The wavefunction \(\psi(L,t)\) is confined to a circle whenever the eigenvalues L of a particle are only nonzero on the points along a circle. When the wavefunction \(\psi(L,t)\) associated with a particle has non-zero values only on points along a circle of radius \(r\), the eigenvalues \(p\) (of the momentum operator \(\hat{P}\)) are quantized—they come in discrete multiples of \(n\frac{ℏ}{r}\) where \(n=1,2,…\) Since the eigenvalues for angular momentum are \(L=pr=nℏ\), it follows that angular momentum is also quantized.
The Eigenvectors of any Hermitian Operator must be Orthogonal
In this lesson, we'll mathematically prove that for any Hermitian operator (and, hence, any observable), one can always find a complete basis of orthonormal eigenvectors.
Origin of Structure and "Clumpyness"
This video was produced by David Butler.\(^{[1]}\) For the transcript of this video, visit: https://www.youtube.com/watch?v=JpNaIzw_CZk
Fundamental questions which have been asked since the beginning of humanity are: How did structure arise in the cosmos? Why is there something rather than nothing? Why is there “clumpyness” in the cosmos (in other words, why is there more stuff here than there)? For a long time, answering such questions would have been impossible. But in the past few decades cosmologists, by using the scientific method and advanced technology, have unraveled these deep and profound mysteries which humans have long pondered since their origin. Let’s start out by answering the second question: Why is there something rather than nothing? The fundamental principles of quantum mechanics predict a myriad of mathematical and physical consequences. One of them is the time-energy uncertainty principle which predicts that over any given time interval, the observed energy of a particle is always uncertain and will be within a range of values because you must give it a “kick” to observe it. The energy of something can never be exactly zero (because then \(ΔE=0\) and the time-energy uncertainty principle would be violated). Therefore, not even the vacuum of empty space can have no energy. This energy of empty space can be transformed into mass (a property of matter) according to Einstein’s mass-energy equivalence principle and the rules of particles physics. Since the possible energy values of any region of space are random within a certain range of values, it follows that the distribution of energy, mass, and matter also must be random throughout space. But if the fundamental principles of quantum mechanics and the time-energy uncertainty principle in particular explain the origin of matter and energy in the universe, then how did the cosmos attain structure and clympyness? Over the next few paragraphs, we’ll begin a discussion which starts with analyzing the predicament (namely, applying quantum mechanics alone predicts that the universe should have been too uniform for galaxies and other structures to arise); then, after that, I’ll take you through a very brief account of how the universe developed structure and clumpyness.
In the earliest epochs of our universe, all of the matter and energy existed in the form of virtual particles popping in and out of existence. If the distribution of this matter was random then, on average, there would be just as many virtual particles over here as over there. But in order for structure to arise in the cosmos, the distribution of matter must start out clumpy and not almost completely uniform as the uncertainty principle predicts. The uncertainty principle predicts that the distribution of matter and virtual particles throughout space starts out more or less uniform. This seems to contradict today’s observations since if all of the matter density is initially uniform, it’ll more or less stay uniform. The Big Bang theory is widely regarded as one of the greatest triumphs in science of the 20th century. It successfully predicted an enormous range of things: the CMBR; where the light elements came from as well as the correct amounts and proportion of each of them; and the observed redshifts and recessional velocities of distant galaxies. But there were many things which the Big Bang theory could not account for such as how the universe got to its present size, how the recorded temperatures of the CMBR are so uniform, and how the CMBR has slight fluctuations to one part in \(10^5\). Cosmologists generally agree that we must postulate the existence of a physical mechanism which correctly predicts all of these observed features. This is precisely what inflationary theory accomplishes. Many have criticized the physical mechanisms postulated by inflationary theory as ad hoc despite the success in inflationary theory of predicting what has been experimentally observed.
The universe is presently 13.8 billion years old and we are causally connected with all of the matter and energy within a sphere whose radius is 13.8 billion light years. When the universe was only one second old, its temperature was \(10^{10}K\); the photons buzzing around back then were 3.7 billion times hotter than their present temperature today of \(2.725K\). By Boltzman’s relationship \(E∝T\), Plank’s relationship \(E∝λ\), and the relationship between wavelength and the scaling factor (\(λ∝a^{-1}\)), it follows that the entire observable universe must have been 3.7 billion times smaller than it is today—or about only 3.7 light years in radius. Back in that distant epoch, since the universe was only one second old, photons emitted by charged particles could, at most, have travelled a distance of one light second. The gravitational field generated by energetic particles propagates at the speed of light and, therefore, would have had enough time to only spread out a distance of one light second. Any signal emitted by an energetic particle therefore could not travel farther than one light second and could not communicate with other energetic particles more than one light second away. Therefore, any pair of energetic particles separated by more than one light second from one another were causally disconnected.
The causal disconnection between matter and energy in a region of the universe 3.7 light years in radius when it was only one second old is significant because it implies that the energy density of regions separated by more than one light-second should be radically different. Therefore, the CMBR should not be so uniform. However, it successfully explains the origin of the observed fluctuations in matter and energy density: all of the particles within very small regions move towards one another due to gravity without effecting distant regions. This is how the first fluctuation in matter density originated. The clumpyness of matter was initially very slight; but over enormous time intervals (billions of years), gravity amplified this clumpyness to form the cosmic web, filaments, galaxy superclusters and clusters and groups, starts, planets, moons, comets, asteroids, and all of the large-scale structure and clumpyness we see in the universe today. On a medium size scale, the force of electromagnetism pulls and subsequently binds together atoms and molecules to form complex chemistry—the basis of all living organisms. This is the origin of structure and clumpyness on a medium size scale. The strong nuclear force binds together protons and neutrons in order to keeps atoms held together and form structure and clumpyness on an even smaller scale. The slight non-uniformities in matter density imparted by quantum fluctuations eons ago and three fundamental forces (gravity, electromagnetism, and the strong force), together, seeded and eventually developed all of the structure and clumpyness in the cosmos we see today.
This article is licensed under a CC BY-NC-SA 4.0 license.
References
1. David Butler. "Classroom Aid - Cosmic Inflation". Online video clip. YouTube. YouTube, 11 September 2017. Web. 11 November 2017.
2. Gott, Richard. The Cosmic Web: Mysterious Architecture of the Universe. Princeton University Press, 2016.
3. Goldsmith, Donald; Tyson, Niel. Origins: Fourteen Billion Years of Cosmic Evolution. Inc. Blackstone Audio, 2014.
How initial states of definite energy change with time
Let’s ask the question: do the wavefunctions which take the particular form \(\psi(x,t)=F(t)G(x)\) satisfy Schrödinger’s time-dependent equation? We can answer this question by substituting \(\psi(x,t)\) into Schrödinger’s equation and check to see if \(\psi(x,t)\) satisfies Schrödinger’s equation. Schrödinger’s time-dependent equation can be viewed as a machine where if you give me the initial wavefunction \(\psi(x,0)\) as input, this machine will crank out \(\psi(x,t)\) and tell you how that initial wavefunction will evolve with time. We are asking the question: if some wavefunction starts out as \(\psi(x,0)=F(0)G(x)\), is \(\psi(x,t)=F(t)G(x)\) a valid solution to Schrödinger’s equation? We know that \(\psi(x,0)=F(0)G(x)\) is a valid starting wavefunction. It is easy to imagine wavefunctions \(\psi(x,0)=\psi(x)=F(0)G(x)=(constant)G(x)\); this is just a wavefunction \(\psi(x)\) that is a constant times some function of \(x\). But does \(\psi(x,t)=F(t)G(x)\) satisfy Schrödinger’s equation? Let’s substitute it into Schrödinger’s equation and find out
$$iℏ\frac{∂}{∂t}(F(t)G(x))=\frac{-ℏ^2}{2m}\frac{∂^2}{∂x^2}(F(t)G(x))+V(x)F(t)G(x).\tag{1}$$
This can be simplified to
$$iℏG(x)\frac{d}{dt}(F(t))=F(t)(\frac{-ℏ^2}{2m}\frac{d^2}{dx^2}G(x)+V(x)G(x)).\tag{2}$$
The term \(\frac{-ℏ^2}{2m}\frac{d^2}{dx^2}G(x)+V(x)G(x)\) is just the energy operator \(\hat{E}\) acting on \(G(x)\); thus we can rewrite Equation (2) as
$$iℏG(x)\frac{d}{dt}F(t)=F(t)\hat{E}(G(x)).\tag{3}$$
Let’s divide both sides of Equation (3) by \(G(x)F(t)\) to get
$$\frac{iℏ}{F(t)}\frac{d}{dt}F(t)=\frac{\hat{E}(G(x))}{G(x)}.\tag{4}$$
Since the left hand side of Equation (4) is a function of time and the right hand side of Equation (4) is a function of position, it follows that
$$\frac{iℏ}{F(t)}{d}{dt}F(t)=\frac{\hat{E}(G(x))}{(G(x)}=C\tag{5}$$
where \(C\) is a constant. From Equation (5), we see that
$$\hat{E}(G(x))=CG(x).\tag{6}$$
In other words, \(G(x)\) must be an eigenfunction of the energy operator \(\hat{E}\) with eigenvalue \(C=E\). Thus, we can rewrite Equation (6) as
$$\hat{E}(\psi_E(x))=E\psi_E(x).\tag{7}$$
What this means is that the initial wavefunction \(\psi(x,0)=F(0)G(x)=(constant)\psi_E(x)\) must be an energy eigenfunction to satisfy Schrödinger’s equation. (Note that a constant times a wavefunction corresponds to that same wavefunction.) We also see from Equation (5) that
$$\frac{d}{dt}(F(t))=\frac{E}{iℏ}F(t).\tag{8}$$
The solution to Equation (8) is
$$F(t)=Ae^{Et/iℏ}.\tag{9}$$
It is easy to verify this by plugging Equation (9) into Equation (8). \(\psi(x,t)=F(t)G(x)\) is indeed a valid solution to Schrödinger’s time-dependent equation if \(G(x)=\psi_E(x)\) and \(F(t)=Ae^{Et/iℏ}\). If a system starts out in an energy eigenstate, then the wavefunction will change with time according to the equation
$$\psi(x,t)=Ae^{Et/iℏ}\psi_E(x).\tag{10}$$
Schrödinger’s equation describes how a wavefunction evolves with time if the system is “undisturbed” without measuring the system. If we measure the energy \(E\) of a system and collapse its wavefunction to the definite energy eigenstate \(\psi_E(x)\), and then simply just “leave the system alone” for a time \(t\) without disturbing it, the wavefunction \(\psi_E(x)\) will change to \(Ae^{Et/iℏ}\psi_E(x)\). But multiplying \(\psi_E\) by \(Ae^{Et/iℏ}\) does not change the expectation value \(〈\hat{L}〉\). Also, if \(|\phi⟩\) is any arbitrary state, then \(|〈\phi|\psi_E(x)〉|^2\) is the same as \(|〈\phi|Ae^{Et/iℏ}\psi_E(x)〉|^2\). Thus the probability of measuring any physical quantity associated with the system does not change with time. For an atom in its ground state with a definite energy \(E_0\) which is not disturbed, \(〈\hat{L}〉\) will not change with time. You are equally likely to measure any particular position, momentum, angular momentum, and so on at any time \(t\).
CRISPR-CAS9 Gene Editing
This article is a technical discussion of how the CRISPR/CAS9 system can be used to modify the genome of a mouse.
P-Series Convergence and Divergence
In this article, we discuss various different ways to test whether or not a p-series diverges or converges.
Derivation of Snell's Law
The law of reflection had been well known as early as the first century; but it took longer than another millennium to discover Snell's law, the law of refraction. The law of reflection was readily observable and could be easily determined by making measurements; this law states that if a light ray strikes a surface at an angle \(θ_i\) relative to the normal and gets reflected off of the surface, it will be reflected at an angle \(θ_r\) relative to the normal such that \(θ_i=θ_r\). The law of refraction, however, is a little less obvious and it required calculus to prove. The mathematician Pierre de Fermat postulated the principle of least time: that light travels along the path which gets it from one place to another such that the time \(t\) required to traverse that path is shorter than the time required to take any other path. In this lesson, we shall use this principle to derive Snell's law.
Harvesting Resources from Saturn and Titan
After humanity has colonized and begun terraforming Mars, the next likely destination would be Saturn's moon Titan. One of the drawbacks of Mars is that it lacks nitrogen which is needed to grow food and also to create a breathable atmosphere. Enormous aerostats or NIFT spacecraft could harvest nitrogen and other resources from Titan's atmosphere and then transport these resources to a nuclear propulsion spacecraft. Given that Titan's gravity is only one-sixth as strong as that of the Earth's, we could use Titan's indigenous resources to construct a space elevator where nuclear propulsion spacecraft could be launched from. These spacecraft would carry their payload to Mars and, after arriving, all of that nitrogen could be deposited into Mars' atmosphere. It would also be very useful to send nitrogen to the Moon and to settlements along the asteroids in the inner-asteroid belt where nitrogen could be used for growing food. And after settling Titan, humans would likely go on to harvest helium-3 (which could be used to power nuclear fusion spacecraft) from Saturn and the other gas giants in the solar system which will be used to power nuclear powered spacecraft to the Kuiper belt, the Oort cloud, and perhaps even to the stars..
How Fast Does Water Rise Up a Cone?
If water is being poured into a cone at a constant rate, what is the rate-of-change of the height of the water inside of the cone with respect to time? To answer this question, we'll need to use the chain rule.
Chaos and Fractals
According to Newtonian classical mechanics, if you knew the initial conditions of the entire universe (meaning that you know the initial positions and momenta of every particle in the universe) you could predict the entire past and future of the whole universe with infinite precision. But during the nineteenth-century, mathematicians studying the three-body problem—a problem which alluded Newton—concluded that the behavior of the solar system becomes completely unpredictable after sufficiently long periods of time. The reason why this was the case was because of something known as sensitivity to initial conditions. In the 1950s, an MIT professor named Edward Lorenz discovered that the weather, like the solar system, loses all predictability after a sufficient length of time (about one month). This lead to the birth of chaos theory. But, despite the apparent randomness of chaotic phenomena, it turns out that there is an underlying geometric order to such phenomena called fractal geometry. We shall introduce chaos theory and fractal geometry in this lesson.
Introduction to Partial Derivatives
In previous lessons, we learned how the derivative \(f'(x)\) gives us the steepness at each point along a function \(f(x)\). In this lesson, we'll discuss how using the concept of a partial derivative we can find the steepness at each point along a surface \(z=f(x,y)\). To find the partial derivative we treat one of the variables as a constant and then take the ordinary derivative of \(f(x,y)\). Using this concept, we can specify how steep a surface \(f(x,y)\) is along the \(x\) direction and along the \(y\) direction at each point along the surface. In other words, for every point along the surface, there is a steepness of the surface associated with both the \(x\) and the \(y\) directions at that point.
Quasars
In this article, we'll discuss the history behind the discovery of one of the most exotic and remarkable objects ever discovered in science—a quasar. Quasars consist of an accretion disk made of ultra-hot gas and dust surrounding a supermassive black hole; two immense strands of super-heated plasma extended in a direction perpendicular to the disk for millions of the light-years through the mostly empty void of intergalactic space. Quasars were ubiquitous in the early, young universe and were located at the center of most galaxies; but today, most quasars are gone because all of the matter comprising the accretion disk eventually got gobbled up by the super massive black hole. For example, our home galaxy—the Milky Way—now only has the left-over remnant of a quasar at its center—a super massive black hole.
Volume of an Oblate Spheroid
In this lesson, we'll discuss how by using the concept of a definite integral one can calculate the volume of something called an oblate spheroid. An oblate spheroid is essentially just a sphere which is compressed or stretched along one of its dimensions while leaving its other two dimensions unchanged. For example, the Earth is technically not a sphere—it is an oblate spheroid. To find the volume of an oblate spheroid, we'll start out by finding the volume of a paraboloid . (If you cut an oblate spheroid in half, the two left over pieces would be paraboloids.) To do this, we'll draw an \(n\) number of cylindrical shells inside of the paraboloid; by taking the Riemann sum of the volume of each cylindrical shell, we can obtain an estimate of the volume enclosed inside of the paraboloid. If we then take the limit of this sum as the number of cylindrical shells approaches infinity and their volumes approach zero, we'll obtain a definite integral which gives the exact volume inside of the paraboloid. After computing this definite integral, we'll multiply the result by two to get the volume of the oblate spheroid.
Our Future as Cyborgs
Undoubtedly, if our wisdom and foresight rises to be commensurate with our science and technology, the future of humanity in the 21st century is Utopian. We will have re-engineered the surface of the Earth with cities, transportation and communication systems, and new energy infrastructure which are designed and constructed to have optimal efficiency according to known science. We will have also spread across much of the solar system and, perhaps, have sent robotic spacecraft off to the nearest star system, Alpha Centauri. Aside from re-engineering the Earth and other worlds in our solar system, we will also likely re-engineer ourselves as we merge with our technology and machines. This will be the subject of discussion in this article.
Gravitational Force Exerted by a Disk
To find the gravitational force exerted by a disk on a particle a height \(h\) above the center of the disk, we must use Newton's law of gravity and the concept of a definite integral.
Shkadov thruster
A Shkadov thruster is a type of megastructure which involves constructing a gargantuan orbital mirror next to a star. In this lesson, we'll start off by discussing how such a stellar engine works. The orbital mirror is placed in a position next to a star where it acts as a statite: that is, the star's gravity acting on the mirror is canceled out by the star's radiation pressure acting on the mirror. This allows the mirror to stay in a position that is stationary relative to the star's surface. The mirror bounces some of the star's light back at itself; when that reflected light collides with the star, it exerts a thrust on the star which causes it to accelerate and move. This will bring us to the second main focus of this lesson: namely, what are the possible uses of a Shkadov thruster? As we'll discuss, since a Shkadov thruster can move the star and since all of the planets, moons, comets, and asteroids in the star system is gravitationally bound to the star, not only does the star move but the entire solar system moves away also. In the distance future, our Sun will eventually die. But we might be able to use a Shkadov thruster to move the Earth to another solar system, but this would take many millions of years.