Hello there! Welcome to lecture 32: quantum physics.
The wave-like nature of matter brings about some bizarre phenomena that we don’t experience on the macro-scale in classical physics. This lecture will introduce you to some of the biggest topics in quantum physics.
Each of the following concepts will be discussed in this video: electron waves, quantum mechanics, the observer effect, quantum tunneling, and quantum entanglement.
We discussed atomic models in lecture 11. From Democritus to Rutherford to Bohr, we learned how each of the atomic models came to be, and what experimental evidence led to changes in how we understand atoms over the past several centuries.
Bohr’s model of electrons orbiting atoms in discrete orbits given by energy levels was prompted by the measurements of atomic emission spectra that exhibit discrete frequencies, discussed in lecture 30.
However, treating electrons as particles that orbit a nucleus such as planets orbit stars is not an accurate depiction of what we know to be true about electrons. Firstly, planets can take on an orbit of any arbitrary distance from a star. This is not true for electrons orbiting the nucleus of an atom. They do not orbit at arbitrary distances, but at discrete and well-defined distances that can be calculated by recording atomic emission spectra.
Secondly, a classical model of an electron orbiting a nucleus would indicate an acceleration of a charged particle. Maxwell’s theory of electromagnetism states that accelerating electrons emit light. Bohr’s model indicates that light is only emitted when electrons decay from high-energy states to low-energy states.
These shortcomings of a purely “planetary” and particle-like model of electrons can be explained by understanding the wave-nature of electrons. In lecture 31, experimental validation of matter waves was discussed. Electrons exhibit wave-like nature. That’s true when electrons are orbiting an atom as well as when electrons travel freely through space.
Consider electrons to act like standing waves when they orbit the nucleus. Only waves that have a particular radius, given by the constructive interference of waves, can be supported in orbits around the atom. This demo shows how this works. If each orbit is constituted by an integer value of wavelength, which would support constructive interference, then those orbits have very well-defined sizes. This is analogous to the standing waves we discussed in lecture 19.
In fact, electron waves are more complicated than this, but this conceptual understanding helps to describe the quantum picture of electrons, and addresses the drawbacks of the classical model of electron orbits.
Quantum mechanics is the study of mechanics, the topics we discussed from lectures 2 to 10, when things become so small that classical physics breaks down and stops working adequately. Quantum objects have three major differences from classical objects. First, the properties of quantum objects (such as energy and momentum) appear in discrete, quantized values. Second, quantum objects exhibit wave-particle duality. And third, quantum objects have fundamental limits to the accuracy we can have in predicting certain quantities, given by the uncertainty principle.
What we find in quantum mechanics is that quantum objects can be described with a mathematical quantity known as the wave function. If the wave function is multiplied by itself, we can calculate the probability of the outcome of different measurements that can be made on a quantum object. For example, a wave function can be calculated regarding the energy of an electron, or the position of a photon, or other properties of other quantum objects.
Instead of obtaining exact values of energy, position, and momentum, like we can calculate in classical physics, the wave function gives us probabilities. A wave function for an electron orbiting a nucleus will give us probabilities that we could find an electron existing at any one position around the nucleus of an atom.
To solve for this wave function, Schrödinger’s equation is used. I’d like to point out in advance that while I am going to display Schrödinger’s equation in this video, it is for demonstration purposes only, and in no way will I expect you to use this equation in class! It is extremely complicated and goes way beyond the math I expect you to be comfortable with. That’s the point: quantum mechanics is a very difficult model of physics that we don’t use on the macro scale because we don’t have to! Why use this complicated equation when we can use simple equations like F equals m times a?
Here goes. Schrödinger’s equation is very complicated. The wave function is represented by the capital Greek letter psi. The left-hand side of the equation contains terms that have to do with how the wave function changes in space. The right-hand side of the equation contains terms that have to do with how the wave function changes in time. We see Planck’s constant divided by two pi, h-bar, twice. M is the mass of the quantum object. The upside-down triangle and squiggly d are calculus operators. I is the square root of negative one, and leads to wave properties when doing calculus.
As far as atoms go, Schrödinger’s equation has only been solved by hand for the hydrogen atom, which consists of one proton and one electron. It can also be solved by hand for individual particles such as a single proton, or electron, in different environments. When using Schrödinger’s equation to model complicated particles or atoms that have more elements than hydrogen, then computers need to be used to numerically calculate wave functions.
The major limitations of Schrödinger’s equation are that it is very complicated to use, to the point where solving it by hand becomes quickly impossible, and that at the end, the wave function only tells us probabilistic information about quantum particles. The quantum universe is indeed much different from the classical universe!
The observer effect
The observer effect occurs when measuring a quantum process. When making an observation, we influence that process in some way and therefore affect the outcome. This doesn’t actually only happen in quantum systems, it also happens in classical systems. To measure the tire pressure in my car or bicycle tire, I actually have to let some of the air in the tire out. This changes the measurement. Whatever I measure is going to be slightly less than the pressure that existed in the tire before I measured it.
Most of the time, our influence on the outcome of a classical or macro-scale measurement is negligible. When using an ultrasonic sensor to measure the speed of a cart in a physics experiment, a sound wave is bounced off of the cart. There is going to be energy and momentum transfer between the sound wave and the cart, but the speed and momentum of the cart will not change to any noticeable degree as a result of using sound waves to measure it. We can essentially ignore the observer effect in cases such as this.
In particle physics, to measure the speed of an electron requires hitting it with a photon. Unlike the sound wave and the cart, an electron and a photon are both subatomic particles that have very similar energies and momenta. When I measure the electron by hitting it with a photon, I am going to be affecting the electron to a great degree.
There’s a great quote in the book Perpetual Motion by Alec Thompson Stewart that I think does a great job summing this up:
“Perhaps the most gentle method of searching for an electron in an atom in order to measure its speed is to shoot another electron through the atom. If the exploring electron hits an electron in the atom, it is deflected and from the angle it may be possible to obtain the velocity of the electron it hit. From the atom’s point of view, this technique is not exactly subtle. It would be like closing one’s eyes and driving full speed through a street intersection to find out whether traffic was moving on the cross street. If a collision occurred, one could assume that there was a car on the cross street.”
When we measure a quantum object, we both affect its outcome and cause the wave function to take on one particular value. This is known as collapsing the wave function. Whereas prior to our measurement an electron could have one of many different values for momentum, after measurement, we know exactly what that value is. The wave function has collapsed.
Relevant to this topic is a thought experiment known as Schrödinger’s cat. First, I’d like to point out that this is a thought experiment, and was not actually performed, nor were any cats harmed. The thought experiment takes a cat, a flask of poison, a Geiger counter, and a radioactive source in a box. If the radioactive source emits a radioactive particle that’s measured by the Geiger counter, the flask of poison is shattered and the poison is released, killing the cat.
Without opening the box, is the cat alive or is the cat dead? One interpretation of quantum physics states that, after a time, the cat is both alive and dead. It exists in a so-called superposition of both states. It is not until the box is opened that we observe the cat to be in one or the other state, collapsing the wave function.
There are a lot of arguments and interpretations of this thought experiment, and it has led to much rigorous debate between physicists.
Consider throwing a bouncy ball at a wall. Assuming you don’t throw the ball so fast that it smashes through the wall, it will predictably bounce back to you. This is a classical particle: a large object that obeys the laws of classical physics. Its wave nature is so trivial that it can be completely ignored.
If instead you throw an electron at a wall, it is possible that the electron will tunnel through. During that process, the wall won’t be damaged or changed, but the electron will simply pass through it. This is due to the wave-like nature of quantum objects. This phenomenon is known as quantum tunneling.
This simulation shows the wave function of a quantum object as it travels toward a barrier. While most of the object reflects off of the barrier, a small amount tunnels through.
If you think this sounds bizarre, consider that the solid-state memory used in your USB drives, smart phone, and possibly computer or laptop, uses quantum tunneling to store data in its flash memory. Electrons tunnel through an insulating barrier and get stored to save your files, text messages, and photos, as binary data.
Quantum entanglement describes the process during which two or more quantum objects interact in such a way that their quantum properties become interrelated. For example, two photons can be generated such that one has a vertical polarization and the other has a horizontal polarization. The two photons have mutually perpendicular polarizations. This means that if I were to measure the polarization of one of the photons, I will instantly know the polarization of the other, as the two are perpendicular to each other.
The aspect of quantum entanglement that is particularly fascinating has to do with the implications of having simultaneous knowledge of both particles. Say I entangle two photos in polarization. I keep one photon traveling in an optical setup nearby, and send the other one to a friend on another planet, or even on the other side of the Earth. If I measure the polarization of my photon, I instantly know the polarization of the other photon. Scientists were initially concerned that this seems to imply that knowledge moves faster than the speed of light. If the second photon is on Jupiter, I know its polarization without having to travel all the way there to measure it!
The fact is, however, that by the time I call my friend with the second photon to tell them the results of the polarization measurement, the transmission of that information to my friend will obey the universal speed limit. My friend will not know the outcome faster than light speed, as my message must travel at or below the speed of light to reach them. While I may know information about both particles, that information cannot be transmitted faster than the speed of light.
Other properties that can be entangled include momentum, position, and particle spin. This is an active area of research for use in quantum computing. Due to the amount of information that can be encoded into entangled quantum particles, the hope is that quantum computing can solve problems that would take classical computers an enormous amount of time to solve.
Thanks for taking the time to learn about quantum physics! Until next time, stay well.