On the other hand, we should also not demand too little from our theories. There are things – like the positions of “pointers” on lab equipment, whether or not a bomb has exploded, and the vitality of a certain cat – which are directly available in ordinary sense perception, and which any candidate account of the microscopic world should, in principle, be able to account for. Theories might validly postulate all kinds of crazy-sounding and counter-intuitive things, but at the end of the day, if a theory predicts the wrong thing for where pointers should point (or somehow cannot account for the existence of pointers that point at all) it cannot be correct.
26 1 Pre-Quantum Theories
x px
Fig. 1.10 The “trajectory” of a free particle (moving with constant momentum) through phase space: att = 0 the particle begins, at a negative value ofx, with a positive momentum which stays constant as the particle moves. So its “path” through phase space looks like the solid line and would continue indefinitely to the right as long as the particle continues moving with unchanged momentum
Fig. 1.11 The “trajectory”
of a one-dimensional harmonic oscillator through phase space is a closed elliptical curve
x px
phase space point moves around the ellipse. (Hint: whenpxis positive, isxincreasing, or decreasing?)
Another type of “abstract space” that is sometimes useful to think about in the context of classical physics – and which we will confront again in the context of quantum mechanics – is “configuration space”. For a single particle, configuration space is just the same as physical space: it is all the places the particle could be.
But in a situation involving (say) two particles (moving, for simplicity, in one spatial dimension), there is the set of all places (call themx1) where the first particle might be, and then also the set of all places (call themx2) where the second particle might be. The configuration space is then the set of all possible configurations of the two particles jointly – that is, configuration space is a (here) two-dimensional space whose axes arex1andx2.
Let’s again consider two examples, one really simple and one at least slightly more interesting. Suppose, for example, that we have two particles moving in (the same) one-dimensional space. Except, let’s say, they aren’t actually moving, but are instead both just sitting there. In particular, particle 1 is sitting at a small positive value ofx, and particle 2 is sitting at a somewhat larger negative value ofx. Figure1.12shows this situation, both in (regular) physical space, and then again in configuration space.
x1
x2
x= 0 x
particle 1 particle 2
Fig. 1.12 On theleft are two particles (“particle 1” and “particle 2”) sitting at different points along thex-axis. This same situation is represented in the abstract configuration space as a single dot whose coordinates correspond to the positions (in physical space) of the particles
x1
x2
x x= 0
Fig. 1.13 The same two particles as before, but now particle 2 (which starts on theleft) is given a kick att=0 so it moves to theright(its initial velocity is indicated by thedark gray arrow) until it collides with particle 1. The post-collision velocities are indicated with thelight gray arrows. This same process is shown, on theright, in the abstract configuration space. Note thedashed lineat x1=x2corresponding to the particles being at the same location, i.e., colliding
As a slightly more interesting example, suppose now that somebody comes in and gives particle 2 a kick so that it starts moving to the right. Eventually it runs into particle 1 and, let’s suppose, they collide elastically. Suppose particle 1 has a greater mass than particle 2, so that after the collision particle 2 recoils back out to the left, whereas particle 1 drifts off slowly to the right. This process is depicted – both in physical space and in the abstract configuration space – in Fig.1.13.
What is the point of discussing these abstract spaces? Well, as I said, they are sometimes useful ways to depict a certain physical process to gain some intuition.
And there are even complete reformulations of Newtonian mechanics where things are formulated in terms of one of these abstract spaces – for example, if you’ve taken a course in classical mechanics, you have probably encountered “Hamiltonian mechan- ics” which is basically a way of re-writingFnet =main terms of energy quantities.
The basic dynamical equations in this Hamiltonian formulation of mechanics are first-order (in time) differential equations for the coordinates of a system in phase space. There is also something called the “Hamilton-Jacobi” formulation of mechan- ics which involves something like a time-dependent “field” on configuration space,
28 1 Pre-Quantum Theories which influences or guides the actual configuration point (representing the positions of all the particles composing the system) through that space. We won’t go into these things in any detail here, but it is good to be aware that they exist.
At this stage, the main take-home lesson from this discussion is this: don’t confuse any of these abstract spaces with regular old physical space! For example, if you are studying a one-dimensional harmonic oscillator, and sketch its phase space trajectory as in Fig.1.11, you should not ask: “What force provides the centripetal acceleration which holds the particle in this elliptical orbit?” Or similarly, you should not ask, about the particle collision process depicted in Fig.1.13, “That dotted line in the picture that the particle bounced off of... what’s itmade of??” Those sorts of questions don’t actually make any sense, and would seem to be based on simply forgetting that the space in question is an abstract one. If, in these kinds of situations, you want to know what is really going on, physically, you need to translate the abstract representation back into direct, literal, physical-space terms. For example, the dotted line in the previous Figure is not really a “thing” at all, but instead a kind of abstract representation of the strong repulsive forces that the two particles exert on each other if they get too close together in physical space. And the only force present in the case of the simple harmonic oscillator is the force exerted on the particle by a spring (or whatever) as it moves back and forth along a line – there is nothing like an elliptical orbit at all, if that means some literal material particle moving along a certain curved path through a two-dimensional physical space.
This of course all seems so clear and obvious as to be almost embarrassing to have to say. But as you’ll see (especially in Chap.5) confusion will arise around these kinds of issues when we get to quantum mechanics – to which we will turn very soon!
Projects:
(1.1) Show that a spherically-symmetric distribution of mass (e.g., a thin spherical shell composed of innumerable massive Particles) exerts the same gravita- tional force on an outside particle as the force that would be exerted by a single Particle with the same total mass as the shell and located at the center of the shell. Explain how this is an example of the theory explaining how and why a large (spherically symmetric) assemblage of Particles can be treated as a particle.
(1.2) Show that a rigid body (like a bunch of Particles glued together) will obey Newton’s 2nd law (F = ma) if the individual particles do. Note that a rigid body canrotatein addition to moving translationally, so you should not assume that all the individual Particles have the same acceleration a.
Indeed, the main problem here is to figure out precisely how “the accelera- tion of the rigid body” can be defined to make something like Newton’s 2nd law true. Hint: it might be helpful to consider thecenter of masscoordinate,
R= M1total
imiri, and its various time-derivatives. Explain how this is an example of the theory explaining how and why a large assemblage of Particles can be treated as a particle.
(1.3) Consider two equal-mass stars in a binary star system, each making a circular orbit about their mutual center-of-mass point. Now suppose that gravitational forces are given by Eq. (1.1), but withrjbeing the position of the distant mass at a slightly earlier time (such that a signal emitted by it at that earlier time would just arrive at the mass in question now). Draw a careful diagram show- ing the gravitational forces acting on each star at some particular moment.
Do the forces respect Newton’s third law? Will the total momentum of the two-star system be conserved (assuming no forces act from the outside)? How about the total angular momentum? Can you think of another similar situation in which the total translational momentum would not be conserved? What do you make of all this?
(1.4) Give a simple example of a system of masses interacting via Newtonian grav- itational forces, and show that/how the motion of the masses would be differ- ent if one used a different slicing of space-time into simultaneity slices. (That is, show that Newtonian mechanics with instantaneous, action-at-a-distance gravitational interactions is incompatible with relativity, since it requires a dynamically privileged notion of simultaneity.)
(1.5) The mathematical parallel between Newton’s law of gravity and Coulomb’s law of electrostatics suggests that a relativistic theory of gravity could be developed by, in effect, copying Maxwell’s equations. Play around with this and see how far you can get. (Hint: the gravitational analog of the electric fieldE is the gravitational field g, which has units of acceleration. So the gravitational analog of Gauss’ Law should be something like ∇ · g ∼ ρm whereρmis the mass density. What should the proportionality constant be in order to reproduce Newtonian gravity? After you figure out the gravitational analog to Gauss’ Law, you can try to work out consistent gravitational analogs to the three other Maxwell equations as well!)
(1.6) Flesh out the equivalence between Coulomb’s law and Gauss’ law by explain- ing in detail how to solve forEin Gauss’ law whenρ=qδ3(x).
(1.7) Work through the details of deriving wave equations for E and B from Maxwell’s equations. (Assume empty space, i.e.,ρ= 0 and j =0.) Show that there are plane-wave solutions of the formE( x,t)= E0sin(k· x−ωt), and similarly forB. Are the waves transverse, or longitudinal? How do you know? What is the relationship between|k|andω? What are the phase and group velocities of the waves in terms of0andμ0?
(1.8) Let’s try to understand better how Eq. (1.24) is a solution of the wave equation with a delta-function source. For simplicity, suppose the source point is at
x=0 andt=0 so that the differential equation in question reads
∇2ψ− 1 c2
∂2ψ
∂t2 =δ3(x)δ(t). (1.45) The solutionψshould be spherically symmetric, i.e., should be a function of randtonly. (a) Show that, forr=0, any function of the general form
30 1 Pre-Quantum Theories ψ(r,t)=g(t−r/c)
r (1.46)
solves Eq. (1.45). (b) The correct functiong should also satisfy Eq. (1.45) atr =0. To ensure this, integrate both sides of Eq. (1.45) over a spherical volume of radiusRcentered atr =0. Show that taking theR→0 limit then gives−4πg(t)=δ(t), i.e.,
ψ(r,t)= − 1 4π
δ(t−r/c)
r (1.47)
as claimed in the text.
(1.9) When he is presenting his formulation of locality, Bell gives an analogy to the boiling of an egg. You put the egg into the boiling water and set the timer for (say) 5 min. Then, 5 min later, “[t]he ringing of the alarm establishes the readiness of the egg.” That is, the two events arecorrelated. However, the ringing of the alarm does notaffectthe egg. Bell explains: “if it is already given that the egg was nearly boiled a second before, then the ringing of the alarm makes the readiness no more certain.” Draw a spacetime diagram; connect
“the ringing of the alarm”, “the readiness of the egg”, and its being “already given that the egg was nearly boiled a second before” with the terminology χ1,χ2, andC; and explain how the last sentence captures the locally causal character of the physical processes involved in this example.
(1.10) An unstable particle is heading for a particle detector which will “click”
if the particle hits it. Given the state of the particle at some earlier time, suppose there is a 50% probability of itsnotdecaying first and hence hitting the detector: P[click|C] = 1/2. On the other hand, if the particle does decay before arriving at the detector, the decay products might themselves be detected and hence indicate that the original particle will not be detected by the original detector. So, for example,P[click|C,χ2]=0, whereχ2denotes the successful detection of one of the decay products. See Fig.1.14. Does the non-equality of the two conditional probabilities here imply a violation of Bell’s locality condition? (One would hope not, since there is clearly nothing nonlocal happening here. On the other hand, this seems to be a case where information from outside the past light cone of the event in question, does affect the probabilities assigned to that event.) Explain.
(1.11) Make up an example, maybe along the lines of the example involving New- tonian gravity from the main text, to show that Maxwellian electrodynamics gets (correctly) diagnosed as “local” both by Bell’s formulation and the mod- ified formulation.
(1.12) Is there a way of choosing a gauge such that the vector potentialApropagates with infinite speed, the wayφdoes in the Coulomb gauge? Explain.
(1.13) A standard introductory physics problem involves analyzing the “ballistic pendulum” in which a block of mass M hanging from a string of lengthL absorbs an incoming bullet of massmmoving at some unknown initial speed
Fig. 1.14 Space-time diagram for the events described in Project 1.10
Σ
χ2
χ1="Click!"
v0. The bullet-and-block then swing up together, with the string eventually making some maximum angleθwith respect to the vertical. By observing θ one can thereby determine the initial speed of the bullet. Work out this relationship (i.e., solve forv0as a function ofθ,m, M, andL) and explain how this method of measuring the bullet’s speed fits into the general scheme introduced in the text in which the outcome is registered in some directly observable “pointer”.
(1.14) Pick a measuring apparatus of interest to you (maybe something that you’ve used in a physics lab course or research, maybe the accelerometer in your iPhone, or just something else you’re interested in) and learn more about how it actually works. Explain whether (and, assuming so, how) the actual device fits into the general scheme introduced in the text in which the outcome is registered in some directly observable “pointer”.
(1.15) Two equal-mass gliders are floating on a (nearly) frictionless air track in an introductory physics classroom. The track is equipped with elastic bumpers on both ends, and the two gliders have elastic-collision attachments so they will bounce when they collide. One glider is initially at rest, while the other is given an initial velocity. Draw a space-time diagram showing world lines for both gliders as well as the two bumpers on the ends of the track. Now draw the “trajectory” of the two-glider system through the two-dimensional configuration space.
References
1. I. Newton,Opticks(Dover Publications Inc, New York, 1952)
2. J. Andrew, Newton’s Philosophy, inThe Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta (2009),http://plato.stanford.edu/archives/win2009/entries/newton-philosophy/
3. A. Einstein, Autobiographical Notes, inAlbert Einstein: Philosopher-Scientist, ed. by P.A.
Schilpp (Harper and Row, New York, 1959)
4. J.D. Jackson,Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975)
5. A. Einstein,Relativity: The Special and the General Theory(Penguin Classics, New York, 2006) 6. J.S. Bell, La Nouvelle Cuisine, reprinted inSpeakable and Unspeakable in Quantum Mechanics,
2nd edn. (Cambridge University Press, Cambridge, 2004) 7. O.L. Brill, B. Goodman, Am. J. Phys.35, 832–837 (1967)
Chapter 2
Quantum Examples
In this chapter we review quantum theory (at the level of wave mechanics) and develop a toolbox of simple quantum mechanical examples that we will use, in the following chapters, to discuss a number of the issues raised in Chap.1: locality, ontology, measurement, etc.