We said a couple times while talking about flames that all matter gives off electromagnetic radiation (EMR or light) according to its temperature. Flames contain hot particles and so they glow. The law that describes how EMR is given off as a function of temperature is called Planck’s Law, and any matter that follows Planck’s Law perfectly is called a black body.
The Planck’s Law equation above gives you qλ, which is the energy flux at a given wavelength. (Remember EMR has a wavelength.) On the right side you plug in the temperature T and the wavelength λ. All the rest of the symbols are famous constants.
Max Planck devised his law in 1900, and this was the first observation that led to quantum mechanics.

We said a couple times while talking about flames that all matter gives off electromagnetic radiation (EMR or light) according to its temperature. Flames contain hot particles and so they glow. The law that describes how EMR is given off as a function of temperature is called Planck’s Law, and any matter that follows Planck’s Law perfectly is called a black body.

The Planck’s Law equation above gives you qλ, which is the energy flux at a given wavelength. (Remember EMR has a wavelength.) On the right side you plug in the temperature T and the wavelength λ. All the rest of the symbols are famous constants.

Max Planck devised his law in 1900, and this was the first observation that led to quantum mechanics.

Electromagnetic radiation, i.e. light, can be thought of as both a wave and a particle. We’ve touched on this before concerning Planck’s idea that light has a minimum quantity, which is the basis of quantum mechanics. Einstein postulated that this minimum amount traveled in one direction and acted like a particle, called a photon.
Light unquestionably acts like a wave too, though. One property of waves is that they can interfere with each other: if one wave’s peak corresponds to another wave’s trough, they will cancel out. Albert Abraham Michelson invented the interferometer in the 1880’s, in which a beam of light is split in half, sent two different directions, and then recombined at the detector.
By varying the distances to the two mirrors, the peaks and troughs of the light waves can be superimposed in any way when the light beams recombine. For example, you could set them up to cancel out, or to add together again. If a single photon is put through this device, it will interfere with itself (wave behavior) as if it went both directions, but still be detected as a single photon (particle behavior). This makes no sense … that’s part of the point—however, it’s true.

Electromagnetic radiation, i.e. light, can be thought of as both a wave and a particle. We’ve touched on this before concerning Planck’s idea that light has a minimum quantity, which is the basis of quantum mechanics. Einstein postulated that this minimum amount traveled in one direction and acted like a particle, called a photon.

Light unquestionably acts like a wave too, though. One property of waves is that they can interfere with each other: if one wave’s peak corresponds to another wave’s trough, they will cancel out. Albert Abraham Michelson invented the interferometer in the 1880’s, in which a beam of light is split in half, sent two different directions, and then recombined at the detector.

By varying the distances to the two mirrors, the peaks and troughs of the light waves can be superimposed in any way when the light beams recombine. For example, you could set them up to cancel out, or to add together again. If a single photon is put through this device, it will interfere with itself (wave behavior) as if it went both directions, but still be detected as a single photon (particle behavior). This makes no sense … that’s part of the point—however, it’s true.

Erwin Schrödinger was the Austrian physicist who came up with the Schrödinger equation in 1926. It is empirical, meaning it was introduced because it got the right answers. And it has been tested countless times, and is correct.
The Clear Science languages expert tells us the correct pronunciation sounds something like Shrayn-jer, but in America we generally say Shro-din-jer, because that’s how it looks. And the Clear Science staff advocates going with the flow on such matters. (Edit: wait there’s a correction.)
In this case, we treat Schrödinger as essentially having a heavy metal umlaut, which is the most awesome kind of umlaut.

Erwin Schrödinger was the Austrian physicist who came up with the Schrödinger equation in 1926. It is empirical, meaning it was introduced because it got the right answers. And it has been tested countless times, and is correct.

The Clear Science languages expert tells us the correct pronunciation sounds something like Shrayn-jer, but in America we generally say Shro-din-jer, because that’s how it looks. And the Clear Science staff advocates going with the flow on such matters. (Edit: wait there’s a correction.)

In this case, we treat Schrödinger as essentially having a heavy metal umlaut, which is the most awesome kind of umlaut.

We introduced the Schrödinger Wave Equation, and now we have substituted in the operator H for an electron in a harmonic oscillator situation. The energy eigenvalues are shown, with a valid solution for every whole number value of n.
The graph shows the wave function (Ψ) plotted for various eigenvalues—the lowest is in red. If you would like to go through the solution yourself, this very nice website describes it in detail, and we have borrowed their plot of the solution. There are three important things to note:
The electron cannot have 0 energy. An eigenvalue of 0 gets added to 1/2 and gives a value of some energy.
The wave function give us probable places to find the electron, with higher probability when the wave is far above or below the zero line for the solution. These are the only solutions: if the electron is in state n = 0 and absorbs a photon, it jumps instantly to n = 1. There is no in between.
The places where the wave function crosses the zero line are places it cannot be found. So: it can go from one place to another along the wave function, but without passing through the spots of zero probability. That’s weird. It is.

We introduced the Schrödinger Wave Equation, and now we have substituted in the operator H for an electron in a harmonic oscillator situation. The energy eigenvalues are shown, with a valid solution for every whole number value of n.

The graph shows the wave function (Ψ) plotted for various eigenvalues—the lowest is in red. If you would like to go through the solution yourself, this very nice website describes it in detail, and we have borrowed their plot of the solution. There are three important things to note:

  1. The electron cannot have 0 energy. An eigenvalue of 0 gets added to 1/2 and gives a value of some energy.
  2. The wave function give us probable places to find the electron, with higher probability when the wave is far above or below the zero line for the solution. These are the only solutions: if the electron is in state n = 0 and absorbs a photon, it jumps instantly to n = 1. There is no in between.
  3. The places where the wave function crosses the zero line are places it cannot be found. So: it can go from one place to another along the wave function, but without passing through the spots of zero probability. That’s weird. It is.

Waves? Particles?

The work of Planck showed that light, normally thought of as a wave, can have properties like a particle. As in: there are discrete quantities of light, called photons, much like there are discrete balls on a billiard table.

The uncertainty principle showed that things we think of as particles are not located in a definite place, but rather have a set of places they can be located in, within our “certainty.” Or, put another way, they have wave-like behavior.

This is Wave-Particle Duality.

We have seen that Planck’s constant is important for several reasons: it defines the quanta (minimum amount) for the energy of light, and therefore also the energies of electrons in atoms. It also factors into something called the Uncertainty Principle, discovered by Werner Heisenberg in 1927.
Say there is a body (particle) as shown above. This particle is located somewhere (has a position, given by x), and is moving with a certain velocity (given by p, which is momentum, the velocity x mass). The Uncertainty Principle shows that one cannot know the x and p values to a precision greater than h-bar/2. The value h-bar is just Planck’s constant divided by 2π, or h/2π. (The “uncertainty” is shown by a Δ.)
You can think of this as if there is a resolution. Your computer monitor has a resolution, because the image is made up of discrete (quantized) pixels. However, the Uncertainty Principle is not talking about a simple distance resolution, like a monitor has. Rather, it is a more complicated concept, a resolution based on momentum and position. Light has a resolution (that would be one photon), all information originates with light, and this is related to the limit to knowledge about particles’ whereabouts. Note that this is not an uncertainty because of bad observation. Rather, it is a real uncertainty that really exists.

We have seen that Planck’s constant is important for several reasons: it defines the quanta (minimum amount) for the energy of light, and therefore also the energies of electrons in atoms. It also factors into something called the Uncertainty Principle, discovered by Werner Heisenberg in 1927.

Say there is a body (particle) as shown above. This particle is located somewhere (has a position, given by x), and is moving with a certain velocity (given by p, which is momentum, the velocity x mass). The Uncertainty Principle shows that one cannot know the x and p values to a precision greater than h-bar/2. The value h-bar is just Planck’s constant divided by 2π, or h/2π. (The “uncertainty” is shown by a Δ.)

You can think of this as if there is a resolution. Your computer monitor has a resolution, because the image is made up of discrete (quantized) pixels. However, the Uncertainty Principle is not talking about a simple distance resolution, like a monitor has. Rather, it is a more complicated concept, a resolution based on momentum and position. Light has a resolution (that would be one photon), all information originates with light, and this is related to the limit to knowledge about particles’ whereabouts. Note that this is not an uncertainty because of bad observation. Rather, it is a real uncertainty that really exists.

Discussing how quantum mechanics works, we first defined mechanics, and then we defined a quantum. It turns out light is not the only thing quantized. This is why quantum mechanics is so important: it describes how the matter in atoms, and therefore everything, behaves.
You can think of the nucleus and electrons in an atom as analogous to the sun and the planets. However, there is a major difference: atoms are small, and solar systems are large. The math for solar systems completely breaks down at the atomic level. Around the turn of the 20th Century, scientists were grappling with this.
Neils Bohr proposed that an electron’s orbital levels in an atom were quantized. This makes things work out*: electrons can’t have any energy level (corresponding to an orbiting distance). Instead, they can only have certain well-defined energies. This is quantum behavior, just like observed in light. The two phenomena are related, because for an electron to change energy levels, it either absorbs or gives off a photon, which is the minimum unit of light. 
*The Bohr Model is not exactly correct, but is still taught because it contains all the basics of QM and is easy to understand. It really works out only for the hydrogen atom.

Discussing how quantum mechanics works, we first defined mechanics, and then we defined a quantum. It turns out light is not the only thing quantized. This is why quantum mechanics is so important: it describes how the matter in atoms, and therefore everything, behaves.

You can think of the nucleus and electrons in an atom as analogous to the sun and the planets. However, there is a major difference: atoms are small, and solar systems are large. The math for solar systems completely breaks down at the atomic level. Around the turn of the 20th Century, scientists were grappling with this.

Neils Bohr proposed that an electron’s orbital levels in an atom were quantized. This makes things work out*: electrons can’t have any energy level (corresponding to an orbiting distance). Instead, they can only have certain well-defined energies. This is quantum behavior, just like observed in light. The two phenomena are related, because for an electron to change energy levels, it either absorbs or gives off a photon, which is the minimum unit of light. 

*The Bohr Model is not exactly correct, but is still taught because it contains all the basics of QM and is easy to understand. It really works out only for the hydrogen atom.

But what is a quantum, and what does it have to do with quantum mechanics? A quantum is any quality that must occur in discrete, non-continuous increments. Consider the two rows of color shown above. In the top one, light blue transitions to dark blue continuously, moving through all the in-between values on the way. If you characterized the shades of blue with numbers, you might say the color of that bar could have a blue of 1, a blue of 1.001, a blue of 1.002, all the way up to a blue of 5.
On the bottom, however, the blue is quantized, occurring in only 5 values: 1, 2, 3, 4, or 5. That’s it. There can be no in-between values. What Planck discovered about light was that its energy was like the blue in the bottom row: it could only have distinct values, and going from one to the other meant adding another photon. (Photons emitted by the metal in the photo are making it glow.)
So what does this have to do with mechanics? Perhaps light is not the only thing that has quantized energy. In fact this is the case: the energy levels of the particles that make up atoms, and therefore everything, also have discrete quanta.

But what is a quantum, and what does it have to do with quantum mechanics? A quantum is any quality that must occur in discrete, non-continuous increments. Consider the two rows of color shown above. In the top one, light blue transitions to dark blue continuously, moving through all the in-between values on the way. If you characterized the shades of blue with numbers, you might say the color of that bar could have a blue of 1, a blue of 1.001, a blue of 1.002, all the way up to a blue of 5.

On the bottom, however, the blue is quantized, occurring in only 5 values: 1, 2, 3, 4, or 5. That’s it. There can be no in-between values. What Planck discovered about light was that its energy was like the blue in the bottom row: it could only have distinct values, and going from one to the other meant adding another photon. (Photons emitted by the metal in the photo are making it glow.)

So what does this have to do with mechanics? Perhaps light is not the only thing that has quantized energy. In fact this is the case: the energy levels of the particles that make up atoms, and therefore everything, also have discrete quanta.

In 1900, Max Planck solved a long-standing problem of physics by suggesting that the energy of light emitted by a body of matter was described by a simple equation E = hν. (Energy equals a constant h times the frequency of the light.) (The symbol ν is the Greek “nu,” which often means frequency.) This finding was the beginning of quantum mechanics.
But why? Planck showed that matter at a given temperature does not emit or absorb light continuously, but instead only in discrete quantities of hν. The constant h is quite small, so the discrete quantities allowed light are quite small, too. Still, light cannot have any value of energy. Einstein then postulated that this minimum quantity of light moved only in one direction, like a particle. This is a quantum of light, and was named a photon.

In 1900, Max Planck solved a long-standing problem of physics by suggesting that the energy of light emitted by a body of matter was described by a simple equation E = hν. (Energy equals a constant h times the frequency of the light.) (The symbol ν is the Greek “nu,” which often means frequency.) This finding was the beginning of quantum mechanics.

But why? Planck showed that matter at a given temperature does not emit or absorb light continuously, but instead only in discrete quantities of hν. The constant h is quite small, so the discrete quantities allowed light are quite small, too. Still, light cannot have any value of energy. Einstein then postulated that this minimum quantity of light moved only in one direction, like a particle. This is a quantum of light, and was named a photon.

When trying to write the equations to describe the mechanics of atoms, it was discovered that the classical mechanics of Newton wouldn’t work. This is because the pieces of an atom, the nucleus, electrons, etc, are quite small, and on that scale, there are qualities of how physical bodies behave which we don’t notice in everyday life, because everything we are used to seeing is much bigger.
If you take a class on quantum mechanics, there are a handful of problems that you end up considering over and over. One is the harmonic oscillator, in which a body is attached to a spring and oscillating back and forth (the other end of the spring is hooked to something bigger that doesn’t move). Another is the rigid rotor, which is two bodies connected at a fixed distance, like a dumbbell, and spinning. 
These problems are ways of modeling atoms and their parts. The harmonic oscillator could be an electron wiggling back and forth in relation to the nucleus. The rigid rotor could be a hydrogen molecule, which is two hydrogen atoms bound together.

When trying to write the equations to describe the mechanics of atoms, it was discovered that the classical mechanics of Newton wouldn’t work. This is because the pieces of an atom, the nucleus, electrons, etc, are quite small, and on that scale, there are qualities of how physical bodies behave which we don’t notice in everyday life, because everything we are used to seeing is much bigger.

If you take a class on quantum mechanics, there are a handful of problems that you end up considering over and over. One is the harmonic oscillator, in which a body is attached to a spring and oscillating back and forth (the other end of the spring is hooked to something bigger that doesn’t move). Another is the rigid rotor, which is two bodies connected at a fixed distance, like a dumbbell, and spinning. 

These problems are ways of modeling atoms and their parts. The harmonic oscillator could be an electron wiggling back and forth in relation to the nucleus. The rigid rotor could be a hydrogen molecule, which is two hydrogen atoms bound together.