QUANTUM MECHANICS: a synopsis
function and Schrodinger equation for a free particle
Schrodinger's equation for a particle in one dimension:
i hbar psi_t = - (hbar^2/2m) psi_xx
where hbar is Planck's constant h divided by 2Pi, m is the mass of the particle, and psi is the wave function. Because of the factor of i on the left hand side, all solutions to the Schrodinger equation must be complex. Numerically, hbar ~= 2/3 eV-fs = (6.63/2Pi ) x 10^(-34) J-s. For macroscopic systems hbar is a TINY number, but for atomic systems it is of order unity. More specifically, the binding energy of an electron in an atom is of the order of eV (electron volt) and the time to complete one orbit around the nucleus is of the order of one fs (1 fs = femtosecond = 10^(-15) s).
The physical meaning of psi is a "probability density amplitude", that is, the probability of finding the particle between x=a and x=b at time t is the integral from a to b of |psi(x,t)|^2, the squared modulus of psi(x,t). Since the particle must be SOMEWHERE the integral of |psi(x,t)|^2 over all space must be 1. This condition is automatically preserved in time for solutions of the Schrodinger equation. A particle doesn't have a definite position, it has only a probability of being found in any given region.
The above equation describes a force-free particle, so it is to quantum mechanics what Newton's first law is to particle mechanics. For a wave function of the form Aexp(-iwt)exp(ikx), Schrodinger's equation implies the dispersion relation hbar w = (hbar k)^2/2m. This is the same as the relation E = p^2/2m between energy and momentum for a free particle, if one adopts the correspondence E = hbar w and p = hbar k. Equivalently, E = h nu and p = h/lambda, where nu and lambda are the frequency and wavelength. Lambda is called the de Broglie wavelength of the particle. In terms of E and p the above free particle Schrodinger wave is A exp(- iEt/hbar) exp(ipx/hbar). Note that although the Schrodinger equation does not at first look like a wave equation, since the time derivative is only first order instead of second order, its solutions are nevertheless like those of the wave equation. This is due to the magic of i. Replace the i by 1 on the left hand side and you get a completely different beast: the diffusion equation.
packets, the uncertainty relation, and velocity
A particle with a definite momentum is evenly spread out in space, equally likely to be found anywhere, since |Aexp(-iwt)exp(ikx)|=|A|. (Actually this is a bit of an embarassment, since the total probability of being anywhere must be unity. No value of the amplitude A will produce this normalization. Therefore a particle with a precisely defined momentum is not physically meaningful. There must always be at least a very small spread in momentum.) A more localized particle is described by a wave packet containing a spread of momenta. A particle localized perfectly at one point is described by the Dirac delta function, which consists of ALL momenta. (This extreme case is also not physically meaningful.) In general , the spread in position is at least inversely proportional to the spread in momentum: (Delta x) (Delta p) >= hbar/2 (it turns out the exact minimum is hbar /2 if Delta x and Delta p are defined as the standard deviations from the mean). This is called the Heisenberg uncertainty relation. As a wave packet peaked at the wave number k evolves, it spreads out, but its center moves with the group velocity dw/dk evaluated at k. This velocity is hbar k/m , swhich corresponds to p/m , the particle velocity. So the velocity of a classical particle corresponds in fact to the group velocity of the quantum wave packet for the particle! (The phase velocity is half of this.)
If a force is acting on the particle, the Schrodinger equation must be modified, If the force is -dV/dx for some potential energy function V(x), the term V psi must be added to the right hand side of the Schrodinger equation, i hbar psi_t = - (hbar^2/2m) psi_xx + V(x) psi. This equation is to quantum mechanics what Newton' second law (F = ma) is to particle mechanics.
Although a quantum particle can not have a definite position, it can have a definite energy. For example in an atom, the allowed definite energies form a discrete set and are called the quantizedenergy levels of the atom. Since energy corresponds to frequency, the energy levels correspond to solutions of the Schrodinger equation with a definite frequency, together with some spatial dependence: psi(x,t) = exp(-iEt/hbar) f(x). The spatial part f(x) must satisfy the time-independent Schrodinger equation - (hbar^2/2m) f_xx + V(x) f = E f, written here in one dimension.Only for particular values of the energy E are there solutions to the spatial equation for f satisfying the boundary conditions. For a quantum particle, the "boundary condition" on f is that the total probability is one, i.e. \int |f|^2 dx = 1. These definite energy level solutions are in perfect analogy with the normal modes of a continuous vibrating system, like a string for example. So the energy levels of a quantum particle are the normal modes of the Schrodinger wave.