The story of quantum mechanics really begins at the end of the 1800s. A time when scientists thought that the properties of light were completely understood. Their understanding, one which is still fully relevant today, was that visible light is a manifestation of electromagnetic waves. Light has been shown in experiments to be a wave corresponding to the changing electric and magnetic fields. Maxwell had introduced his equations describing the behavior of light waves. There was one problem that remained in that wave description of light, and it really troubled physicists. The standard description of light waves did not correctly describe the light emitted from hot objects. An emission known as blackbody radiation. We know that if we turn on a stove, the element will become hot and glow red. But when we consider light using the physical laws laid out in the 1800s, we would instead predict that our stove would emit more blue light than red, and even more ultraviolet light than blue. The theory of light created in the 1800s predicted the wrong spectrum of colors from blackbody emitters. Theory predicted that the frequency of light increases as the light becomes more and more intense, contributing more and more energy to the spectrum. This is clearly wrong as it results in the stove emitting an infinite amount of energy. When you turn on your stove, it does become hot and the energy emitted in turn heats your food. However, the energy emitted from your stove is not infinite. Otherwise, turning it on would destroy the planet. The problem of supposedly infinite energy being emitted from blackbodies came to be known as the Ultraviolet Catastrophe. The ultraviolet catastrophe was a huge problem for theoreticians of the era, and its resolution became the turning point in the history of physics. In 1899 Max Planck was commissioned to study the efficiency of incandescent light bulbs, which shine due to their heat. That same year, he proposed a theory that resolved the problem of the infinite energy emitted by black bodies by introducing a new idea. Electromagnetic waves can transport only a special amount of energy called a quantum, instead of any arbitrary amount of energy. This was the key to explaining why the energy emitted by a light bulb or any other hot object is not infinite. The new equations Planck developed with a mathematical foundation to resolving the ultraviolet catastrophe. They correctly describe the color and energy emission properties of a blackbody. This new concept of a quantized packet of light energy, was called the photon. To accompany this concept, Planck also introduced a new physical constant. Which is now named after him called, Planck's constant. The letter h is often used for Planck's constant, and that is equal to 6.6 times 10 to the minus 34 Joules seconds. The minus 34 tells you that Planck's constant is a very small number. It's hard to wrap your head around how small this number actually is, or what it means, but we'll investigate its implications in more detail soon. Planck's introduction of the tiny constant h was mainly a trick to make the maths work properly for blackbody radiation. He didn't suggest that it had any true physical meaning. However, Albert Einstein realized that Planck's constant does imply a new physical reality, that light can be thought of as a particle. In a brilliant paper published in 1905, Einstein showed that particles of light or photons come in units that are called quanta, where a specific amount of energy is related to each color of light. If we have light of a certain color, we know its wavelength and its frequency. Einstein introduced a simple equation for the energy of a photon of that color. The energy of a photon is equal to h, which is Planck's constant, times frequency. This radical idea led to Einstein receiving the Nobel Prize for his work in physics in 1922. So, why was the concept of a photon such a radical idea? It's radical because it claims that there is a smallest indivisible amount of energy for light of any particular color. Think about our modern understanding of matter, we know that matter is made up of fundamental particles that are indivisible. For instance, an electron is a particle that has a specific amount of electrical charge and a specific amount of mass. It is possible to have zero electrons, or one, or two, or any whole number of electrons. However, you cannot have half an electron. Matter comes in clumps. The concept of a photon is similar when you see light of some color, the light is arriving in a group of photons. There might be one, or two, or five million photons, but never half a photon. Light also comes in clumps. Light is clumps of energy, while matter is clumps of mass. Since the value of h is so tiny, the amount of energy carried by most photons of light is also tiny. Normally, when we see light, the number of photons is enormous and so we don't notice the clumpy nature of light. For instance, if you're a couple of meters away from a standard incandescent light bulb, about one trillion photons of yellow light enters your eye every second. The particle nature of light is more obvious when you do experiments with very short wavelength radiation like X-rays or gamma rays. When radiation has a long wavelength such as radio waves, the wave nature of light is more apparent since the individual photons in the radio spectrum have a wavelength much larger than a human. The quantized nature of light is a fundamental principle of modern physics. Lasers for instance, can only be understood if we acknowledge the existence of photons. The interesting thing about light is that it can be described using the properties of both waves and particles. Our being able to describe light in these two ways is called wave-particle duality of light. In fact, we are not just able to describe photons, but all fundamental particles as both particles and waves. Wave-particle duality is one of the foundational ideas of quantum mechanics. Einstein came up with the radical idea that photons can have both wave and particle properties. So, what about matter? How can matter ever behave like a wave? In 1923, a graduate student named Louis de Broglie, asked this question and hypothesized that a particle such as an electron could have some wave-like features. One of the most fundamental features of a wave is that it has a distance over which properties repeat. The distance is what we call the wavelength and we use the Greek letter Lambda to represent wavelength. De Broglie hypothesis was this, if a particle has a known mass and velocity then it will also have a wavelength. Lambda equals Planck's constant divided by mass multiplied by velocity. We call this, the de Broglie wavelength. Since the mass and velocity are in the denominator of the equation, the wavelength becomes large if either the particles mass or velocity are small. So, since an electron has a smaller mass than a proton, if they both are travelling at the same speed the electron will have a larger wavelength. By the way, setting the speed to zero in this equation makes no sense since anything that has a temperature above absolute zero has some motion. We don't ever actually have particles with zero speed. One of the important properties of waves is that it doesn't make sense to talk about where a wave is located. Say, I asked you to point to the location of a water wave traveling across an ocean, could you point to a specific location that defines where the wave is? It can be difficult to define. When a wave is forced to travel through a small region like this hole in the wall it spreads out. The resulting circular wavelengths can be seen in a photo of ocean waves passing through the hole in a rock wall. We call the effect of spreading waves diffraction. But can matter really act like a wave. When you are considering distances that are smaller than the wavelength for a clump of matter, as is often the case in quantum mechanics, some strange patterns begin to emerge. In this animation, we see a stream of particles travelling towards a wall with two holes. In this picture, the de Broglie wavelength of the particle is tiny compared to the distance between the slits. Since the wavelength is very tiny, the particles move in a way you would expect particles to move. They land in two stripes on the far wall. Now, we have the same experimental setup but we choose particles with a large de Broglie wavelength compared to the slit separation. We could do this by either choosing slower speeds or smaller mass particles. When the particles pass through the slits they spread out in the same way that water or light waves would spread out and they interfere with each other. The resulting pattern when they hit the wall has a series of stripes called an interference pattern. The interference pattern is not obvious at first, but over time as more particles arrive and interfere with each other a series of bright and dark stripes are seen. An example of this would be in the use of an electron microscope. An electron microscope is similar to an optical microscope. But instead of photons to see the detail in an image, an electron microscope shines a beam of electrons. Actually, since the electron's matter wavelength is so small compared to the wavelengths of the photons, an electron microscope has more powerful magnification than regular light-based microscopes. The fact that matter can also be described using the properties of waves tells us that it wouldn't make sense for us to know the exact location of a matter particle. Instead, we say that a particle is likely inside an envelope bounded by its de Broglie wavelength. But the de Broglie wavelength depends on how fast the particle is moving. If the particle is moving fast then its wavelength is small, if the speed is slow then its wavelength is large. This idea led Werner Heisenberg to introduce a concept called the Uncertainty Principle. It is difficult to say exactly where a wave is or what exactly its momentum is. In fact, there is a limit to exactly how well we can figure out these quantities. In other words, the position and momentum of any wave has uncertainty. Heisenberg bridged this idea of uncertainty to particles using his formula for matter waves. Which resulted in the creation of the Heisenberg Uncertainty Principle. Here's an example, let's say we are studying a beam of protons and we want to know where they are and how fast they are going. The uncertainty principle sets a limit on how well we can know each quantity. This is a strange aspect of quantum mechanics. The more you know about a particles position in space the less you know about its momentum through space. For instance, if we could pin down a protons location to a small interval called Delta x, we could only determine the protons momentum to within a small interval according to the Heisenberg Uncertainty Principle. Where Delta x times Delta p is greater than or equal to h_bar over two. Here, the little squiggly Ds are the Greek letter Delta, which we use to denote small values in mass and momentum. In this equation, we also use the symbol h_bar, this is not a typing error, this is the symbol that represents the reduced Planck's constant. H_bar equals h divided by two Pi, since some physicists think that writing h_bar is easier than dividing by two Pi. The important thing that the uncertainty principle tells us is that we can't know exactly where a particle is located and what its speed is. If you are very certain about where a particle is located then you can't accurately know the particles speed and vice versa. There is also an energy time version of the uncertainty principle. The equivalent energy-time version of Heisenberg's Uncertainty Principle is, Delta E times Delta t is greater than or equal to h_bar divided by two. We will explore what this means for black holes in the following section.
