Tag: Max Planck

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physics

The Uncertainty Principle

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Heisenberg’s uncertainty principle is of those topics where people feel confident that they understand it but many people only understand a portion of it. The uncertainty principle is likely the first thing people will mention if you ask them about quantum mechanics. As with many things in science there is actually a much deeper and richer explanation of this phenomenon than you might be aware of. So let’s jump in and learn about the history and applications of one of the most famous and misunderstood topics in physics.

Werner Karl Heisenberg was born on December 5th, 1901 in the city Wurzburg, Germany. Heisenberg displayed an aptitude for mathematics from an early age and it wasn’t until he attended the University of Munich in 1920 that he began his study of physics in earnest. At the tender age of 25 he was appointed professor of theoretical physics in Leipzig. During this time Heisenberg worked with Max Born, Neils Bohr, Arnold Sommerfeld and others to develop and refine the field of quantum mechanics.

In 1925 Werner Heisenberg submitted a paper that in his words “seeks to establish a basis for theoretical quantum mechanics founded exclusively on relationships between quantities which in principle are observable.” This paper helped describe quantum mechanics using the mathematical principles of matrices. This method of calculating the quantum behavior of particles was a tremendous breakthrough in the emerging field of quantum mechanics.

History of the Uncertainty Principle

Before we describe exactly what the uncertainty principle states and describe its applications let’s take a moment to discuss how Heisenberg arrived at this monumental insight. In 1926 a debate was raging between those that held to Heisenberg’s matrix version of quantum theory and Erwin Schrodinger who believed in the wave theory of quantum mechanics. Most scientists of the day preferred the wave theory due in, no small part, to the ease and familiarity with the mathematical equations presented by Schrodinger.

Erwin Schrodinger presented his findings that the wave theory and the matrix theory gave identical mathematical results. As a result of this finding Paul Dirac and Pascual Jordan developed a set of unified equations collectively known as transformation theory which became the foundation for quantum mechanics. While Heisenberg studied how to measure the variables in the unified equations, and with input from Wolfgang Pauli, Heisenberg detected a problem with trying to measure the variables.

Heisenberg noticed that in trying to determine the precise position and momentum of a particle, at the same time, imprecisions appeared. These imprecisions occurred when trying to measure the time and energy variables at the same moment as well. Heisenberg presented his results in a paper to Pauli dated February 23rd, 1927. Upon receiving positive and encouraging feedback from Pauli, Heisenberg formalized and submitted his findings for publication.

What Does the Uncertainty Principle Mean?

Werner Heisenberg attempted to use his matrix mechanics, in a thought experiment to describe the path of an electron in a cloud chamber. He realized that the path the electron traversed was made visible by the condensation of droplets of water that were actually larger than the electrons they were trying to detect. (J Baggott, The Quantum Story p91). The consequence of this meant that the instantaneous position and velocity of the electron can only be approximately known. We will come back to the electron and cloud chamber thought experiment later.

Let’s discuss a few terms that we will need in our explanation of the uncertainty principle. Planck’s constant represented by the letter h and has a value of 6.6262E -34 Joule-second. “Planck’s constant defines the amount of energy that a photon can carry, according to the frequency of the wave in which it travels.” (https://science.howstuffworks.com/dictionary/physics-terms/plancks-constant.htm) Next is momentum which is the product of mass multiplied by velocity. This is important because you will see momentum used in some descriptions of the uncertainty principle and in others you will see velocity. It is more common to use momentum as all particles such as electrons, have a mass associated with them.

Heisenberg had discovered a fundamental property of nature which states “the uncertaintites is position and momentum cannot be smaller than Planck’s constant.” (J Baggott, The Quantum Story p91) This results in a limit to the amount of precision we can simultaneously have regarding the position and momentum of a particle. This fundamental limit does not hold true in our everyday experiences of classical mechanics.

Courtesy of clutchprep.com

The image above is the equation for the uncertainty principle where x represents position, p represents momentum and the triangle represents the uncertainty. The equation states the product of the uncertainty of the position and momentum must be greater than or equal to the Planck’s constant divide by 4pi. This equation can be rewritten to include the uncertainty of time and energy in addition to position and momentum.

Recall the electron and cloud chamber thought experiment from earlier. One way to measure the position and momentum would be to use a microscope. An issue here is that every time a photon bounces off the electron the momentum and position of the electron is changed. One might be able to determine the electrons instantaneous position, however the large interaction between the electron and the device we are using to measure its position means we are unable to determine its momentum. You may be asking yourself can we use a device with lower energy photons, that is those with a lower frequency or longer wavelength ? This would allow us to calculate the electron’s momentum but we would not be able to accurately measure its position. This thought experiment is useful to give you a conceptual understanding of the uncertainty principle but it is more useful than truthful.

What is really happening here is that because of the wave particle duality it is impossible to know the exact location and momentum of an object. The reason for this is because “what we can measure is limited by the fact that position and momentum are undefined until we measure them in the quantum realm. In the thought experiment with the electron and the cloud chamber the electron has a definite position and momentum prior to making a measurement. In the quantum world, due to the wave-particle duality the precise position and momentum do not exist. (C Orzel, How to Teach Quantum Physics to Your Dog p44). Here is a quick conceptual explanation of the uncertainty principle: https://youtu.be/m8VQue1Nffw

Misconceptions of the Uncertainty Principle

One of the most common misconceptions of the uncertainty principle is that it is caused by the act of measurement. The idea here is that the act of measuring the object causes a change in the position or speed of the object as in the thought experiment. The reality is that the principle is a result of the dual nature of quantum objects.

A second misconception is that current technology limits our ability to determine both the position and momentum of a quantum object. It turns out that the uncertainty principle is a fundamental limit set by nature rather than a technological or observational limit.

What it All Means

The point of all this is that the uncertainty principle, also known as the indeterminacy principle, is a fundamental limit of nature and is due to the wave-particle nature of quantum particles. The concept of an exact position and exact momentum of a quantum particle is meaningless. “Any attempt to measure precisely the velocity of a subatomic particle, such as an electron, will knock it about in an unpredictable way, so that a simultaneous measurement of its position has no validity. This result has nothing to do with inadequacies in the measuring instruments, the technique, or the observer; it arises out of the intimate connection in nature between particles and waves in the realm of subatomic dimensions. (https://www.britannica.com/science/uncertainty-principle) As stated earlier this applies not just to position and momentum but to energy and time as well. Well, I certainly hope you enjoyed this post.

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physics

The Ultraviolet Catastrophe

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So what comes to mind when you hear the term catastrophe? It certainly sounds ominous and dreadful. According to merriam-webster.com a catastrophe is: a momentous tragic event ranging from extreme misfortune to utter overthrow or ruin and/or utter failure. It is the “utter failure” description that more accurately describes this particular catastrophe. So just what is the ultraviolet catastrophe and what was its significance to the birth of quantum mechanics? Keep reading to find out.

As the 19th century drew to a close, there was among the scientific community, an belief that most of the physical phenomena could be described and explained through classical physics. Nature, however, had other ideas. A series of discoveries would usher in a new branch of physics that classical or Newtonian physics was not able to describe. The photoelectric effect was one of these discoveries. This phenomena was accurately described by Albert Einstein which demonstrated that light travels as a wave and interacts with matter as particles. This is often described as the dual nature of light. The double slit experiment showed that particles, such as electrons, could produce interference patterns by interfering with themselves as they traveled through a narrow set of slits. This remarkable experiment showed that the behavior of individual particles could travel as particles and still demonstrate wave like behavior. Oddly enough, when scientists attempted to determine which slit the particle traveled through the interference pattern collapsed. The result of this collapse is what would be expected from a series of single particles being fired though one slit or another. Both of these topics have been discussed in earlier blog posts on my site. Feel free to go read those posts if you haven’t already.

So both of the above experiments helped describe discrete amounts of fundamental units. These units were considered to be quantized that is containing the smallest whole number amount of a specific material. For example, a quanta of light is a photon. So let’s see how the ultraviolet catastrophe relates to all this.

Blackbody and Blackbody Radiation

It will be helpful to be familiar with some of the vocabulary associated with this phenomenon. A blackbody is an idealized or theoretical object that absorbs all electromagnetic radiation that falls on it. A blackbody, according to Kirchoff, is a body that is able to “…completely absorb all incident rays, and neither reflects nor transmit any.” A blackbody emits blackbody radiation which is also known as thermal radiation. Many objects such as people, heating elements, flames, and the sun approximate blackbodies.

So you may be wondering how would a person experimentally measure blackbody radiation? You could set up a hollow container with a small hole in it. This cavity allows incident radiation to get in while the design makes it unlikely that much radiation will escape.

A blackbody cavity courtesy of https://phys.libretexts.org/

A small amount of emitted radiation will pass back out through the cavity opening where it can be measured. Thermal equilibrium occurs when two objects are in direct contact or close contact with each other but no net energy is transferred between them. They may gain energy from one another but no net energy is transferred. The amount of emitted radiation is small enough that it will not disturb the thermal equilibrium inside the cavity. As it turns out, the radiation a blackbody emits depends upon the temperature of the body.

Catastrophe

In science a theory and the mathematical representation of the theory must match the experimental results in order for the theory to be valid. Using classical physics, Lord Rayleigh and Sir James Jeans, working independently developed a law to predict the amount of radiation intensity emitted at a given wavelength. They were trying to come up with a rule that would allow them to accurately predict the color a blackbody would radiate based on temperature. The Rayleigh-Jeans Law predicted that the intensity of radiation was proportional to the wavelength so as the intensity increased the wavelength should decrease.

Courtesy of researchgate.net

In the graph above you can see that as the temperature in Kelvin increases the peak of the curve is shifted to the left. At 5000 K the peak of the curve is in the red portion of the spectrum and at 6000 K the peak is in the yellow portion of the spectrum. The catastrophe occurred in the ultraviolet portion of the spectrum and it stated that at low wavelengths the intensity should become infinite. This obviously was a violation of the law of conservation and because this occurred in the ultraviolet portion of the spectrum it was named the ultraviolet catastrophe.

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The graph above shows the experimental results versus the classical theory predictions regarding the intensity of radiation versus wavelength. You can see that at higher wavelengths the classical theory closely matches experimental results but at lower wavelengths there is no agreement.

A Quantum Way Out of Catastrophe

Max Planck solved the ultraviolet catastrophe in 1900. Planck developed a model in which the electromagnetic radiation in the cavity is absorbed by simple harmonic oscillators or resonators. These oscillators were merely a description and he didn’t argue that they actually exist in the walls of the cavity. Planck determined that the energy absorbed by these resonators needed to be in discrete packets or quantized. These discrete packets of energy became known as quanta of energy. This idea of quanta of energy was the birth of quantum theory. He developed a formula to relate the energy absorbed to the frequency of oscillation: E = hv where E is the energy absorbed, v is the frequency of oscillation and h is Planck’s constant 6.626 x 10^-34 J s

Max Planck derived his blackbody radiation law working backward from Wein’s distribution law. Planck realized that Wein’s distribution did not agree with experimental data at long wavelengths, that is wavelengths in the infared portion of the spectrum. The results from Planck’s blackbody radiation law are in agreement with experimental data.

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The above equation was derived from Wein’s distribution law. Planck’s equation differs in that it allows the units of measure to be derived entirley based on the universal physical constants or the four fundamental constants of nature, Planck’s constant (h), Boltzman’s constant (kB), speed of light (C), and the universal gravitational constant (G).

 Planck’s theoretical result (continuous curve) and the experimental blackbody radiation curve (dots). Courtesy of https://phys.libretexts.org/

Max Planck was able to work through the ultraviolet catastrophe and develop an theory that matched the experimental results. His realization that the radiation in the cavity must be absorbed in discrete quantities ushered in a brand new branch of physics called quantum mechanics. Here is a short video describing Planck’s discovery of light being quantized and the birth of quantum mechanis: https://youtu.be/i1TVZIBj7UA Ironically, many physics students probably refer to this as the quantum catastrophe!

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physics

The Photoelectric Effect

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A Brief History of the Photoelectric Effect

In 1887 scientist Heinrich Hertz discovered the photoelectric effect while experimenting with a device called the spark gap generator which is a precursor to the radio. What Hertz found while using this device was “…sparks generated between two small metal spheres in a transmitter induce sparks that jump between between two different metal spheres in a receiver.” (https://physics.info/photoelectric/) The sparks that were jumping across the gap were, in fact electrons, which were receiving energy from ultraviolet light. In other words “when ultraviolet light shines on two metal electrodes with a voltage applied across them, the light changes the voltage at which sparking takes place.” (https://www.britannica.com/science/photoelectric-effect) This was an important finding because up until this point people where unaware of the relationship between light and electricity.

JJ Thompson, who many remember for his “plum pudding” model of the atom, discovered that the particles that were freed in the photoelectric effect were the same particles observed in the cathode rays he had been working with. His research using the cathode ray tube led to the discovery of corpuscles which we now know as electrons.

A depiction of a cathode ray tube. Courtesy of study.com

In 1902 Philipp Lenard made a shocking discovery. He found that as the frequency of light increased so to did the energy of the electrons. The expected result was that as the intensity or brightness of light increased the energy of the electron would increase. The experimental observation did not match the accepted theory of the time. So what does all this mean and who could make sense of it all? Before we get to that let’s discuss what specifically the photoelectric effect is.

What is the Photoelectric Effect?

So we know a little about the history of the photoelectric effect and we know that it has something to do with electrons and the frequency of light. So what is it exactly? Great question, glad you asked. The photoelectric effect is a phenomenon which occurs when a light of a high enough frequency is shown onto a photo-sensitive metal resulting in the ejection of electrons from that metal. If the threshold frequency is not high enough then no electrons will be ejected. The threshold frequency varies for different metals and is the minimum frequency required for electrons to be ejected.

Increasing the intensity of the light will result in more electrons being ejected provide the frequency is at or above the threshold frequency. A high intensity light at a frequency below the threshold frequency will not result in the ejection of electrons, in other words the photoelectric effect will not be observed. Increasing the frequency of light results in an increase in kinetic energy of the photons. The intensity of light had no impact on the kinetic energy of the photons only on the number of photons being ejected.

Einstein and the Photoelectric Effect

Einstein is of course known for his theory of general and special relativity, his work with quantum mechanics, and his famous equation E=mc^2. What you might not realize is that Albert Einstein was awarded the 1921 Nobel Prize in physics for “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” (https://www.nobelprize.org/prizes/physics/1921/summary/)

As you now know, Einstein was not the first to observe the photoelectric effect, so why is he so often credited and associated with it rather than Hertz, Lenard, or Thompson? He was the first to accurately describe how it occurs and to make the ground-breaking discovery relating waves and particles.

Einstein realized two facts regarding the photoelectric effect: 1) light is made of particles called photons and 2) the metal can only absorb the entire photon and not any other portion of it, think of it as an all or nothing proposition. Some of the energy of the photon that is absorbed is used to free the electron and the rest is converted to kinetic energy of the photon. The energy required to free the electron is called the work function. The strict definition of the work function is “energy (or work) required to withdraw an electron completely from a metal surface.” (https://school.eb.com/levels/high/article/electronic-work-function/32337)

What does all this mean? The results that Einstein observed were not in agreement with the accepted theory of the time. A new model of light was needed to match observation. Einstein had the insight to recognize that sometimes light acted as a wave and sometimes it acted as a particle. This was a stunning revelation that shocked the scientific community. Sir Isaac Newton thought light must be made of particles in order for it to experience reflection and refraction while Robert Hooke had argued that light had wave like behavior. Finally Einstein came along and settled the argument: light acts as both a wave and a particle.

Today we refer to this as the wave-particle duality or the dual nature of light. In general light travels as a wave and interacts with matter as particles called photons. Light is quantized meaning it is packaged in discrete units or particles which are called photons. In 1900 Max Planck, the father of quantum mechanics derived the equation for the energy of electromagnetic radiation, including light. Here is a short video describing Einstein and his contributions to the photoelectric effect: https://youtu.be/0b0axfyJ4oo

Courtesy of socratic.org

If we use the above equation and compare it to the known work function of a specific metal we can determine if the photoelectric effect will occur. The energy of the photon must be greater than the work function in order for the photoelectric effect to be observed. Interestingly enough, the electrons that were ejected from the metal end up falling back into the metal almost immediately.

What Did We Learn from the Photoelectric Effect?

We now know some of the history of the photoelectric effect and what the photoelectric effect is all about. Let’s take a moment and summarize what we learned from this scientifically significant discovery. Most importantly we leaned that classical physics can not accurately predict what happens at the atomic level. Classical physics predicted that increasing the intensity of light should increase the energy of the photons. What actually happened was that increasing the frequency of light resulted in an increase in the energy of a photon while increasing the intensity of light only resulted in an increases in the number of photons being ejected. Einstein was able to determine the energy of a photon by the equation E=hv and that the energy of the photon must be greater than the work function in order to be ejected from the metal. The photoelectric effected demonstrated the dual nature of light, that is it has both wave and particle behavior, in general light travels as a wave and interacts with matter as particles.

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Uncategorized

The Double Slit Experiment

Thomas Young performed his famous “double slit experiment in 1801. The results of this experiment demonstrated the dual nature of light. The term “dual” in the dual nature of light means that light, which is electromagnetic radiation, has both wavelike and particle properties. While the initial experiment was done with light it has since been repeated with electrons, atoms, and even molecules as large as 60 carbon atoms called a Buckyball. All of these experiments show the same dual nature results.

Common terminology

Before we get into the actual experiment and its results lets discuss some of the vocabulary. The first term we should discuss is quantized. If something is quantized then it exists only in discrete continuous amounts. Light is quantized into quanta called photons. Quanta is the fundamental size unit the smallest, indivisible portion of a property. A small packet or chunk of light, for example, which can’t be broken down any further.

Next we need to discuss some terminology about waves. These terms can be used with water waves, light waves, sound waves, basically any type of wave. There are different types of waves, transverse and longitudinal, that both have the same features. The amplitude of a wave is the height of a wave and is directly proportional to the energy of the wave. The higher the wave, the greater the amplitude. Greater amplitude corresponds to a more energetic wave. The top of a wave is called the crest and the low point of a wave is called the trough.

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An image of a transverse wave. Courtesy of study.com

Diffraction is the term used to describe how waves bend around an opening or barrier.

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Note how the pattern changes based on the size of the aperture.
Courtesy of s-cool.co.uk

Wave interference

Waves can interact with other waves or even with itself and this is called interference. There are two basic types of interference. When the peak also called crest of one wave interacts with the crest of another wave then constructive interference occurs. If the trough of one wave interacts with the trough of another wave then the same constructive interference occurs. Constructive interference results in a larger single wave, as if the two parts of the wave that interacted constructed a new single larger wave. If the crest of one wave interacts with the trough of another wave then the waves cancel one another out in a process called destructive interference.

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Examples of constructive and destructive interference. Courtesy of researchgate.net

Wave Function

The wave function in quantum mechanics is an equation that describes all the characteristics of a particle.

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Schrodinger’s wave equation.

The square of a wave function represents the probability of finding a particle at a given time and place. The square of the wave function becomes significant when trying to determine the location of an object during the double slit experiment.

The Experiment

Thomas Young designed and conducted an experiment which had shocking results. He was attempting to show that light behaved as waves rather than the accepted notion that light behaved as individual particles. Young knew that waves created interference patterns when they interacted with each other. To test his hypothesis he shined a light source at a barrier with two slits which were fairly close to one another. The light passed through the slits and shone on a screen behind the barrier. If light was acting as a wave he would expect to see an interference pattern on the screen. If light was acting as a particle he would expect to see an image of the the two slits as the particles passed through the barrier.

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A depiction of Thomas Young’s double slit experiment. Courtesy of curiosity.com

The Results of the experiment

When Young performed his experiment an interference pattern emerged on the screen behind the barrier. This seemed to confirm Young’s idea that light behaved as a wave. In the early part of the 20th century quantum mechanics was born. Scientists like Max Planck, the father of quantum mechanics and Albert Einstein argued that light behaved as both a particle, the photon and a wave. The double slit experiment was done again only this time scientists were able to fire photons one by one at the barrier. In this case one would think that the pattern that emerged would not be an interference pattern but rather a band of light at through each slit. Think of spraying paint though a two slit barrier. Bands would only be present at the site of the slits, similar to a stencil.

Image result for double slit experiment particle results
The top image is what would be expected if light behaved as particles with the photons present directly in line of sight of the slits. The bottom image is what would be expected if light behaved as a wave. The square of the wave function predicts most likely location of the photons. Courtesy of medium.com

This experiment has been repeated many times since Young’s experiment. Scientists have even repeated the experiment using electrons which they could fire one at a time to try to determine wave versus particle behavior. Even when the electrons were fired one by one, eventually an interference pattern emerged. When the electrons are fired one at a time they can not be interacting with other electrons to produce an interference pattern. This means the electrons are interacting with themselves to create an interference pattern.

Here is where it gets weird. In order to for an interference pattern to occur the barrier must have two slits. How can a single electron be aware of another slit? One solution to this question is that the electron splits and goes through each of the two slits and interacts with itself. Another alternative is that the electron does not split but goes through both slits at the same time. Both of these scenarios vary greatly from our everyday experiences dealing with Newtonian mechanics.

With advances in technology we can now perform the experiment using either wave or particle detectors to once and for all determine if an electron behaves as a particle or a wave. Quantum weirdness, however, rears it head once again. If we try to use a detector to see which route, that is which slit, the electron went through, then no interference pattern is detected and the result is consistent with particle behavior. The act of observing, in this case by use of a detector, destroys the interference pattern by collapsing the wave function. The implication is that the act of observing which slits the electrons goes through causes the interference pattern to collapse and the particle pattern to be demonstrated. If, however, no detector is used then the wave interference pattern prevails. These results demonstrate the dual nature of many things once thought of as particles and that observation at the quantum level can have a dramatic impact on the very results one is tying to measure.