From: Grant¹ 9/12/99 18:58:03 Subject: Tutorial 4 - Zero Point Energy post id: 14862 In this tutorial (and in following ones) I'm going to brand some parts in red text. These parts are more technical, and not necessary to understanding the tutorial, but will provide more colour for inclined readers. The choice is left to the reader whether to skip the red sections or not. :o) Cheers Chris Two equations we'll need to revise… To understand where the notion of the Zero Point Energy (ZPE) comes from, we'll need to revise a couple of central concepts. Mass-energy relationship.. The mass energy relationship is the general form of Einstein's well-known E=mc2 relation. The full relation is: E2 = m2c4 + p2c2 Here energy is E, momentum is p, m is rest mass and c a constant equal to the speed of light in a vacuum. Notice that if p = 0, then the above relation becomes: E2 = m2c4 + 02c2 E2 = m2c4 + 0 E = m c2 (with which we are more familiar). Basically the relation says that the total energy of a particle (E) is a function of the energy associated with its motion (pc) and the "rest" energy associated with it's mass (mc2). In relativistic mechanics we can specify the energy we're talking about. For example we might talk about the kinetic energy of a particle, W, such that: W2 = m2c4 + p2c2 We could rearrange this equation to look like this: W2 / c2 - p2 - m2c2 = 0 - - - - - (1) This equation will become useful later on. Technical section: The equation W2 = m2c4 + p2c2 is also applicable in the quantum realm. We simply let the LHS of the equation operate on the wave function Y, and understand that W = ih/t and pr = - ih/xr (for r 1,2,3). It is by making this equation linear in W and p that we achieve the matrix form of the Dirac equation. Heisenberg's Uncertainty Principle... This principle is one of the two cornerstones of the Copenhagen Interpretation of quantum theory, and it is a basic statement for operations in the quantum world. It says that you interfere with a particle or system when you attempt to observe it or measure it. It also says that there is a limit to the precision with which you can "know" a quantum system. Stated mathematically, the common form of the principle is: Dp · Dx ³ h / 4p - - - - - (2) This says that the uncertainty in your measurement of momentum (Dp) multiplied by the uncertainty in your measurement of position (Dx) is never less than a constant. Which is pretty earth shattering stuff! From what we know about classical (non-quantum) physics, we would expect the right hand side of the equation above to be zero, which would give us unlimited precision in our measurements (in theory). But, suppose I was able to measure the momentum of a particle with infinite precision. The relation above with its non-zero right hand side means that the uncertainty in its position is infinite - ie it could be anywhere in the universe! Neils Bohr (the "first" quantum physicist) explains this with his principle of complementarity. Simply put, some things are complementary - knowledge of the two at the same time is mutually exclusive. Another pair of such "mutually exclusive variables" is energy and time. The corresponding relation is: DE · Dt ³ h / 4p - - - - - (3) These two equations will prove useful later on as well. ZPE from a particle perspective… The quantum vacuum is not exactly empty! Because of the uncertainty principle above, the quantum vacuum is alive with fluctuations and virtual particles. What are these? Well, from equation (3) above, we can see that there is a limit to which we can specify the energy of the quantum vacuum at any one time. If I measure a small enough time interval, that is I become very precise in my measurement of Dt, then the imprecision in energy grows correspondingly large. In fact it might grow large enough for two particles to pop out of the vacuum for a short time (the interval Dt) before disappearing again. These particles are virtual (non-"real") and the quantum vacuum is teeming with them. Technical example: How long a time could a virtual electron - positron pair exist? The mass of the electron is 9.109*10-31kg. The rest energy of an electron - positron pair is: E = 2(9.109*10-31kg) * (299792458m/s)2 E = 1MeV (1.64*10-13J) By Heisenberg, DE · Dt ³ h / 4p We need an uncertainty, DE, of at least 1 MeV to make the pair. Dt £ h / 4p · DE Dt £ 3.22*10-22s The important thing to remember that these variations and fluctuations are local, higher energy here, lower energy there - and the whole thing may change in the next instant. But it seems to be very tidy: the energy to make a virtual pair is "borrowed" from the vacuum, then a fraction later it is paid back. How does the ZPE arise from this? The answer is that it doesn't necessarily. Our examination of the virtual particles shows that the quantum vacuum is not static or strictly stable (at least in the classical sense of the word). Now any particle can interact with this vacuum and induce changes in the energy here or there: it could be an electrical or magnetic interaction, or something more complicated. If we consider the classical kinetic energy of a particle in an external field (in the non-relativistic limit), we would describe it as: KE = ½ mv2 + V(x) where V(x) is a potential energy due to the particle's position (x) in field V. To minimise the particle's energy, we might consider the particle at rest (v = 0) and at the position in the field where the potential V(x) is smallest. This tells us the ground state or lowest energy for the particle. But there is a problem in our reasoning. To achieve the ground state, we have had to specify both the particle's velocity and position - which Heisenberg's principle (2) forbids! In practice, then, the quantum ground state of the particle is an amount higher than the classical figure. The difference is the ZPE. ZPE from a field theory perspective… What is quantum field theory? Well, we know there are interactions about us which we describe in terms of fields. These include electrical fields, magnetic fields, gravitational fields, and more. We come up with a mathematical description of the field which means that if we place a particle at some point x within the field we will be able to predict what will happen to it. For example we can describe the gravitational field around the earth rather well, and we can predict what will happen to you should we place you in the earth's gravitational field at, say, 10km above sea level. But, as we say time and time again, things are a tad different in the quantum world. To turn a field theory into a quantum field theory, we look very closely at the field's interaction at the quantum level. We identify the smallest indivisible "chunk" of the field (called the field's quantum) and we model the way that quantum interacts with other particles. Such modelling can be difficult, particularly when lots of particles are interacting with each other. One trick particle physicists use is to first model the particles and field when not interacting with each other. This is called a "free field" model. Then you can work the interactions of the particles in, giving you an "interacting field" model. In either case the vacuum is considered to be the ground state, or lowest energy state for the system. But the trick is that the ground states for the free field and interacting field (for the same system) turn out to be different! The difference is the ZPE. Thinking about the difference between the two field models, one begins to get a feeling that the ZPE could be explained in terms of the particle interactions explained above, which should be the case. The value is that we have reached the explanation for the ZPE from a different direction (converging explanations are always good! :o) So how do we extract it? It turns out to be a trivially easy thing to do. All interactions between quantum entities will interact with the quantum vacuum as well, influencing the local ground state. But we're talking quantum scale here, not the fabulous dreams of the "free lunch crowd" (see next section). Particle interactions everyday account for and include the effect of vacuum energy and the ZPE. Still it would be nice to have a concrete experiment confirm the vacuum energy conclusively. It so happens that we do have such an experiment. The Casimir effect is a small pressure effect felt between two parallel conducting plates placed a short distance apart. By "short distance" I mean on the order of a few nanometres! The plates will experience a force which pushes them together, despite no external force being applied. The reason is that the plates are close enough together such that the distance between them is on the same scale as the wavelengths of virtual photons in the quantum vacuum. A series of standing waves is induced between the plates for particles whose wavelengths are whole or half multiples of the distance between the plates. All those wavelengths which are not even multiples will destructively interfere and not exist between the plates. As a result more possible particles are allowed outside the plates than between them, and therefore there is a pressure gradient from outside the plates to within them. The plates are forced together. In essence, the Casimir plates extract energy from the quantum vacuum to move towards each other, experimental verification of vacuum fluctuations, and vacuum or zero point energy. A Free Lunch? In a word - No! :o) The "free lunch" crowd are the modern evolutionary descendants of the "perpetual motion" crowd (although the latter crowd still persists in smaller numbers). Determined to make untold fortunes through getting something for nothing, they plunder the realms of physics for possible sources of free energy - never bothering with the trifling details of understanding what they're on about. Despite the fact that the ZPE definitely exists, and can be extracted from the vacuum, it cannot be extracted in useful quantities. The reason for this is that - as stated above - quantum fluctuations are local. The total vacuum energy is very close to - if not exactly - zero. Some physicists argue it is exactly zero, others identify it with (a non zero) cosmological constant. Some non-zero predictions run on the order of 10-123J - a staggeringly small amount. This comes about because fluctuations in the vacuum can vary above or below the ZPE level. Remember equation 1 above? The energy term W must be either greater than or equal to mc2, or less than or equal to -mc2. This arises from the W2 factor having two roots. The effect of the two together brings the total fluctuations back towards zero. Conclusion… I have seen the ZPE described this way: suppose the bank arbitrarily decided tomorrow that instead of setting your lowest bank balance (ie the "zero") at $1000 instead of $0. Effectively they're simply calling the number 0 as 1000 now. When you check your balance and see that you have $1000, you may get excited and rush out to try and spend it. However your bank manager is still going to tell you you're broke - he just raised the bar. I think there is some merit to this explanation - particularly when dealing with free lunchers. But it doesn't reflect that the ZPE is still a real effect. Perhaps if we add to this that you can spend 1c or 2c - the smallest quantum of monetary exchange being 5c - of the $1000 every now and again, but never more than that, we might be close. Hope this helps! Chris As always, please post questions and comments to the tutorial 4 question thread following this thread, rather than posting them here This forum is un-moderated. The views and opinions expressed are those of the individual poster and not the ABC. The ABC reserves the right to remove offensive or inappropriate messages.