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8.051: Quantum Physics II Introduction 1 February 3, 2020 8.051: Quantum Physics II Lecturer: Professor Barton Zwiebach Notes by: Andrew Lin Spring 2020 Introduction An MITx page for this class will be up by this afternoon. That’s where most of the content for this class will come from (regular class meetings are recitations). Lecture videos on the MITx page are divided into 7 to 25 minute sections, each covering some particular topic. After each block, there are some “lecture exercises:” this class is structured in a way so that before each recitation, we should have finished going over the material designated for it. We’ll also have real problems in problem sets – there are about 11 of these over the semester, and they’re also going to be online. Homework is the best predictor of how we’re going to do in this course – if we basically understand the homework completely, we’re extremely likely to do very well. Homework on the MITx page is given to us problem by problem, part by part, but we’ll also be given a PDF of the problems that we can complete like a regular problem set. And we’ll have some tests in the evenings of the exams instead of recitations on those days. This class also has a graduate TA (Matt Hodel) and an MITx site administrator (Michelle Tomasik), and we should ask them if we have questions. Grading is 10 percent lecture questions, 30 percent online homework, 15 percent for each of the two midterms (March 11 and April 22), and 25 percent from the final. It’s pretty likely we’ll be able to drop one problem set. Because this is an online course, we have a bit more flexibility - if we need a one-day extension, it’s easy for the system to give that. Beyond that, we should go through S3. In principle, no textbook is required for this class, but Griffiths and Shankar are good references – it’s good to read things in a different way. There will be lecture notes, and students seem to like them. MIT has been a bit reluctant to accept that online classes are equivalent to normal lecture-based classes, so in general, coming to recitation is generally mandatory. When this class runs in previous years, a rule has not had to be imposed, but attendance sheets are passed around. It will make a difference, and it’s important! Scheduled office hours will be posted by the end of today – there will be about 2 to 3 hours per week. Other times are also welcome by appointment. 1 February 3, 2020 8.05 is being changed a bit. The issue being corrected is that the addition of angular momentum is usually the last topic of the semester, and it’s always a little rush even though it’s a bit difficult. So we’ll shift it a bit forward, and there will be at least three more lectures at the end about the density matrix (which is a topic that jumped between 8.05 and 8.06 and eventually got dropped from both). So the aspects of 8.05 that were 8.04 review will go away: there 1 will be an 8.04 review section in the MITx page, but it won’t be covered anymore. We can use it at our discretion, or we can proceed to unit 1. Fact 1 This class assumes pretty good knowledge of 8.04, so we’ll move to new material very quickly. Wednesday’s exercises will not be due, but next week’s Monday, Wednesday, and Friday lectures will be graded. We’re really hitting the ground running. Fact 2 8.051 is a class based on online lectures, so we go through the lectures at our own pace. However, I will still transcribe notes from each lecture and insert them in the appropriate spots of the class: lectures 1 and 2 will be placed in the notes after this recitation (that is, before recitation 2), and all future lectures will be placed before their corresponding recitations. This means that the flow of the class material approximately corresponds to the flow of the notes. With the last half hour or so of this first recitation, we’ll talk a bit about states in quantum mechanics. Example 3 (Two-state system) Consider a quantum system with two basis states + and −. This means that any state in this system can be written as a superposition of the two basis states. (Recall that the notation tells us that these are [complex-valued] wavefunctions.) Because these are wavefunctions, they are normalized in such a way so that h +; +i = 1; h −; −i = 1 and it’s nice if the two basis states are orthonormal, so that h +; −i = 0: Recall that we often define the inner product between two wave functions as Z hϕ(x); (x)i = dxϕ∗(x) (x); R so that hϕ, ϕi = dxjϕj2 = 1. Question 4. How many real parameters do I need to characterize a general state in this quantum system? That is, how many real parameters are needed to describe the inequivalent states (that is, states that aren’t proportional by some complex constant)? (Survey: 1, 3, 6, 3 votes for 1, 2, 3, and 4.) Let’s go through and solve this: a general superposition looks like (x) = c+ + + c− −; c+; c− 2 C: But many of these states are physically equivalent: right now, we have two complex numbers, which means four real parameters. But a state like is equivalent to 2 , which is equivalent to (1 + 3i) , and so on: we normalize the state to get the physics! So we need h ; i = 1, which means hc+ + + c− −; c+ + + c− −i = 1: 2 We can expand out the inner product, noticing that constants from the left term come out with a conjugate, to find that ∗ h i ∗ h i ∗ h i ∗ h i c+c+ +; + + c+c− +; − + c−c+ −; + + c−c− −; − = 1: And now using our orthonormality conditions, this tells us that 2 2 jc+j + jc−j = 1: This is a real constraint, so it removes one parameter. But are we done? Let’s use our new information to rewrite iα+ iα− = e a + + e b − 2 2 where a; b ≥ 0 and α+; α− are real. So now our condition tells us that a + b = 1, and a; b ≥ 0, but now we can see another constraint: remember that we can normalize in the phase direction as well! So this is actually equivalent to i(α−−α+) = a + + e b −; which means we really only have 2 free parameters: one for the phase difference and one for the magnitude of +. In summary, let’s rewrite this in a slightly nicer way: iβ = a + + e b −; where a; b ≥ 0; a2 + b2 = 1, which can also be written as iβ = cos(χ) + + e sin(χ) −; 2 2 π where β [0; 2π] and χ 0; 2 (because both cos and sin have to be positive). Here’s one way to never forget the characterization of this state: think about spherical coordinates. Every point has a coordinate θ and ϕ, and in spherical coordinates we have θ 2 [0; π] and ϕ 2 [0; 2π]. So we could write θ iϕ θ = cos + e sin −: 2 + 2 So we can think of the space of states as the surface of a 2-sphere in 3-dimensional space: for every direction of our 2-sphere, we have a state, which is a linear combination of our “up state” + and “down state” −. 2 The Variational Method and Introduction to Stern-Gerlach We’ll start this class by discussing the variational problem, which might be new to some of us. This is connected to a field called the calculus of variations – this is a pretty complicated topic, and we won’t get into a lot of the difficulties. The main idea of calculus of variations is to look at maxima and minima of a functional instead of a function. (In ordinary calculus, we try to find a point where some quantity is maximized, but here we’ll be trying to find a function that maximizes some quantity.) So things are a bit more challenging! 3 Fact 5 Calculus of variations seemed to have originated from Newton. The question he considered was to start with a cross-sectional area and try to taper it in a way such that the resistance from a viscous fluid flowing through is minimized. And this is complicated, because our ultimate goal is to find a shape rather than just a single number or point (as we do in ordinary calculus). Fact 6 Another famous problem, called the Brachistochrone problem, asks for us to design a curve from point A to point B in a plane such that an object (under the influence of gravity) traverses the path the fastest. This was a difficult problem for many mathematicians back then, including Leibniz, Newton, andthe Bernoullis! So how is this calculus of variations idea related to quantum mechanics? It seems that special functions are often the minimizer of some important quantity, and we have some special functions in quantum mechanics: the energy eigenfunctions. It’s then natural to ask whether there’s a quantity that they minimize, and indeed the answer is yes. So let’s start setting up our problem: we’re trying to solve the energy eigenstate equation H^ = E ; where H^ is some (any-dimensional) Hamiltonian and is a wavefunction (which can be of a vector ~x or just a single x in one dimension).
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