Quantum Mechanics: Commutators & Uncertainty - OPTI 511, Spring 2009 (Problem Set 3), Assignments of Chemistry

Download Quantum Mechanics: Commutators & Uncertainty - OPTI 511, Spring 2009 (Problem Set 3) and more Assignments Chemistry in PDF only on Docsity! OPTI 511, Spring 2009 Problem Set 3 Prof. R.J. Jones 1. Commutators Recall that an operator is an instruction to do something with the function that follows, in general producing a new function. Here we define a new operator α̂ ≡ x̂p̂x, where x̂ is the position operator and p̂x is the operator for momentum along the x direction. Since we will work in coordinate space in the variable x, we have the following representations for these two operators: x̂ = x p̂x = −ih̄ d dx . In this problem, we’ll first want to evaluate α̂ψ(x), where ψ(x) = De−x 2/w2 (D,w are constants). So, we’d write α̂ψ(x) = x̂p̂xDe−x 2/w2 . This is evaluated by starting with the operator farthest to the right: first, p̂x = −ih̄ ddx operates on ψ(x). Then x̂ = x operates on that new result via simple multiplication of the position variable x. (a) Using the above definitions, evaluate α̂ψ(x). Is ψ(x) an eigenfunction of α̂? (b) Assume another new operator is defined: β̂ ≡ p̂xx̂. Evaluate β̂ψ(x). Remember that order of operation matters, and you must take the derivative after multiplying by x. Is ψ(x) an eigenfunc- tion of β̂? (c) We’ll now define yet another operator: Ĉ ≡ x̂p̂x − p̂xx̂. Evaluate Ĉψ(x), and simplify your answer as much as possible. [Hint: (α̂− β̂)ψ(x) = α̂ψ(x)− β̂ψ(x).] (d) You should now see that Ψ(x) is an eigenfunction of Ĉ. (If not, you should find your mis- takes.) What is the eigenvalue? There is another way to write Ĉ: Ĉ = [x̂, p̂x] ≡ x̂p̂x − p̂xx̂. This quantity with the square brack- ets is called the commutator of x̂ and p̂x, and although it may look a bit funny, it is an operator. More generally, for any two operators  and B̂, the commutator of  and B̂ is [Â, B̂] ≡ ÂB̂ − B̂Â. If [Â, B̂] = 0 (i.e., for any physical wavefunction Ψ, if ÂB̂Ψ = B̂ÂΨ) we say that that the operators  and B̂ commute. Otherwise they do not commute. (Recall the algebraic property of commutation of two variables u and v: uv = vu.) (e) Prove that any function f(x) is an eigenfunction of [x̂, p̂x], and thus we can write the equal- ity [x̂, p̂x] = const., with the constant being the eigenvalue from part (d) above. (f) Prove the identity [ÂB̂, Ĉ] = Â[B̂, Ĉ] + [Â, Ĉ]B̂ for any three operators Â, B̂, and Ĉ. 1 2. Heisenberg’s Uncertainty Principle revisited We have previously encountered the Heisenberg Uncertainty Principle (HUP) in the form σAσB ≥ h̄/2, where σA and σB are standard deviations for conjugate variables A and B. We encountered the spe- cific example of x and px. Here we restate the HUP in more general terms. The HUP is formally stated as follows: For any two physical quantities (observables) A and B and their associated linear op- erators  and B̂, σ2Aσ 2 B ≥ ( 1 2i 〈[Â, B̂]〉)2. This statement makes use of the commutator defined in the previous problem. (a) Restate the HUP above for position and momentum observables x and px. Making use of your answers from the previous problem, evaluate this expression and arrive at the HUP in the form for x and px that was discussed in class. You should notice that the expectation value of a constant is equal to that same constant, even if that constant is imaginary (and hence not representative of a physical observable such as position or momentum.) (b) When we’re working with the time dependence of a quantum state, we can associate the to- tal energy E of the state with an operator equivalent ih̄ ddt . In other words, we can say Ê = ih̄ d dt . (This is not the usual operator equivalent of energy, but we’ll use it here.) Here you will evaluate [Ê, t̂] for any stationary state, using t̂ = t as the operator for time. As mentioned in class, we generally have to be careful when working with time in quantum mechanics, as it’s not always straightforward to interpret what we’re doing! For the moment though, pretend like you can use t̂ = t as an operator for time without having to worry about it. To do this problem, you will need to know that any stationary-state wavefunction can be ex- pressed in the form Ψ(~x, t) = ψ(~x)e−iEct/h̄. Treat Ec as a constant. Notice that the probability density |Ψ(x, t)2| is time independent, even though Ψ(x, t) does have time dependence. Now evaluate [Ê, t̂], simplify as much as possible, and conclude this part of the problem by stating the HUP for energy and time. While it might not yet make sense to have an uncertainty relationship between E and t, for now we will just interpret this relationship as stating that there are limitations on how well the energy difference between two quantum states can be specified, and that those limitations depend on the time available for a measurement (i.e., how long an atom stays in an excited state before it decays). We will return to this concept later in the course when we discuss the natural lifetime of atoms and spontaneous emission of radiation: the natural lifetime of an excited atomic state (a time) is inversely proportional to the natural linewidth (an energy or a frequency) of the atomic transition. (c) Consider an arbitrary 3-dimensional stationary-state wavefunction of the form f(x, y, z)e−iEt/h̄. Evaluate [ŷ, p̂x] (first try to guess the answer). Notice the subscript on p, and let ŷ = y. Do y and px commute? Is there a limitation to simultaneous minimization of uncertainties in y and px? 2 4. A Reading Assignment This is another problem where this is nothing to compute. Instead, you are simply to read the final “Afterword” chapter out of Griffiths. This chapter has been scanned and posted on the course website in case you don’t have access to Griffiths. Based on the questions in class about timescales of repeated measurements (“How fast is fast? What does immediately after mean?”), and questions regarding the nature of measurements, this should be a worthwhile read. However, some of you may find a few diffi- cult spots in the chapter; that’s OK, just read the entire chapter and try to get an overall impression of the various aspects being discussed. (Some of these issues may come up later in the course.) One particular item that comes up in the chapter is the issue of “spin.” We will talk about spin later on in the course, so for now, here is a quick summary that may make the Afterword more readable for now. How to think about Spin for now : 1. Spin is a type of angular momentum associated with quantum particles. It is a type of angu- lar momentum that only appears in quantum mechanics. Its closest classical mechanics analog is the rotation of a sphere about an axis, such as the Earth’s rotation about its axis. The Earth’s rotational angular momentum is different than its angular momentum associated with its orbit around the Sun. Similarly, an electrons spin angular momentum is different than its orbital angular momentum about a nucleus in an atom. The analogy breaks down, but for now that’s not a problem. 2. Although we will discuss later that it is somewhat wrong to do so, for now think of every electron in the universe (even those confined in atoms) as spinning about an axis that runs through the electron. 3. Suppose you want to measure or specify the value of the electron’s spin angular momentum about some axis, say the z axis. Once you’ve chosen the your coordinate system, an electron’s spin about the z direction can only take on two values, either h̄/2 or −h̄/2 (we’ll see why later). The values don’t matter for right now; what’s important is the sign. If the electron is “spinning” one way about the z axis, we call that “spin up.” If it’s “spinning” the other way, we call that “spin down”. As a classical analog: if you define a z axis to extend from the south pole of Earth to the north pole (so positive z is south to north), the Earth is spinning in the “up” direction. But with electrons, the spin can be either up or down along z. 4. The electron’s spin can also be measured about any other axis in space, not just z, with up spin and down spin possible results. If you think this is starting to get weird, you’re right: the elec- tron waits until you’ve decided on your measurement direction before it decides whether or not it will be up or down along that direction. In contrast, the Earth’s direction of spinning is fixed in space, and no measurement will induce it to spin in any other direction. 5. Suppose that you pick a coordinate system axis. The assigned chapter in Griffiths uses an up arrow ↑− to represent an electron in a spin up state about that axis, and ↓− to represent an electron with spin down about that axis. The symbols ↑+ and ↓+ are up and down spin states of a positron (not a proton!), which is a particle with the same characteristics as an electron but with a positive charge. 6. The notation ↑−↓+ represents a combination state: an electron with spin up about some spec- ified axis, and a positron with spin down about that same axis. 7. What if you have an electron and a positron, and you know one to be spin up about some specified 5 axis, and the other to be spin down about that axis, but you don’t know which one is up and which one is down? Both combinations are possible, with equal probability. A state of this type would be represented by the expression on line [A.1] in Griffiths (page 375 of the edition scanned and placed on the course website). Don’t worry about the minus sign or the word “singlet;” the things that really matter are: (a) along any arbitrary axis that you choose, when one particle has spin up, the other has spin down, and (b) both combinations are possible, so both appear in the quantum state representation. 8. The chapter should now be easier to interpret. It will still take some effort, but you should have learned enough by now in this course to get some sense of the meaning of the discussions. 6