The Quantum World

Bohr's model of the atom worked well for hydrogen. Now, here's the bad news. Wait for it...

It ONLY worked to explain the line spectra for the hydrogen atom. Scientists went back to the drawing board. Some 14 years later, quantum theory was born.

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The central tenant of quantum theory is the wave nature of the electron. Louis de Borglie first proposed that electrons behaved as waves in 1924, and it was later confirmed by experiments in 1927. It was crazy at the time since electrons were thought of as particles and known to have mass. Much like light, the wave nature of the electron is most apparent in its diffraction. An interference pattern, similar to that observed for light, is seen when a beam of electrons passes through slits. Further experiments revealed some very strange results. I'll let Dr. Quantum explain (see video on the right).

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When an unobserved electron passes through two slits, it behaves as a particle AND a wave simultaneously. However, when it is observed, it must choose which state to occupy. The odd idea that matter can occupy two states at once is where Schrodiner used a cat, poison, radioactive material, and a Geiger counter in a box to create a thought experiment to show how absurd this idea that matter could have wave-like and particle-like properties. Schrodinger's cat is said to be both dead and alive in the box until the box is opened and the cat is forced to occupy one state or the other (just like the electron is forced to be a wave or a particle when it is observed).

The abstract idea that electrons can behave as waves has a couple implications. Part of de Broglie's proposal was that the wavelength of an electron traveling as a wave is related to its mass and velocity. This relationship is known as the de Broglie relation:

$\lambda = \frac f {mv}$

In fact, all matter is said to have wave-like behavior. But, the mass of everyday objects is so large that the wavelength is incredibly small and unobservable. When the mass of an object is comparable to its wavelength, the wave-particle duality is apparent.

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A consequence of the wave behavior is that the more you know about the velocity, the less you know about the position. Since we cannot observe the electron simultaneously as both a particle and a wave, we cannot simultaneously measure the position and velocity with infinite precision. Werner Heisenberg formalized this idea with the equation:

$\Delta x \times m\Delta v \geq \frac h {4\pi}$

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$\Delta x$ is the uncertainty in the position, $\Delta v$ is the uncertainty in velocity, m is the mass of the particle, and h is Plank's constant. Heisenberg's uncertainty principle states that the product of $\Delta x$ and $m \Delta v$ must be greater than or equal to a finite number ($h/4 \pi$). In other words, the more accurately you know the position of an electron (smaller $\Delta x$), the less accurately you know its velocity (the bigger $\Delta v$) and vice versa.

In quantum mechanics, we cannot know precisely the location of an electron if we know the velocity. However, we can determine a region where there is a higher probability of finding an electron. These regions of space are known as orbitals. More often, we will be interested in an electron's energy. Many of the properties of an element depend on the energies of its electrons. Recall that energy is related to velocity (kinetic energy equals $\frac1 2 mv^2$). Therefore, we can also say that the more we know about the energy of an electron, the less we know about the position. The mathematical derivation of energies and orbitals for electrons in atoms comes from solving the Schrodinger equation for the atom of interest. The general form of the Schrodinger equation is:

$H\Psi = E\Psi$

The symbol H stands for the Hamiltonian operator, a set of mathematical operations that represents the total energy (kinetic and potential) of the electron within the atom. The symbol E is the actual energy of the electron. The symbol $\Psi$ (Greek letter psi, pronounced "sigh") is the wave function, a mathematical function that describes the wave-like nature of the electron. A plot of $\Psi^2$ represents an orbital, a probability distribution map of the electron. To better understand what an orbital is, we can compare them to a cloud:

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