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!

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.

Bohr's model of the hydrogen atom: quantization of electronic structure

Bohr’s model of the hydrogen atom started from the planetary model, 

but he added one assumption regarding the electrons. 

What if the electronic structure of the atom was quantized? Bohr suggested that the perhaps the electrons could only orbit the nucleus in specific orbits or shells with a fixed radius. Only shells with a radius given by the equation below would be allowed, and the electron could not exist in between these shells. Mathematically, we could write the allowed values of the atomic radius as $\mathrm{ <b>\; r</b>}$(n)=n2⋅r(1)r(n)=n^2\cdot r(1)r(n)=n​2​​⋅r(1)r, left parenthesis, n, right parenthesis, equals, n, start superscript, 2, end superscript, dot, r, left parenthesis, 1, right parenthesis,

where $n$nn is a positive integer, and $r(1)$r(1)r, left parenthesis, 1, right parenthesis is the Bohr radius, the smallest allowed radius for hydrogen.

He found that $r(1)$r(1)r, left parenthesis, 1, right parenthesis has the value

$\text{Bohr radius}=r\left(1\right)=0\mathrm{.}529\times 10^{-10}\text{m}$
$\text{Bohr radius}=r(1)=0.529 \times 10^{-10} \,\text{m}$By keeping the electrons in circular, quantized orbits around the positively-charged nucleus, Bohr was able to calculate the energy of an electron in the $n$nnth energy level of hydrogen: $E(n)=-\dfrac{1}{n^2} \cdot 13.6\,\text{eV}$E(n)=−​n​2​​​​1​​⋅13.6eVE, left parenthesis, n, right parenthesis, equals, minus, start fraction, 1, divided by, n, start superscript, 2, end superscript, end fraction, dot, 13, point, 6, space, e, V, where the lowest possible energy or ground state energy of a hydrogen electron—$E(1)$E(1)E, left parenthesis, 1, right parenthesis—is $-13.6\,\text{eV}$−13.6eVminus, 13, point, 6, space, e, V. Bohr radius=r(1)=0.529×10​−10​​m




Note that the energy is always going to be a negative number, and the ground state, $n=1$n=1n, equals, 1, has the most negative value. This is because the energy of an electron in orbit is relative to the energy of an electron that has been completely separated from its nucleus, $n=\infty$n=∞n, equals, infinity, which is defined to have an energy of $0\,\text{eV}$0eV0, space, e, V. Since an electron in orbit around the nucleus is more stable than an electron that is infinitely far away from its nucleus, the energy of an electron in orbit is always negative.

Atomic line spectra

Atomic line spectra are another example of quantization. When an element or ion is heated by a flame or excited by electric current, the excited atoms emit light of a characteristic color. The emitted light can be refracted by a prism, producing spectra with a distinctive striped appearance due to the emission of certain wavelengths of light.

For the relatively simple case of the hydrogen atom, the wavelengths of some emission lines could even be fitted to mathematical equations. The equations did not explain why the hydrogen atom emitted those particular wavelengths of light, however. Prior to Bohr's model of the hydrogen atom, scientists were unclear of the reason behind the quantization Light emitted or absorbed by single atoms contributes only very little to the colours of our surroundings. Neon signs (or other gas discharge tubes) as used for advertising, sodium or mercury vapour lamps show atomic emission; the colours of fireworks are due to it. The aurora borealis (northern light) is very rare at our latitudes, and to appreciate the colours of cosmic objects, powerful telescopes are necessary. Neon, which gives red colour in a gas discharge, is a colourless gas. If the light of the sun is spread out into different colours by a simple glass prism, the narrow absorption lines cannot be seen.
of atomic emission spectra.