----------------------------------- S4f. How real is the wave function? ----------------------------------- In thought experiments one often assigns a state to a single particle. How defendable is this, and what is the meaning of the state? In a statistical interpretation - see the section on measurements -, this would make no sense, since there the state is a property of the ensemble of particles generated by a given source. But then it is difficult to visualize what happens in each single case. Thus many people prefer the 'realistic' language of particles having definite states. So let us discuss some of its implications. Suppose that the particle is in the pure state represented by the wave function psi. It is possible to give the wave function, or rather its absolute valued squared, a geometric interpretation: m(x)=m|psi(x)|^2 is the mass density and e(x)=e|psi(x)|^2 the charge density. Thus while the wave function itself has no tangible interpretation, certain fields computable from it have. This extends - but not quite in the obvious way - to multiparticle systems: For a system of several, say n particles, the wave function is 3n-dimensional psi(x_1,...,x_n), each x_i being an ordinary 3-dimensional position vector, but the correct densities are still 3-dimensional, obtained by integration: m(x) = sum_a m_a integral dx_1...dx_n delta(x-x_a)|psi(x_1:n)|^2, e(x) = sum_a e_a integral dx_1...dx_n delta(x-x_a)|psi(x_1:n)|^2. This reduces for n=1 to the above, and is consistent with the definition of mass and charge density in quantum field theory as m(x) = , e(x) = , where Psi_0(x) is the time component of the relevant matter field. These formulas are the common starting point for the derivation from first principles of the semiconductor equations in solid state physics. It is also what chemists draw as molecular shapes, using a cutoff where m(x) and e(x) are negligible to delineate the boundary. Indeed, chemists use such an interpretation all the time when visualizing molecules in terms of orbitals, and with great success. The charge distribution of the electron cloud of a molecule is one of the important outputs of quantum chemistry packages such as GAUSSIAN (commercial) http://www.gaussian.com/ MOLPRO (commercial) http://www.molpro.net/ GAMESS (free after registration) http://www.msg.ameslab.gov/GAMESS/pcgamess.shtml In the ground state (but also in definite excited states), the mass or charge distribution is spread out over an infinite region, although it becomes negligibly small outside a tiny core region (or, sometimes, such as in Stern-Gerlach experiments, where the wave function is multimodal, outside a few disconnected core regions). The infinite extension invites apparent paradox in that upon collapse (e.g., due to hitting a detector screen), the particle contracts from its infinite extension to a single spot. This seems to violate the central tenet of relativity that information cannot flow faster than the speed of light. However, special relativity only restricts the observational consequences of theory. Since most of the wave function of an individual particle is unobservable, there is no contradiction. (It is like the nonlocality in tests of Bell's inequalities. Nonlocality is unavoidable in QM, but the observable consequences respect the bound relativity puts on the speed of information flow.) For example, on a TV set, one observes just 3 position degrees of freedom of each electron reaching the screen, while - in contrast to the case of a classical particle - the wave function characterizing a pure state of the electron sits in a space of functions of 3 variables, which has infinitely many degrees of freedom. Thus one observes only a tiny little bit about the electron's state. It is like knowing the velocity of the wind (a 3-dimensional vector field) in the earth's atmosphere at a single point (giving a velocity vector with 3 coordinates)! This unobservability of most of the state causes a problem for those who require that everything a theory is talking about is observable. But this requirement is not satisfied anyway in current microphysics - no one ever observed a quark, but it is generally believed that they make up most of the matter in our universe. Thus, while it is reasonable to require that theory has observable consequences in agreement with Nature, it is not reasonable to require that everything the theory talks about is observable. Then the unobservability of most of the state of a single particle is harmless. On the other hand, one can probe the state of particles in detail if one has a large ensemble of identically prepared particles (to make sure that they have the same state). These are usually created by a carefully calibrated source, such as a laser. Then one can subject them to different kinds of measurements from which one can reconstruct a reasonable approximation of the state by quantum tomography. In theory, one can make the approximation arbitrarily good. Similarly a particle bound to a surface in a stationary state will be measurable repeatedly if after the measurement the particle returns to its state (which is natural if the bound system is in equilibrium). Therefore one can measure equilibrium properties quite accurately. In this sense one can say that the state of a single particle is indeed real, and objective. Note that single particles can nowadays be routinely prepared and studied; see, e.g., D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Reviews of Modern Physics 75 (2003), 281-324. S.M. Reimann and M. Manninen, Electronic structure of quantum dots, Reviews of Modern Physics 74 (2002), 1283-1342.