation connects the wave-like properties of light with the particle-like properties of the photon.

In a great conceptual advance, de Broglie conjectured that the same relation was true for all particles, be it a photon of light or an electron. An electron of mass m moving with velocity v would, by de Broglie's way of thinking, have a corresponding wavelength

λde Broglie = h/p = h/(mv)

Quantum mechanical theory of a "particle in a box"

The line spectra of the hydrogen atom may seem to be the simplest chemical system that we could study that would be of any relevance to chemistry. After all, the Periodic Table starts with hydrogen, so nothing could be simpler. Not so! A still simpler, and no less important, model for understanding chemical systems is that of a single particle, such as an electron, confined in a "box."

How is the particle-in-a-box simpler than the hydrogen atom? For one thing, the hydrogen atom exists in three dimensions. The electron has kinetic energy and potential energy, as it is attracted to the nucleus by the Coulombic inverse-square force. And the electron can be "bound" to the nucleus, in a "quantized" state of the neutral hydrogen atom, or it can be "free" to leave the nucleus, in one of an infinite number of continuum states, when the hydrogen atom is ionized.

Suppose we simplify things by putting the electron in a "box." We can make the box one-dimensional so that the electron moves freely on a line over a distance L, the length of the box. We say that the potential energy of the electron is infinite outside the box and zero inside the box. An electron confined to a box of length L is free to move within the box, but it can't leave.

You might say, sure the particle-in-a-box is simpler, but it has nothing to do with chemistry. But it does! It can be a model for electrons solvated in liquids, observed in the lovely blue color of a solvent "still" of tetrahydrofuran. It can be a model of a conducting electron in a wire. It can also be a model of an electron trapped in a "vacancy" defect in an ionic crystal, a so-called F-center. And it can even be a model for an electron moving in large organic molecules, such as β-carotene or a porphyrin.

Computing the allowed energies of the particle-in-a-box

Let's think of the electron as a particle. It has a certain mass, m, and a velocity, v. The momentum of the particle is the mass times the velocity, p=mv. The energy of the electron is its kinetic energy

E = ½ m v2 = p2/2m

Now let's think of the electron as a de Broglie particle wave. In a box of length L, the allowed wavelengths are given by our result for the standing waves formed by a string of length L

λn = 2L/n         n = 1,2,3...

Only certain allowed wavelengths are possible. Note that the number of nodes is n-1, which increases with growing n to indicate the increases curvature of the wave and higher kinetic energy.

If we combine this result with deBroglie's relation, we find that only certain allowed momenta are possible

pn = h/λn = nh/2L         n = 1,2,3...

That result is reminiscent of Bohr's assumption that only certain values of the angular momentum are possible for the electron in the one-electron atom. But this different! It rests on a firm foundation - the fact that only certain values of the momentum are allowed is a direct consequence of the wave-like nature of the electron.

Let's go one step further. For each allowed value of the momentum, pn, there is a corresponding allowed energy

En = pn2/2m = n2h2/(8mL2)         n = 1,2,3...

That is a very interesting result. It states that if we model our electron confined to a box of length L as a wave, only certain allowed wavelengths are possible. And relating the wavelength to the momentum, we find that only certain allowed energies are possible. Like Bohr's atom, the energy is quantized. Only certain energies are allowed, and only certain frequencies of light will be emitted or absorbed, determined by the energy differences between the allowed energy levels.

A quantum mechanical particle in a three-dimensional box

It is not hard to extend this model to a particle in a three-dimensional box. Suppose that the box is rectangular with sides Lx, Ly, and Lz. The momenta in the separate x, y, and z directions are px, py, and pz. The total energy is the sum of the kinetic energies due to motion in the separate x, y, and z directions

E = px2/2m + py2/2m + pz2/2m

The wavelength in each of the directions is restricted by size of the box. The restricted wavelengths are related to the restricted values of the momenta in a given direction by the de Broglie relation λdeBroglie=h/p. Substitution leads to the allowed energies

Enx, ny, nz = (nx2/Lx2 + ny2/Ly2 + nz2/Lz2) (h2/8m)         nx = 1,2,3..., ny = 1,2,3..., and nz = 1,2,3...

We find that in one-dimension there is one quantum number defining the allowed energies of the system. In three-dimensions there are three quantum numbers, defining the degree of excitation in each of the three dimensions.

Emission and absorption spectroscopy of the particle-in-a-box

Let's take the allowed energies to be those of the particle-in-a-box and relate the frequency of light emitted or absorbed to the change in energy of the particle-in-a-box. We understand the absorption of light in terms of the conservation of energy

Eparticle before + Ephoton = Eparticle after

It follows that the frequency of the absorbed light is

ν = (Efinal - Einitial)/h = (h/8 m L2) (nfinal2 - ninitial2)

The energy of the particle increases when light is absorbed, and nf > ni.

For the emission of light, the conservation of energy dictates that

Eparticle before = Ephoton + Eparticle after

so that the frequency of the emitted light is

ν = -(Efinal - Einitial)/h = (h/8 m L2) (ninitial2 - nfinal2)

The energy of the particle decreases when light is emitted, and nf < ni.

Note! It is absolutely essential to understand that the emission or absorption of light always corresponds to a change in energy between two different quantum states. The energy of the photon is never equal to the energy of the quantum mechanical system. The energy of the photon is always equal to a difference in energy between two allowed quantum mechanical states of the system.

The "F-center" of an ionic crystal modeled as a particle-in-a-box

Question: In an ionic solid, there can be defects in the orderly crystal structure due to missing ions. Those defects are called "vacancies." When there is a vacancy defect due to a missing anion, that site can be occupied by an electron. The electron can absorb light in the visible spectrum, leading a normally transparent crystal to appear colored. The lattice defect is called an "F-center" where "F" stands for Fabre, the German word for color.

F-centers in potassium chloride crystals cause the crystals to appear magenta (which is an equal mixture of red and blue light). An experiment shows that the electron, in making a transition from its ground state to its first excited state, absorbs light of wavelength 510 nm. Model the electron localized by the surrounding ions as a particle in a three-dimensional box. Calculate the length of one side of the cubic box (in meters).

The F-centers are also observed in crystals of potassium bromide. The crystals are different in color than the crystals of potassium chloride. You are told that a box contains crystals of potassium bromide. On opening the drawer, you find crystals of two different colors: yellow and blue. Only one of the crystals is the potassium bromide. Which one is it? Explain the relative color difference in terms of the differing size of the vacancies in the solids, using the three-dimensional particle-in-a-box model.

You can check your answers here.

Modeling the "solvated electron" as a particle-in-a-box

Question: An electron is injected into a sample of dense liquid helium at a temperature of 309K. After a short time, the electron is localized in a cavity in the fluid. An experiment shows that light of wavelength 9.8 x 10-7 m is absorbed by the electron in making a transition from its ground state to its first excited state.

Model the electron localized by the surrounding helium atoms as a particle in a three-dimensional box. Calculate the size of the box, in meters.

You can check your answers here.