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.

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.

Quantization and photons

By the early 1900s, 

scientists were aware that some phenomena occurred in a discrete, as opposed to continuous, manner. 

Physicists 

                 Max Planck      and 

                                                     Albert Einstein 

had recently theorized that electromagnetic radiation not only behaves like a wave, 

but also sometimes like particles called photons. 

Planck studied the electromagnetic radiation emitted by heated objects, and he proposed that the emitted electromagnetic radiation was "quantized" since the energy of light could only have values given by the following equation $E_{\text{photon}}=nh\nu$ E​photon​​=nhν, where $n$nn is a positive integer, $h$hh is Planck’s constant—$6.626 \times10^{-34}\,\text {J}\cdot \text s$6.626×10​−34​​J⋅s6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, J, dot, s—and $\nu$ν is the frequency of the light, which has units of $\dfrac{1}{\text s}$​s​​1​​start fraction, 1, divided by, s, end fraction.

As a consequence, the emitted electromagnetic radiation must have energies that are multiples of $h\nu$hν. Einstein used Planck's results to explain why a minimum frequency of light was required to eject electrons from a metal surface in the photoelectric effect.

quantized

When something is quantized, it means that only specific values are allowed, such as when playing a piano. Since each key of a piano is tuned to a specific note, only a certain set of notes—which correspond to frequencies of sound waves—can be produced. As long as your piano is properly tuned, you can play an F or F sharp, but you can't play the note that is halfway between an F and F sharp.

 

The planetary model of the atom

The planetary model of the atom

At the beginning of the 20th century, 

a new field of study known as 

                                quantum mechanics 

emerged. One of the founders of this field was Danish physicist Niels Bohr,

                          who was interested in explaining 

                              the discrete line spectrum 

                         observed when light was emitted 

                                by different elements. 

 Bohr was also interested in the structure of the atom, which was a topic of much debate at the time. Numerous models of the atom had been postulated based on experimental results including the discovery of the electron by J. J. Thomson and the discovery of the nucleus by Ernest Rutherford. Bohr supported the planetary model, in which electrons revolved around a positively charged nucleus like the rings around

However, scientists still had many unanswered questions:$$

• Where are the electrons, and what are they doing?
• If the electrons are orbiting the nucleus, why don’t they fall into the nucleus as predicted by classical physics?
  

  According to classical physics, a negatively charged electron moving around in the positive electric field created by the nucleus should emit electromagnetic energy. The electron would continue to lose energy as it orbited the nucleus until it eventually collapsed into the nucleus. Unfortunately, this reasoning would suggest that all atoms are inherently unstable!
• How is the internal structure of the atom related to the discrete emission lines produced by excited elements?
  Bohr addressed these questions using a seemingly simple assumption: what if some aspects of atomic structure, such as electron orbits and energies, could only take on certain values?