Wave-particle duality and the de Broglie wavelength

Another major development in quantum mechanics was pioneered by French physicist Louis de Broglie. Based on work by Planck and Einstein that showed how light waves could exhibit particle-like properties, de Broglie hypothesized that particles could also have wavelike properties.

[What are wavelike properties?]

Examples of observable wavelike behavior are interference and diffraction. For example, when light is shined through a barrier with two slits, as in Young's double-slit experiment, the light waves will diffract through the slits. The destructive and constructive interference between the light waves produces a pattern of dark and light areas on the detector.

De Broglie derived the following equation for the wavelength of a particle of mass $\text m$mm (in kilograms $\text{kg}$kgk, g), traveling at velocity $\text v$vv (in $\dfrac{\text m}{\text s}$​s​​m​​start fraction, m, divided by, s, end fraction), where $\lambda$λlambda is the de Broglie wavelength of the particle in meters and $h$hh is Planck's constant, $6.626 \times 10^{-34} \,\dfrac{\text{kg} \cdot \text m^2}{\text s}$6.626×10​−34​​​s​​kg⋅m​2​​​​6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, start fraction, k, g, dot, m, start superscript, 2, end superscript, divided by, s, end fraction:

$\lambda=\dfrac{h}{\text {mv}}$λ=​mv​​h​​lambda, equals, start fraction, h, divided by, m, v, end fraction

Note that the de Broglie wavelength and particle mass are inversely proportional. The inverse relationship is why we don't notice any wavelike behavior for the macroscopic objects we encounter in everyday life. It turns out that the wavelike behavior of matter is most significant when a wave encounters an obstacle or slit that is a similar size to its de Broglie wavelength. However, when a particle has a mass on the order of $10^{-31}$10​−31​​10, start superscript, minus, 31, end superscript kg, as an electron does, the wavelike behavior becomes significant enough to lead to some very interesting phenomena.

Concept check: The fastest baseball pitch ever recorded was approximately 46.7 $\dfrac{\text{m}}{\text s}$​s​​m​​start fraction, m, divided by, s, end fraction. If a baseball has a mass of 0.145 kg, what is its de Broglie wavelength?

 

Example 1: Calculating the de Broglie wavelength of an electron

The velocity of an electron in the ground-state energy level of hydrogen is $2.2\times10^6\,\dfrac{\text{m}}{\text s}$2.2×10​6​​​s​​m​​2, point, 2, times, 10, start superscript, 6, end superscript, space, start fraction, m, divided by, s, end fraction. If the electron's mass is $9.1\times10^{-31}$9.1×10​−31​​9, point, 1, times, 10, start superscript, minus, 31, end superscript kg, what is the de Broglie wavelength of this electron?

We can substitute Planck's constant and the mass and velocity of the electron into de Broglie's equation:

λ=hmv=6.626×10−34kg⋅m2s(9.1×10−31kg)(2.2×106ms)=3.3×10−10 m

The wavelength of our electron, $3.3\times10^{-10}\,$3.3×10​−10​​3, point, 3, times, 10, start superscript, minus, 10, end superscript, space meters, is on the same order of magnitude as the diameter of a hydrogen atom, ~$1\times 10^{-10}\,$1×10​−10​​1, times, 10, start superscript, minus, 10, end superscript, space meters. That means the de Broglie wavelength of our electron is such that it will often be encountering things with a similar size as its wavelength—for instance, a neutron or atom. When that happens, the electron will be likely to demonstrate wavelike behavior!

Review of Bohr's model of hydrogen

As we have seen in a previous article, the emission spectra of different elements contain discrete lines. The following image shows the visible region of the emission spectra for hydrogen.

 

The quantized emission spectra indicated to Bohr that perhaps electrons could only exist within the atom at certain atomic radii and energies. Recall that quantized refers to the fact that energy can only be absorbed and emitted in a range of allowable values rather than with any possible value. The following diagram of the Bohr model shows the electron existing in a finite number of allowed orbits or shells around the nucleus.

 

From this model, Bohr derived an equation that correctly predicted the various energy levels in the hydrogen atom, which corresponded directly to the emission lines in the hydrogen spectrum. Bohr's model was also successful at predicting the energy levels in other one-electron systems, such as $\text{He}^+$He​+​​H, e, start superscript, plus, end superscript. However, it failed to explain the electronic structure in atoms that contained more than one electron.

While some physicists initially tried to adapt Bohr's model to make it useful for more complicated systems, they eventually concluded that a completely different model was needed

Introduction to the quantum mechanical model

"We must be clear that when it comes to atoms, language can only be used as in poetry." —Niels Bohr

Matter begins to behave very strangely at the subatomic level. Some of this behavior is so counterintuitive that we can only talk about it with symbols and metaphors—like in poetry. For example, what does it mean to say an electron behaves like a particle and a wave? Or that an electron does not exist in any one particular location, but that it is spread out throughout the entire atom?

If these questions strike you as odd, they should! As it turns out, we are in good company. The physicist Niels Bohr also said, "Anyone who is not shocked by quantum theory has not understood it." So if you feel confused when learning about quantum mechanics, know that the scientists who originally developed it were just as befuddled.

We will start by briefly reviewing Bohr's model of hydrogen, the first non-classical model of the atom.

introduction to quantum mechanics

Key points

• Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength $\lambda$λlambda, given by the following equation:

$\lambda=\dfrac{h}{mv}$λ=​mv​​h​​lambda, equals, start fraction, h, divided by, m, v, end fraction

• Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves.
• Schrödinger's equation, $\hat{H}\psi=E\psi$​H​^​​ψ=Eψ, can be solved to yield a series of wave function $\psi$ψ, each of which is associated with an electron binding energy, $E$EE.
• The square of the wave function, $\psi^2$ψ​2​​, represents the probability of finding an electron in a given region within the atom.
• An atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time.
• The Heisenberg uncertainty principle states that we can't know both the energy and position of an electron. Therefore, as we learn more about the electron's position, we know less about its energy, and vice versa.
• Electrons have an intrinsic property called spin, and an electron can have one of two possible spin values: spin-up or spin-down.
• Any two electrons occupying the same orbital must have opposite spins

What have we learned since Bohr proposed his model of hydrogen?

The Bohr model worked beautifully for explaining the hydrogen atom and other single electron systems such as $\text{He}^+$He​+​​H, e, start superscript, plus, end superscript. Unfortunately, it did not do as well when applied to the spectra of more complex atoms. Furthermore, the Bohr model had no way of explaining why some lines are more intense than others or why some spectral lines split into multiple lines in the presence of a magnetic field—the Zeeman effect.

In the following decades, work by scientists such as Erwin Schrödinger showed that electrons can be thought of as behaving like waves and behaving as particles. This means that it is not possible to know both a given electron’s position in space and its velocity at the same time, a concept that is more precisely stated in Heisenberg's uncertainty principle. The uncertainty principle contradicts Bohr’s idea of electrons existing in specific orbits with a known velocity and radius. Instead, we can only calculate probabilities of finding electrons in a particular region of space around the nucleus.

The modern quantum mechanical model may sound like a huge leap from the Bohr model, but the key idea is the same: classical physics is not sufficient to explain all phenomena on an atomic level. Bohr was the first to recognize this by incorporating the idea of quantization into the electronic structure of the hydrogen atom, and he was able to thereby explain the emission spectra of hydrogen as well as other one-electron systems.

Absorption and emission

Bohr could now precisely describe the processes of absorption and emission in terms of electronic structure. According to Bohr's model, an electron would absorb energy in the form of photons to get excited to a higher energy level as long as the photon's energy was equal to the energy difference between the initial and final energy levels. After jumping to the higher energy level—also called the excited state—the excited electron would be in a less stable position, so it would quickly emit a photon to relax back to a lower, more stable energy level. 

 

The energy levels and transitions between them can be illustrated using an energy level diagram, such as the example above showing electrons relaxing back to the $n=2$n=2n, equals, 2 level of hydrogen. The energy of the emitted photon is equal to the difference in energy between the two energy levels for a particular transition. The energy difference between energy levels $n_{high}$n​high​​n, start subscript, h, i, g, h, end subscript and $n_{low}$n​low​​n, start subscript, l, o, w, end subscript can be calculated using the equation for $E(n)$E(n)E, left parenthesis, n, right parenthesis from the previous section:

$\begin{aligned} \Delta E &= E(n_{high})-E(n_{low}) \\ \\ &=\left( -\dfrac{1}{{n_{high}}^2} \cdot 13.6\,\text{eV} \right)-\left(-\dfrac{1}{{n_{low}}^2} \cdot 13.6\,\text{eV}\right) \\ \\ &= \left(\dfrac{1}{{n_{low}}^2}-\dfrac{1}{{n_{high}}^2}\right) \cdot 13.6\,\text{eV} \end{aligned}$​ΔE​​​​​​​=E(n​high​​)−E(n​low​​)​=(−​n​high​​​2​​​​1​​⋅13.6eV)−(−​n​low​​​2​​​​1​​⋅13.6eV)​=(​n​low​​​2​​​​1​​−​n​high​​​2​​​​1​​)⋅13.6eV​​

Since we also know the relationship between the energy of a photon and its frequency from Planck's equation, we can solve for the frequency of the emitted photon:

$\begin{aligned} h\nu &=\Delta E = \left(\dfrac{1}{{n_{low}}^2}-\dfrac{1}{{n_{high}}^2}\right) \cdot 13.6\,\text{eV} ~~~~~~~~~~~~\text{Set photon energy equal to energy difference}\\ \\ \nu &= \left(\dfrac{1}{{n_{low}}^2}-\dfrac{1}{{n_{high}}^2}\right) \cdot \dfrac{13.6\,\text{eV}}{h}~~~~~~~~~~~~~~~~~~~~~~\text{Solve for frequency}\end{aligned}$​hν​​ν​​​=ΔE=(​n​low​​​2​​​​1​​−​n​high​​​2​​​​1​​)⋅13.6eV            Set photon energy equal to energy difference​=(​n​low​​​2​​​​1​​−​n​high​​​2​​​​1​​)⋅​h​​13.6eV​​                      Solve for frequency​​

We can also find the equation for the wavelenth of the emitted electromagnetic radiation using the relationship between the speed of light $\text c$cc, frequency $\nu$ν, and wavelength $\lambda$λlambda:

$\begin{aligned}\text c &=\lambda \nu ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{Rearrange to solve for }\nu . \\ \dfrac{\text c}{\lambda}&=\nu=\left(\dfrac{1}{{n_{low}}^2}-\dfrac{1}{{n_{high}}^2}\right) \cdot \dfrac{13.6\,\text{eV}}{h}~~~~~~~~~~~~~~\text{Divide both sides by c to solve for }\dfrac{1}{\lambda}.\\ \\ \dfrac{1}{\lambda} &=\left(\dfrac{1}{{n_{low}}^2}-\dfrac{1}{{n_{high}}^2}\right) \cdot \dfrac{13.6\,\text{eV}}{h\text c} \end{aligned}$​c​​λ​​c​​​​​λ​​1​​​​​=λν                                                                  Rearrange to solve for ν.​=ν=(​n​low​​​2​​​​1​​−​n​high​​​2​​​​1​​)⋅​h​​13.6eV​​              Divide both sides by c to solve for ​λ​​1​​.​=(​n​low​​​2​​​​1​​−​n​high​​​2​​​​1​​)⋅​hc​​13.6eV​​​​

 Thus, we can see that the frequency—and wavelength—of the emitted photon depends on the energies of the initial and final shells of an electron in hydrogen.