Masses on springs get a lot of use in physics; you see them early in that first year of introductory classical mechanics with Hooke’s law and they come back over and over again after that. Physicists are fond of saying that basically everything in reality reduces to a mass on a spring if you squint at it the right way. I chose the tortured title for this post thinking about how a pendulum bob can be described as a mass on a spring at small angles of deflection and that Schwinger’s method, which is important to the Quantum mechanics problem I’m covering, is almost like a pendulum swinging in an ellipse at a small angle.
If you haven’t guessed, this post is back to Sakurai problem 3.19, finally –of which I’ve spoken previously. What dalliance in quantum mechanics would be complete without spending time on Schwinger’s angular momentum method? Julian Schwinger should be familiar to anyone with a background in 20th century physics since he won the Nobel Prize at the same time as Richard Feynman. His wikipedia entry shows how his approaches contrasted with those of Feynman, but he was certainly no less brilliant.
As a quick refresher, the problem is asking about two of Schwinger’s operators, K+ and K-. “What do these two operators do?”
The quantum mechanical version of a mass on a spring is the ‘quantum simple harmonic oscillator.’ This system differs from the basic ‘mass on a spring’ model in that you really can’t think about it as something moving ‘back and forth’ the way a pendulum bob can. In the quantum version, it would be most accurate to say that the bob tends to be distributed along the range of its swing and that it is more likely to be found highly compressed and highly extended, at the extreme positions of its swing, the greater the energy it contains. In this, the swinging of the pendulum bob can be broken down into a spectrum of energy eigenstates where you can describe the motion as some combination of these states. Eigenstates are, of course, the bread and butter of quantum mechanics and correspond to stationary probability waves which are not overall that different from vibrations in a guitar string –even though it would be very very wrong to draw too close of an analogy here in absence of the math. A probability wave is literally existential, not ‘vibrational’ like a sound wave.
An important structure in this version of the quantum mass-on-a-spring is the existence of the so-called ‘creation’ and ‘annihilation’ operators (a† and a), which are very central to Schwinger’s method. These operators work together in a set where one undoes the action of the other. Together, these operators allow the skilled technician to transit the eigenstate energy spectrum of the mass-on-a-spring, using the creation operator to step from one state to the next higher energy state and the annihilation operator doing the opposite, stepping down in energy between successive states. These operators work sort of like moving your finger on the frets of a guitar string, the annihilator moving toward the tuning pegs and the creator moving down the neck toward the body of the instrument. If you get too close to the tuning pegs, the annihilator can actually cause you to fall off the end of the instrument. Not kidding, really: that’s part of why it earned the name ‘annihilator.’ The creation operator, on the other hand, can get arbitrarily close to the bottom of the string and still find notes, provided you continue to have some way of plucking the string.
Now then, Schwinger’s method takes this collection of ideas and turns it on its head to produce what can only be described as a stroke of genius. This idea follows from the basic observation that if you have an object moving at a constant speed around a circular track, that you can parameterize it using two mass-spring systems at right angles to each other in the plane of the track and have a complete description of the circular motion. Literally, if you’re moving in a circle in the x-y plane, the equation describing the x-position is a harmonic oscillator, as is the one describing the y-position. Schwinger’s brilliance was simply to say, “So, why don’t we do this in quantum mechanics?”
Schwinger’s angular momentum method applies to quantum mechanical rotation: the spinning top. As a physical parameter, angular momentum tends to describe what might be thought of as the ‘strength of a rotation.’ Having more angular momentum tends to correspond to greater speed in rotation, but in quantum mechanics, it also tends to strongly influence how ‘spinning’ objects are distributed within whatever volumes they occupy by giving them distinct ‘orientations.’
Coming back to the Sakurai problem, which I’ve been orbiting at quite a distance, the operators K+ and K- manipulate a state of rotation. K+ is two creation operators and K- is two annihilation operators. Mathematically, K+ increases the binary harmonic oscillator state by one unit of total angular momentum, while K- does the opposite. If you actually consider K+ to be two creation operators (a†a†) you can see it directly in the description of the general two-oscillator eigenstate:
Here, n+ and n- are just the two numbers needed to find the address of any one eigenstate among an infinite number where one mass-on-a-spring is labeled as ‘+’ while the one perpendicular to it is labeled as ‘-‘. The ket (state) on the right |0,0> is just the ground state where some number ‘n’ applications of a† elevates you to any eigenstate in spectrum. The equation, as written, simply tells you where you put your finger on the guitar string(s) to produce any note you want. The Schwinger method is actually where to put your finger on two strings in order to produce a particular kind of rotation in 2 dimensions. Do you see K+ in this equation? If n+ and n- are equal, K+ is just the thing right in the middle!
So, if you read my previous post (and survived far enough to read this), you’ll know that the Sakurai 3.19 problem was asking about matrix elements. Since ‘operators’ in quantum mechanics take states and convert them into other states, the structure of an operator is in a matrix where each element tells how one eigenstate is referenced to another during whatever transformation that operator is supposed to mediate. You could write K+ and K- in such a way that you can tell how any one state is converted to any other state by action of the operator.
This will almost certainly lose readers, but if you don’t actually like physics, you probably won’t like this blog anyway. As I worked problem 3.19, here are the forms I found for the matrix elements of the K+ and K- operators acting on the space of all harmonic oscillator eigenstates.
Solving the matrix element problem is actually quite simple and, as I worked the problem, I delayed executing this step until I had slogged exhaustively through the Schwinger method and was certain I knew what was up. To get the answer listed here, you just take the form for K+ or K- as presented in the previous blog post and act them on a particular ket |n’+,n’->, then sandwich with a particular bra <n+,n-|. This is like looking for the expectation value, but for only one element out of an entire matrix. The primed value of each ‘n’ is understood to be a different number from the unprimed form. Talk of Bra and Ket will sound weird to anyone who has never encountered the Dirac ‘bracket’ notation, which denotes eigenstates as ‘bra’-‘ket’ where the ‘bra’ is the conjugate transpose of the ‘ket’ and the ‘ket’ is a representation-free form of a particular quantum mechanical eigenstate. A matrix element is just a ‘bra’ sandwiched with a ‘ket’ and after the kets and bras are all gone, what’s left are Kroenecker deltas that describe where a particular element is located in a matrix since you only get non-zero elements where the indices of the ‘delta’s are equal. This form can be a very handy alternative to the two dimensional lists that every linear algebra student has learned to hate… in this case, the matrices are infinitely large in their two dimensions and you could never actually write them. With the delta notation, you only need to say which matrix elements are non-zero, thus reducing a matrix which can’t fit on this Earth into a single line expression. What’s written above are both two-matrix things where each matrix acts on one of two ket-spaces and each ket space is the series of eigenstates for one of the harmonic oscillators that Schwinger used in his description of angular momentum.
It certainly may not look it and it definitely doesn’t sound like it, but this does all come down to rotational quantum mechanics.
As an apology to any reader who may stray here, I’m still deciding exactly what the voice of this blog will be. My initial vision was to be as broadly friendly as possible, but I’ve ping-ponged back and forth on this. I expect that there will be articles involving the far more crowd-friendly subject of ‘physics and/or science in popular culture,’ but I also had a desire to create a space where I could store my practicing. A formal truth about education in general is that you don’t keep what you make no effort to keep, which means that if you don’t actively practice, things you care about can disappear out of your life forever… particularly in a subject as hard as physics. I think we live in a world where it seems like everybody wants to believe they have a high level understanding of everything without having actually invested sufficient effort to attain even a basic level understanding of anything. One thing I want here is to serve as an example of what it takes to maintain skill and maybe make people think twice about what’s needed to be good at an intellectual pursuit. I have a desire to one day read an article quoting someone like Jaden Smith or Terrence Howard where they say “I was in this class at school learning how to do physics and it was so cool; I finally understood this and I wish everybody did!” but I fear that this will never happen. I have notebooks that are literally crammed with my physics practicing and I wish I could open them to the world and convince people to stop being afraid of playing at math. In some ways, it’s like working a crossword puzzle and I think anybody who has invested deeply probably can do as well as me. I’m no genius, I’m just a walking testament to hard work and due diligence.