Chapter 6: Quantum mechanics in 1D

Born interpretation of wave function

• The probability that a particle will be found in the infinitesimal interval dx about the point x, denoted by P(x) dx, is

• 

• Notice that phi itself is not a measurable quantity; however, psi^2 is measurable and is just the probability per unit length, or probability density P(x), for finding the particle at the point x at time t.

• Because the particle must be somewhere along the x-axis, the probabilities summed over all values of x must add to 1:

• 

• Any wavefunction satisfying Equation is said to be normalized

• The fundamental problem of quantum mechanics is this. Given the wavefunction at some initial instant, say t 0, find the wavefunction at any subsequent time t.

• In Newton’s mechanics x(t) and v(t) are calculated from Newton’s second law; in quantum mechanics psi(x, t) must be calculated from another law—Schrödinger’s equation.

WAVEFUNCTION FOR A FREE PARTICLE

• A free particle is one subject to no force. This special case can be studied using prior assumptions without recourse to the Schrödinger equation. The development underscores the role of the initial conditions in quantum physics.

• The wavenumber k and frequency of free particle matter waves are given by the de Broglie relations

  

• For nonrelativistic particles is related to k as

• 

• 

• This is an oscillation with wavenumber k, frequency , and amplitude A. Because the variables x and t occur only in the combination kx omegat,* the oscillation is a traveling wave, as befits a free particle in motion.

• The wavefunction psi must satisfy the Schrödinger equation,

• 

• a time-independent wavefunction satisfying the time independent Schrödinger equation.

• 

• The approach of quantum mechanics is to solve the above equation for psi and E, given the potential energy U(x) for the system. In doing so, we must require,

1. that (x) be continuous • that (x) be finite for all x, including x = -infinity/+infinity
2. that psi(x) be single valued
3. that dpsi/dx be continuous wherever U(x) is finite

• Explicit solutions to Schrödinger’s equation can be found for several potentials of special importance.

• A free particle known to be in some range del x is described not by a plane wave, but by a wave packet, or group, formed from a superposition of plane waves. The momentum of such a particle is not known precisely, but only to some accuracy del p that is related to x by the uncertainty principle,

• 

• in a box of L length, the wavefunctions within the box are given by,

• 

• For the harmonic oscillator the total potential energy is,

• 

• The lowest energy is E=1/2omega * h* ; the separation between adjacent energy levels is uniform and equal to h*omega. The wavefunction for the oscillator ground state is,

• 

• The oscillator results apply to any system executing small-amplitude vibrations about a point of stable equilibrium. The effective spring constant in the general case is

• 

• The stationary state waves for any potential share the following attributes:

1. Their time dependence is e ^-i*omega* t .
2. They yield probabilities that are time independent.
3. All average values obtained from stationary states are time independent.
4. The energy in any stationary state is a sharp observable; that is, repeated measurements of particle energy performed on identical systems always yield the same result, E=h*omega.

• For other observables, such as position, repeated measurements usually yield different results. We say these observables are fuzzy. Their inherent “fuzziness” is reflected by the spread in results about the average value, as measured by the standard deviation, or uncertainty. The uncertainty in any observable Q can be calculated from expectation values as,
•